Log-barrier interior point methods are not strongly polynomial
Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, Michael Joswig
aa r X i v : . [ m a t h . O C ] A ug LOG-BARRIER INTERIOR POINT METHODSARE NOT STRONGLY POLYNOMIAL
XAVIER ALLAMIGEON, PASCAL BENCHIMOL, ST´EPHANE GAUBERT, AND MICHAEL JOSWIG
Abstract.
We prove that primal-dual log-barrier interior point methods are not stronglypolynomial, by constructing a family of linear programs with 3 r + 1 inequalities in dimension2 r for which the number of iterations performed is in Ω (2 r ). The total curvature of the centralpath of these linear programs is also exponential in r , disproving a continuous analogue of theHirsch conjecture proposed by Deza, Terlaky and Zinchenko. Our method is to tropicalize thecentral path in linear programming. The tropical central path is the piecewise-linear limit of thecentral paths of parameterized families of classical linear programs viewed through logarithmicglasses. This allows us to provide combinatorial lower bounds for the number of iterations andthe total curvature, in a general setting. Introduction
An open question in computational optimization, known as Smale’s 9th problem [Sma00], askswhether linear programming can be solved with a strongly polynomial algorithm. This requiresthe algorithm to be polynomial time in the bit model (meaning that the execution time ispolynomial in the number of bits of the input) and the number of arithmetic operations to bebounded by a polynomial in the number of numerical entries of the input, uniformly in theirbit length.It is instructive to consider interior point methods in view of this question. Since Karmarkar’sseminal work [Kar84], the latter have become indispensable in mathematical optimization. Path-following interior point methods are driven to an optimal solution along a trajectory called the central path . The best known upper bound on the number of iterations performed by path-following interior point methods for linear programming is O ( √ nL ), where n is the numberof variables and L is the total bit size of all coefficients. Hence, they are polynomial in thebit model (every iteration can be done in strongly polynomial time). It is tempting to askwhether a suitable interior point method could lead to a strongly polynomial algorithm in linearprogramming. In other words, this raises the question of bounding the number of iterations bya polynomial depending only on the number of variables and constraints.Early on, Bayer and Lagarias recognized that the central path is “a fundamental mathe-matical object underlying Karmarkar’s algorithm and that the good convergence properties ofKarmarkar’s algorithm arise from good geometric properties of the set of trajectories” [BL89,p. 500]. Such considerations led Dedieu and Shub to consider the total curvature as an informalcomplexity measure of the central path. Intuitively, a central path with a small total curvatureshould be easier to approximate by linear segments. They conjectured that the total curvature ofthe central path is linearly bounded in the dimension of the ambient space [DS05]. Subsequently,Dedieu, Malajovich and Shub showed that this property is valid in an average sense [DMS05]. Date : August 10, 2017.2010
Mathematics Subject Classification. owever, Deza, Terlaky and Zinchenko provided a counterexample by constructing a redun-dant Klee-Minty cube [DTZ09]. This led them to propose a revised conjecture, the “continuousanalogue of the Hirsch conjecture”, which says that the total curvature of the central path islinearly bounded in the number of constraints.In this paper, we disprove the conjecture of Deza, Terlaky and Zinchenko, by constructinga family of linear programs for which the total curvature is exponential in the number ofconstraints. Moreover, we show that for the same family of linear programs, a significant classof polynomial time interior point methods, namely the primal-dual path-following methods withrespect to a log-barrier function, are not strongly polynomial.More precisely, given r >
1, we consider the linear program LW r ( t ) minimize x subject to x t x tx j +1 t x j − , x j +1 t x j x j +2 t − / j ( x j − + x j ) x r − > , x r > j < r which depends on a parameter t >
0. The linear program LW r ( t ) has 2 r variables and 3 r + 1constraints. The notation LW r ( t ) for these linear programs refers to the “long and winding”nature of their central paths. More precisely, our first main result is the following. Theorem A (see Theorem 25) . The total curvature of the central path of the linear programs LW r ( t ) is exponential in r , provided that t > is sufficiently large. Our second main result provides an exponential lower bound for the number of iterationsof a large class of path-following interior point methods. We only require these methods tostay in the so-called “wide” neighborhood of the central path; see Section 2 for the definition.Remarkable examples of such methods include short or long-step methods, like the ones ofKojima, Mizuno and Yoshise [KMY89a, KMY89b] and Monteiro and Adler [MA89], as well aspredictor-corrector methods like the ones of Mizuno, Todd and Ye [MTY93] and Vavasis andYe [VY96].
Theorem B (see Corollary 31) . The number of iterations of any primal-dual path-followinginterior point algorithm with a log-barrier function which iterates in the wide neighborhood of thecentral path is exponential in r on the linear programs LW r ( t ) , provided that t > is sufficientlylarge. The proofs of these theorems rely on tropical geometry. The latter can be seen as the(algebraic) geometry on the tropical (max-plus) semifield ( T , ⊕ , ⊙ ) where the set T = R ∪ {−∞} is endowed with the operations a ⊕ b = max( a, b ) and a ⊙ b = a + b . A tropical variety can beobtained as the limit at infinity of a sequence of classical algebraic varieties depending on onereal parameter t and drawn on logarithmic paper, with t as the logarithmic base. This processis known as Maslov’s dequantization [Lit07], or Viro’s method [Vir01]. It can be traced backto the work of Bergman [Ber71]. In a way, dequantization yields a piecewise linear image ofclassical algebraic geometry. Tropical geometry has a strong combinatorial flavor, and yet itretains a lot of information about the classical objects [IMS07, MS15].The tropical semifield can also be thought of as the image of a non-Archimedean field underits valuation map. This is the approach we adopt here, by considering LW r ( t ) as a linearprogram over a real closed non-Archimedean field of Puiseux series in the parameter t . Then,the tropical central path is defined as the image by the valuation of the central path over thisfield. We first give an explicit geometric characterization of the tropical central path, as atropical analogue of the barycenter of a sublevel set of the feasible set induced by the dualitygap; see Section 4.1. Interestingly, it turns out that the tropical central path does not dependon the external representation of the feasible set. This is in stark contrast with the classical ase; see [DTZ09] for an example. We study the convergence properties of the classical centralpath to the tropical one in Section 4.2.We then show that, when t is specialized to a suitably large real value, the total curvature ofthe central path of the linear program LW r ( t ) is bounded below by a combinatorial curvature(the tropical total curvature) depending only on the image of the central path by the nonar-chimedean valuation (Section 5). The linear programs LW r ( t ) have an inductive construction,leading to a tropical central path with a self-similar pattern, resulting in an exponential numberof sharp turns as r tends to infinity. In this way, we obtain the exponential bound for the totalcurvature (Theorem A).A further refinement of the tropical analysis shows that the number of iterations performed byinterior point methods is bounded from below by the number of tropical segments constitutingthe tropical central path; see Section 6. For the family of linear programs LW r ( t ), we show thatthe number of such segments is necessarily exponential, leading to the exponential lower boundfor the number of iterations of interior point methods (Theorem B). We provide an explicitlower bound for the value of the parameter t . It is doubly exponential in r , implying that thebitlength of t is exponential in r , which is consistent with the polynomial time character ofinterior point methods in the bit model. Related Work.
The redundant Klee-Minty cube of [DTZ09] and the “snake” in [DTZ08] areinstances which show that the total curvature of the central path can be in Ω ( m ) for a polytopedescribed by m inequalities. Gilbert, Gonzaga and Karas [GGK04] also exhibited ill-behavedcentral paths. They showed that the central path can have a “zig-zag” shape with infinitelymany turns, on a problem defined in R by non-linear but convex functions.The central path has been studied by Dedieu, Malajovich and Shub [DMS05] via the multiho-mogeneous B´ezout Theorem and by De Loera, Sturmfels and Vinzant [DLSV12] using matroidtheory. These two papers provide an upper bound of O ( n ) on the total curvature averaged overall regions of an arrangement of hyperplanes in dimension n .In terms of iteration-complexity of interior point methods, several worst-case results havebeen proposed [Ans91, KY91, JY94, Pow93, TY96, BL97]. In particular, Stoer and Zhao [ZS93]showed the iteration-complexity of a certain class of path-following methods is governed by anintegral along the central path. This quantity, called Sonnevend’s curvature , was introducedin [SSZ91]. The tight relationship between the total Sonnevend curvature and the iterationcomplexity of interior points methods have been extended to semidefinite and symmetric coneprograms [KOT13]. Note that Sonnevend’s curvature is different from the geometric curvaturewe study in this paper. To the best of our knowledge, there is no explicit relation between thegeometric curvature and the iteration-complexity of interior point methods. We circumvent thisdifficulty here by showing directly that the geometric curvature and the number of iterationsare exponential for the family LW r ( t ).The present work relies on the considerations of amoebas (images by the valuation) of al-gebraic and semi-algebraic sets, see [EKL06, DY07, RGST05, Ale13] for background. Anotheringredient is a construction of Bezem, Nieuwenhuis and Rodr´ıguez-Carbonell [BNRC08]. Theirgoal was to show that an algorithm of Butkoviˇc and Zimmermann [BZ06] has exponential run-ning time. We arrive at our family of linear programs by lifting a variant of this constructionto Puiseux series.Finally, let us point out that the first main result of this article, the exponential boundfor the total curvature of the central path, initially appeared in our preprint [ABGJ14]. Wenext discuss the main differences with the present paper. In the original preprint, we exploiteddifferent tools: to characterize the tropical central path, we used methods from model theory,employing o-minimal structures and Hardy fields, in the spirit of Alessandrini [Ale13]. In thepresent revision, we provide a more elementary proof, avoiding the use of model theory. Theoriginal model theory approach, however, keeps the advantage of greater generality. We expectthat this will allow to extend some of the present results to other kinds of barrier functions.In addition, we have added an explicit lower bound for the number of iterations, our second ain result here. We thank the colleagues who commented on [ABGJ14], in particular A. Deza,T. Terlaky and Y. Zinchenko.2. The Primal-Dual Central Path and Its Neighborhood
In this section, we recall the definition of the central path and introduce the notions related topath-following interior point methods which we will use in the rest of the paper.In what follows, we consider a linear program of the formLP(
A, b, c ) minimize h c, x i subject to Ax + w = b , ( x, w ) ∈ R n + m + in which the slack variables w are explicit. Here and below, A is a m × n matrix, b ∈ R m , c ∈ R n , we denote by h· , ·i the standard scalar product, and by R + the set of non-negative reals.The dual linear program takes a form which is very similar:DualLP( A, b, c ) maximize h b, y i subject to s − A ⊤ y = c , ( s, y ) ∈ R n + m + , where · ⊤ denotes the transposition. For the sake of brevity, we set N := n + m to representthe total number of variables in both linear programs. Let F ◦ be the set of strictly feasibleprimal-dual elements, i.e., F ◦ := (cid:8) z = ( x, w, s, y ) > Ax + w = b , s − A ⊤ y = c (cid:9) . which we assume to be nonempty. In this situation, for any given µ >
0, the system of equationsand inequalities(1) Ax + w = bs − A ⊤ y = c (cid:18) xswy (cid:19) = µex, w, y, s > x µ , w µ , s µ , y µ ) ∈ R N ; here e stands for the all-1-vectorin R N ; further, xs and wy denote the Hadamard products of x by s and w by y , respectively.The central path of the dual pair LP( A, b, c ) and DualLP(
A, b, c ) of linear programs is definedas the function which maps µ > x µ , w µ , s µ , y µ ). The latter shall be referred toas the point of the central path with parameter µ . The equality constraints in (1) define a realalgebraic curve, the central curve of the dual pair of linear programs, which has been studiedin [BL89] and [DLSV12]. The central curve is the Zariski closure of the central path.The primal and dual central paths are defined as the projections of the central path onto the( x, w )- and ( s, y )-coordinates, respectively. Equivalently, given µ >
0, the points ( x µ , w µ ) and( s µ , y µ ) on the primal and dual central paths can be defined as the unique optimal solutions ofthe following pair of logarithmic barrier problems:minimize h c, x i − µ (cid:16)P nj =1 log( x j ) + P mi =1 log( w i ) (cid:17) subject to Ax + w = b , x > , w > , and: maximize µ (cid:16)P nj =1 log( s j ) + P mi =1 log( y i ) (cid:17) − h b, y i subject to s − A ⊤ y = c , s > , y > . The uniqueness of the optimal solutions follows from the fact that the objective functions arestrictly convex and concave, respectively. The equivalence to (1) results from the optimalityconditions of the logarithmic barrier problems. The main property of the central path is thatthe sequences ( x µ , w µ ) and ( s µ , y µ ) converge to optimal solutions ( x ∗ , w ∗ ) and ( s ∗ , y ∗ ) of thelinear programs LP( A, b, c ) and DualLP(
A, b, c ), when µ tends to 0.The duality measure ¯ µ ( z ) of an arbitrary point z = ( x, w, s, y ) ∈ R N + is defined by(2) ¯ µ ( z ) := 1 N (cid:0) h x, s i + h w, y i (cid:1) . ith this notation, observe that the point z belongs to the central path if and only if we have(3) (cid:18) xswy (cid:19) = ¯ µ ( z ) e . In other words, the difference ( xswy ) − ¯ µ ( z ) e indicates how far the point z = ( x, w, s, y ) is fromthe central path. This leads to introducing the neighborhood(4) N θ := n z ∈ F ◦ : (cid:13)(cid:13)(cid:13)(cid:18) xswy (cid:19) − ¯ µ ( z ) e (cid:13)(cid:13)(cid:13) θ ¯ µ ( z ) o , of the central path by bounding some norm of the deviation in terms of a precision parameter0 < θ <
1. Clearly, this neighborhood depends on the choice of the norm k · k . In the contextof interior point methods common choices include the ℓ - or ℓ ∞ -norms. However, here we focuson the wide neighborhood (5) N −∞ θ := n z ∈ F ◦ : (cid:18) xswy (cid:19) > (1 − θ )¯ µ ( z ) e o . This arises from replacing k · k in (4) by the one-sided ℓ ∞ -norm . The latter is the map sendinga vector v to max(0 , max i ( − v i )). This is a weak norm in the sense of [PT14], it is positivelyhomogeneous and subadditive, but it vanishes on some non-zero vectors.Our first observation is that the map ¯ µ commutes with affine combinations. Proposition 1.
Let z = ( x, w, s, y ) and z ′ = ( x ′ , w ′ , s ′ , y ′ ) be two points in F . Then, for all α ∈ R , we have ¯ µ ((1 − α ) z + αz ′ ) = (1 − α )¯ µ ( z ) + α ¯ µ ( z ′ ) . Proof.
We write ∆z := z ′ − z and similarly for the components, i.e., ∆z = ( ∆x, ∆w, ∆s, ∆y ).Since z, z ′ ∈ F , we have A∆x + ∆w = 0 and ∆s − A ⊤ ∆y = 0. Employing these equalities it canbe verified that h ∆x, ∆s i + h ∆w, ∆y i = 0. Therefore, the function α ¯ µ ( z + α∆z ) is affine,which completes the proof. (cid:3) Interior point methods follow the central path by computing a sequence of points in a pre-scribed neighborhood N θ of the central path, in such a way that the duality measure decreases.Here, we do not precisely specify N θ , but we only assume that it arises from the choice of someweak norm k · k . The basic step of the algorithm can be summarized as follows. At iteration k ,given a current point z k = ( x k , w k , s k , y k ) ∈ N θ with duality measure µ k = ¯ µ ( z k ), and a positiveparameter µ < µ k , the algorithm aims at solving the system (1) up to a small error in order toget an approximation of the point of the central path with parameter µ . To this end, it startsfrom the point z k and exploits the Newton direction ∆z = ( ∆x, ∆w, ∆s, ∆y ), which satisfies(6) A∆x + ∆w = 0 ∆s − A ⊤ ∆y = 0 (cid:18) x k ∆sw k ∆y (cid:19) + (cid:18) ∆x s k ∆w y k (cid:19) = µe − (cid:18) x k s k w k y k (cid:19) . Then, the algorithm follows the direction ∆z and iterates to a point of the form z ( α ) := z k + α∆z , where 0 < α
1. The correctness and the convergence of the approach are based on theconditions that, firstly, the point z k +1 := z ( α ) still belongs to the neighborhood N θ and that,secondly, the ratio of the new value µ k +1 := ¯ µ ( z ( α )) of the duality measure with µ k is sufficientlysmall. The following lemma shows that, in fact, the whole line segment between z k and z ( α ) iscontained in N θ . Lemma 2. If z k and z ( α ) are contained in N θ then z ( β ) ∈ N θ for all β ∈ [0 , α ] . roof. The point z ( β ) lies in F ◦ since the latter set is convex. We use the notation z ( β ) =( x ( β ) , w ( β ) , s ( β ) , y ( β )). Using the last equality in (6), we can write (cid:16) x ( β ) s ( β ) w ( β ) y ( β ) (cid:17) = βµe + (1 − β ) (cid:16) x k s k w k y k (cid:17) + β (cid:0) ∆x∆s∆w∆y (cid:1) . The first two equalities in (6) entail h ∆x, ∆s i + h ∆w, ∆y i = 0. Exploiting the last equality, weget ¯ µ ( z (1)) = µ . Then, it follows from Proposition 1 that ¯ µ ( z ( β )) = (1 − β ) µ k + βµ . We deducethat (cid:16) x ( β ) s ( β ) w ( β ) y ( β ) (cid:17) − ¯ µ ( z ( β )) e = (1 − β ) h(cid:16) x k s k w k y k (cid:17) − µ k e i + β (cid:0) ∆x∆s∆w∆y (cid:1) . The same relation holds when β = α . In this way, we eliminate the term (cid:0) ∆x∆s∆w∆y (cid:1) to write (cid:16) x ( β ) s ( β ) w ( β ) y ( β ) (cid:17) − ¯ µ ( z ( β )) e = (cid:0) (1 − β ) − β α (1 − α ) (cid:1)h(cid:16) x k s k w k y k (cid:17) − µ k e i + β α h(cid:16) x ( α ) s ( α ) w ( α ) y ( α ) (cid:17) − ¯ µ ( z ( α )) e i . Since β α
1, the term (1 − β ) − β α (1 − α ) is non-negative. Using the fact that z k and z ( α ) belong to N θ , and the subadditivity and positive homogeneity of the weak norm k · k , wededuce that (cid:13)(cid:13)(cid:13)(cid:16) x ( β ) s ( β ) w ( β ) y ( β ) (cid:17) − ¯ µ ( z ( β )) e (cid:13)(cid:13)(cid:13) (cid:0) (1 − β ) − β α (1 − α ) (cid:1) θµ k + β α θ ¯ µ ( z ( α ))= θ (cid:0) (1 − β ) µ k + β α µ (cid:1) θ (cid:0) (1 − β ) µ k + βµ (cid:1) = θ ¯ µ ( z ( β )) as β α . (cid:3) The implementation of the basic iteration step which we have previously described variesfrom one interior point method to another. In particular, there exists several strategies for thechoice of the neighborhood, the parameter µ with respect to the current value of the dualitymeasure µ k , and the step length α , in order to achieve a polynomial-time complexity. Let usdescribe in more detail the main ones. Considering the large variety of existing path-followinginterior point methods in the literature, we stick to the classification of [Wri97, Chapter 5], andrefer to it for a complete account on the topic.Short-step interior point methods, like [KMY89a, MA89], use an ℓ -neighborhood of pre-scribed size θ , and set µ to σµ k where σ < α to 1. In contrast, long-step interior pointmethods, such as [KMY89b], exploit the wider neighborhood N −∞ θ , allow more freedom forthe choice of µ at every iteration ( µ is set to σµ k where σ < σ min , σ max ]), and take α ∈ [0 ,
1] as large as possible to ensure that z ( α ) ∈ N −∞ θ . Anotherimportant class of methods, the so-called predictor-corrector ones, make use of two nested ℓ -neighborhoods N θ ′ and N θ ( θ ′ < θ ), and alternate between predictor and corrector steps. In theformer, µ is optimistically set to 0 (the duality measure of optimal solutions), while α is chosenas the largest value in [0 ,
1] such that z ( α ) ∈ N θ . The next corrector step aims at “centering”the trajectory by doing one Newton step in the direction of the point of the central path withparameter µ k +1 = ¯ µ ( z ( α )). This means that the duality measure is kept to µ k +1 , and the steplength is set 1. A careful choice of θ depending on θ ′ ensures that we obtain in this way a pointin the narrower neighborhood N θ ′ .The predictor-corrector scheme, initially introduced in [MTY93], has inspired several works.Let us mention the one of Vavasis and Ye [VY96], who made a step towards a strongly polynomialcomplexity by arriving at an iteration complexity upper bound depending on the matrix A only. Their technique has been later refined into more practical algorithms, see [MMT98,MT03, KT13]. The difference of these methods with the original predictor-corrector one isthat they sometimes exploit another direction than the Newton one, called the layered leastsquares direction . However, in such iterations, the step length α is always chosen so that for all0 β α , the point z ( β ) lies in the neighborhood N θ (see [VY96, Theorem 9]).In consequence of Lemma 2 and the previous discussion, the aforementioned interior pointmethods all share the property that they describe a piecewise linear trajectory entirely included n a certain neighborhood N −∞ θ of the central path, where θ is a prescribed value. We stressthat this property is the only assumption made in our complexity result, Theorem B, on interiorpoint methods. More formally, this trajectory is a polygonal curve in R N , i.e., a union of finitelymany segments [ z , z ] , [ z , z ] , . . . , [ z p − , z p ]. Since polygonal curves play an important role inthe paper, we introduce some terminology. We say that a polygonal curve is supported by thevectors v , . . . , v p when the latter correspond to the direction vectors of the successive segments[ z , z ] , [ z , z ] , . . . , [ z p − , z p ]. In the case where we equip the curve with an orientation, weassume that the direction vectors are oriented consistently.3. Ingredients From Tropical Geometry
Tropical geometry provides a combinatorial approach to studying algebraic varieties definedover a field with a non-Archimedean valuation. To deal with optimization issues, we need somevalued field which is ordered. We restrict our attention to one such field, which is particularlyconvenient for our application, to keep our exposition elementary. Our field of choice, which wedenote as K , are the absolutely convergent generalized real Puiseux series. Here ‘generalized’means that we allow arbitrary real numbers as exponents as in [Mar10]. Note that the ordinaryPuiseux series have value group Q , leading to restrictions which are artificial from a tropicalperspective. In some sense, K is the “simplest” real closed valued field for which we can obtainour results.3.1. Fields of real Puiseux series and Puiseux polyhedra.
The field K of absolutelyconvergent generalized real Puiseux series consists of elements of the form(7) f = X α ∈ R a α t α , where a α ∈ R for all α , and such that: (i) the support { α ∈ R : a α = 0 } is either finite orhas −∞ as the only accumulation point; (ii) there exists ρ > t > ρ . Note that the null series is obtained by taking an empty support. When f = 0, the first requirement ensures that the support has a greatest element α ∈ R . We saythat the element f is positive when the associated coefficient a α is positive. This extends to atotal ordering of K , defined by f g if g − f is the null series or positive. Equivalently, therelation f g holds if and only if f ( t ) g ( t ) for all sufficiently large t . We write K + for theset of non-negative elements of K .The valuation map val : K → R ∪ {−∞} is given as follows. For f ∈ K the valuation val( f )is defined as the greatest element α of the support of f if f = 0, and −∞ otherwise. Denotingby log t ( · ) := log( · )log t the logarithm with respect to the base t > f ) = lim t → + ∞ log t | f ( t ) | , with the convention log t −∞ . Observe that, for all f , g ∈ K , this yieldsval( f + g ) max(val( f ) , val( g )) and val( f g ) = val( f ) + val( g ) . (8)The inequality for the valuation of the sum turns into an equality when the leading terms inthe series f and g do not cancel. In particular, this is the case when f and g belong to K + .We point out that K actually agrees with the field of generalized Dirichlet series originallyconsidered by Hardy and Riesz [HR15]. This was already used in the tropical setting in [ABG98].Classical Dirichlet series can be written as P k a k k s , and these are obtained from (7) by substitut-ing t = exp( s ) and α k = log k . It follows from results of van den Dries and Speissegger [vdDS98]that the field K is real closed. The interest in such fields comes from Tarski’s Principle , whichsays that every real closed field has the same first-order properties as the reals. We point out that the ℓ -neighborhood with size θ is obviously contained in N −∞ θ . s a consequence of the previous fact, we can define polyhedra over Puiseux series as usual.In more details, given d >
1, a (Puiseux) polyhedron is a set of the form(9) P = { x ∈ K d : Ax b } , where A ∈ K p × d , b ∈ K p (with p > stands for the partial order over K p . By Tarski’sprinciple, Puiseux polyhedra have the same (first-order) properties as their analogs over R . Inparticular, the Minkowski–Weyl theorem applies, so that every Puiseux polyhedron admits aninternal representation by means of a finite set of points and rays in K d .Using a field of convergent series allows us to think of Puiseux polyhedra as parametricfamilies of ordinary polyhedra. Indeed, to any Puiseux polyhedron P of the form (9), weassociate the family of polyhedra P ( t ) ⊂ R d , defined for t large enough, P ( t ) := { x ∈ R d : A ( t ) x b ( t ) } . The next proposition implies in particular that the family of polyhedra P ( t ) is independent ofthe choice of the external representation of P . Proposition 3.
Suppose that P is the Minkowski sum of the convex hull of vectors u , . . . , u q ∈ K d and of the convex cone generated by vectors v , . . . , v r ∈ K d (here, the notions of convex hulland of convex cone are understood over K ). Then, for t large enough, P ( t ) is the Minkowskisum of the convex hull of vectors u ( t ) , . . . , u q ( t ) ∈ R d and of the convex cone generated byvectors v ( t ) , . . . , v r ( t ) ∈ R d (the notions of convex hull and of convex cone are now understoodover R ).Proof. Let Q ( t ) denote the Minkowski sum of the convex hull of vectors u ( t ) , . . . , u q ( t ) andof the convex cone generated by vectors v ( t ) , . . . , v r ( t ). Since P contains u , . . . , u q togetherwith the rays K + v , . . . , K + v r , we have Au i b and Av j i ∈ [ q ] and j ∈ [ r ]. Itfollows that A ( t ) u i ( t ) b ( t ) and A ( t ) v j ( t ) i ∈ [ q ] and j ∈ [ r ] and for t largeenough. Hence, P ( t ) ⊃ Q ( t ) for t large enough.Let us now consider an extreme point u of P . Then, a characterization of the extreme pointsof a polyhedron shows that the collection of gradients of the constraints A k x b k , k ∈ [ p ] whichare active at point x = u constitutes a family of full rank. This property can be expressed inthe first order theory of K . It follows that, for t large enough, the same property holds forthe collection of the gradients of the constraints A k ( t ) x b k ( t ), k ∈ [ p ] that are active atpoint x = u ( t ). Hence u ( t ) is an extreme point of P ( t ), and so, u ( t ) must belong to the set { u ( t ) , . . . , u q ( t ) } . A similar argument shows that if v ∈ K d generates an extreme ray of P ,the ray generated by v ( t ) is extreme in P ( t ) for t large enough, and so v ( t ) ∈ ∪ ℓ ∈ [ r ] R + v ℓ ( t ). Itfollows that P ( t ) ⊂ Q ( t ) holds for t large enough. (cid:3) Remark . One can show, by arguments of the same nature as in the latter proof, that for all x ∈ K d , x ∈ P ⇐⇒ ( x ( t ) ∈ P ( t ) for t large enough) . (10)Note, however, that the smallest value t such that x ( t ) ∈ P ( t ) for all t > t cannot be boundeduniformly in x .3.2. Tropical polyhedra.
Tropical polyhedra may be informally thought of as the analoguesof convex polyhedra over the tropical semifield T . Note that in this semifield, the zero and unitelements are −∞ and 0, respectively. Given λ ∈ T \ {−∞} , we shall also denote by λ ⊙ ( − the inverse of λ for the tropical multplication, i.e., λ ⊙ ( − := − λ . The tropical addition andmultiplication extend to vectors and matrices in the usual way. More precisely, A ⊕ B :=( A ij ⊕ B ij ) ij , and A ⊙ B := ( L k A ik ⊙ B kj ) ij , where A and B are two matrices of appropriatesizes with entries in T . Further, the d -fold Cartesian product T d is equipped with the structureof semimodule, thanks to the tropical multiplication λ ⊙ v := ( λ ⊙ v i ) i of a vector v with ascalar λ . igure 1. The three possible shapes of tropical segments in dimension 2A tropical halfspace of T d is the set of points x ∈ T d which satisfy one tropical linear (affine)inequality, max( α + x , . . . , α d + x d , β ) max( α ′ + x , . . . , α ′ d + x d , β ′ ) , where α, α ′ ∈ T d and β, β ′ ∈ T . A tropical polyhedron is the intersection of finitely many tropicalhalfspaces. Equivalently, it can be written in the form (cid:8) x ∈ T d : A ⊙ x ⊕ b A ′ ⊙ x ⊕ b ′ (cid:9) where A, A ′ ∈ T p × d and b, b ′ ∈ T p for some p >
0. The tropical semifield T = R ∪ {−∞} isequipped with the order topology, which gives rise to the product topology on T d . Tropicalhalfspaces, and thus tropical polyhedra, are closed in this topology. Note also that the subsettopology on R d ⊂ T d agrees with the usual Euclidean topology.An analogue of the Minkowski–Weyl Theorem allows for the “interior represention” of atropical polyhedron P in terms of linear combinations of points and rays [GK11]. That is tosay that there exist finite sets U, V ⊂ T d such that P is the set of all points of the form (cid:16)M u ∈ U α u ⊙ u (cid:17) ⊕ (cid:16)M v ∈ V β v ⊙ v (cid:17) (11)where α u , β v ∈ T and L u ∈ U α u is equal to the tropical unit, i.e., the real number 0. We shallsay that the tropical polyhedron P is generated by the sets U and V . The term L u ∈ U α u ⊙ u is atropical convex combination of the points in U , while L v ∈ V β v ⊙ v is a tropical linear combinationof the vectors in V . These are the tropical analogues of convex and conic hulls, respectively.Indeed, all scalars α u , β v are implicitly non-negative in the tropical sense, i.e., they are greaterthan or equal to the tropical zero element −∞ . We point out that the “tropical polytopes”considered by Develin and Sturmfels [DS04] are obtained by omitting the term L u ∈ U α u ⊙ u and by requiring the vectors v ∈ V to have finite coordinates in the representation (11).If P is a non-empty tropical polyhedron, the supremum sup( u, v ) = u ⊕ v with respect to thepartial order of T d of any two points u, v ∈ P also belongs to P . If in addition P is compact,then the supremum of an arbitrary subset of P is well-defined and belongs to P . Consequently,there is a unique element in P which is the coordinate-wise maximum of all elements in P .We call it the (tropical) barycenter of P , as it is the mean of P with respect to the uniformidempotent measure.The tropical segment between the points u, v ∈ T d , denoted by tsegm ( u, v ), is defined as theset of points of the form λ ⊙ u ⊕ µ ⊙ v such that λ ⊕ µ = 0. Equivalently, the set tsegm ( u, v )is the tropical polyhedron generated by the sets U = { u, v } and V = ∅ . As illustrated inFigure 1, tropical segments are polygonal curves, and the direction vectors supporting everyordinary segment have their entries in { , ± } [DS04, Proposition 3]. We shall slightly refinethis statement in the case where u v . To this end, for K ⊂ [ d ], we denote by e K the vectorwhose k th entry is equal to 1 if k ∈ K , and 0 otherwise. emma 5. Let u, v ∈ T d such that u v . The tropical segment tsegm ( u, v ) is a polygonal curvewhich, when oriented from u to v , consists of ordinary segments supported by direction vectorsof the form e K , . . . , e K ℓ where K ( · · · ( K ℓ and ℓ d .Proof. Since u v , the set tsegm ( u, v ) is reduced to the set of the points of the form u ⊕ ( µ ⊙ v ),where µ
0. Let K ( µ ) be the set of i ∈ [ d ] such that u i < µ + v i . When µ ranges from −∞ to0, K ( µ ) takes a finite number of values K = ∅ ( K ( · · · ( K ℓ , where ℓ d . It is immediatethat the ordinary segments constituting the tropical segment tsegm ( u, v ) are supported by thevectors e K , . . . , e K ℓ . (cid:3) We now relate tropical polyhedra with their classical analogues over Puiseux series via thevaluation map. The fact that sums of non-negative Puiseux series do not suffer from cancellationtranslates into the following.
Lemma 6.
The valuation map is a monotone and surjective semifield homomorphism from K + to T .Proof. That val is a homomorphism is a consequence of (8) and the subsequent discussion.Monotonicity and surjectivity are straightforward. (cid:3)
This carries over to Puiseux polyhedra in the non-negative orthant:
Proposition 7.
The image under the valuation map of any Puiseux polyhedron P ⊂ K d + is atropical polyhedron in T d .Proof. Let U , V ⊂ K d be two finite collections of vectors such that P is the set of combinationsof the form x = X u ∈ U α u u + X v ∈ V β v v , (12)where α u , β v ∈ K + and P u ∈ U α u = 1. Observe that U and V both lie in K d + since P ⊂ K d + .From Lemma 6, we deduce that val( P ) is contained in the tropical polyhedron P generated bythe sets U := val( U ) and V := val( V ). Conversely, any point in P of the form (11) is the imageunder the valuation map of X u ∈ U Z t α u u + X v ∈ V t β v v , where Z = P u ∈ U t α u is such that val Z = 0. (cid:3) The special case of Proposition 7 concerning “tropical polytopes” in the sense of [DS04] wasalready proved by Develin and Yu [DY07, Proposition 2.1]. One can show that, conversely, eachtropical polyhedron arises as the image under the valuation map of a polyhedron included in K d + ; see [ABGJ15, Proposition 2.6].3.3. Metric properties.
In this section, we establish various metric estimates which will beused in the analysis of the central path in Section 6. These estimates involve different metrics.We start with the non-symmetric metric δ F , defined by δ F ( x, y ) := inf (cid:8) ρ > x + ρe > y (cid:9) , where x, y ∈ T d . Recall that e denotes the all-1-vector. Writing the inequality x + ρe > y as ρ ⊙ x > y reveals that δ F is the tropical analogue of the Funk metric which appears in Hilbert’sgeometry [PT14]. Equivalently, we can write δ F ( x, y ) = max(0 , max k ( y k − x k )) , with the convention −∞ + (+ ∞ ) = −∞ . In this way, we observe that δ F is derived from theone-sided ℓ ∞ -norm k · k which we used to define the wide neighborhood of the central pathin (5), i.e., δ F ( x, y ) = k x − y k . We point out that δ F ( x, y ) < + ∞ if and only if the support of x contains the support of y , i.e., { k : x k = −∞} ⊃ { k : y k = −∞} . he metric d ∞ induced by the ordinary ℓ ∞ -norm is obtained by symmetrizing δ F as follows: d ∞ ( x, y ) := max( δ F ( x, y ) , δ F ( y, x )) . We shall consider another symmetrization of δ F , leading to the affine version of Hilbert’s pro-jective metric : d H ( x, y ) := δ F ( x, y ) + δ F ( y, x ) . The metric d H was shown in [CGQ04] to be the canonical metric in tropical convexity. Forinstance, the projection onto a convex set is well defined and is a best approximation in thismetric. The relevance of Hilbert’s geometry to the study of the central path was alreadyobserved by Bayer and Lagarias [BL89]. Notice that d H ( x, y ) < + ∞ if and only if the supportsof the two vectors x, y ∈ T d are identical.We extend our notation to sets as follows. Given X, Y ⊂ T d we define d H ( X, Y ) := sup x ∈ X inf y ∈ Y d H ( x, y ) and d ∞ ( X, Y ) := sup x ∈ X inf y ∈ Y d ∞ ( x, y ) . These are the directed Hausdorff distances from X to Y induced by d H and d ∞ , respectively.In order to establish the metric properties of this section, we repeatedly use the followingelementary inequalities: if t > γ , . . . , γ p ∈ R + ,(13) max(log t γ , . . . , log t γ p ) log t ( γ + · · · + γ p ) max(log t γ , . . . , log t γ p ) + log t p . We start with a metric estimate over classical and tropical segments.
Lemma 8.
Let S = [ u, v ] be a segment in R d , and let S trop be the tropical segment between thepoints log t u and log t v . Then d ∞ ( S trop , log t S ) log t , Here and below log t S is short for { log t s : s ∈ S } .Proof. Let x := λ ⊙ (log t u ) ⊕ µ ⊙ (log t v ) be a point of the tropical segment S trop , where λ, µ ∈ T are such that λ ⊕ µ = 0. Now the point x ′ := t λ u + t µ vt λ + t µ belongs to S . Using (13) and λ ⊕ µ = 0, we get 0 log t ( t λ + t µ ) log t
2. Similarly, for all i ∈ [ d ],we have x i log t ( t λ u i + t µ v i ) x i + log t
2. We deduce that x i − log t log t x ′ i x i + log t d ∞ ( x, log t x ′ ) log t
2. It follows that d ∞ ( S trop , log t S ) log t (cid:3) We now focus on estimating the distance between tropical polyhedra and related logarithmicdeformations of convex polyhedra. To this end, we consider a Puiseux polyhedron P includedin the non-negative orthant, as well as the associated parametric family of polyhedra P ( t )over R . The following theorem shows that the tropical polyhedron val( P ) is the log-limit of thepolyhedra P ( t ), and that the convergence is uniform. This is related to a result of Briec andHorvath, who established in [BH04] a uniform convergence property for a parametric family ofconvex hulls. Theorem 9.
Let P ⊂ K d + be a Puiseux polyhedron. Then the sequence (log t P ( t )) t of realpolyhedra converges to the tropical polyhedron val( P ) with respect to the directed Hausdorffdistance d H .Proof. By Proposition 3, we can find finite sets U , V ⊂ K d + such that for sufficiently large t ,the real polyhedron P ( t ) is generated by the sets of points U ( t ) := { u ( t ) : u ∈ U } ⊂ R d + andrays V ( t ) := { v ( t ) : v ∈ V } ⊂ R d + . Let u ∈ U . If t is large enough, then u i ( t ) = 0 is equivalentto val( u ) i = −∞ , for all i ∈ [ d ]. Thus δ F (log t u ( t ) , val u ) as well as δ F (val u , log t u ( t )) convergeto 0 when t → + ∞ . The situation is similar for v ( t ) and val v , for any v ∈ V .Moreover, the tropical polyhedron P := val( P ) is generated by the sets val( U ) and val( V ),as shown in the proof of Proposition 7. ow consider x ∈ P ( t ). From Carath´eodory’s Theorem we know that there exist subsets { u k ( t ) } k ∈ K ⊂ U ( t ) and { v ℓ ( t ) } ℓ ∈ L ⊂ V ( t ) with | K | + | L | d + 1 such that the point x can bewritten as x = X k ∈ K α k u k ( t ) + X ℓ ∈ L β ℓ v ℓ ( t ) , where α k , β ℓ > k ∈ K , ℓ ∈ L and P k ∈ K α k = 1. Then, for all i ∈ [ d ], we get(14) M k ∈ K (cid:0) (log t α k ) ⊙ log t u k ( t ) (cid:1) ⊕ M ℓ ∈ L (cid:0) (log t β ℓ ) ⊙ log t v ℓ ( t ) (cid:1) log t x i (cid:20)M k ∈ K (cid:0) (log t α k ) ⊙ log t u k ( t ) (cid:1) ⊕ M ℓ ∈ L (cid:0) (log t β ℓ ) ⊙ log t v ℓ ( t ) (cid:1)(cid:21) + log t ( | K | + | L | ) . Setting γ := max k ∈ K α k , we have | K | γ
1. Now we define x ′ := (cid:0)M k ∈ K α ′ k ⊙ u k (cid:1) ⊕ (cid:0)M ℓ ∈ L β ′ ℓ ⊙ v ℓ (cid:1) , where α ′ k := log t ( α k /γ ), β ′ ℓ := log t β ℓ , u k := val( u k ) and v ℓ := val( v ℓ ). By choice of γ we have L k ∈ K α ′ k = 0 and thus x ′ ∈ P . Further, x i > k ∈ K such that u ki ( t ) > ℓ ∈ L such that v ℓi ( t ) >
0. Provided that t is sufficiently large, this is equivalentto the fact that u ki > −∞ for some k ∈ K , or v ℓi > −∞ for some ℓ ∈ L . This latter propertyamounts to x ′ i > −∞ . Consequently, we have d H (log t x, x ′ ) < + ∞ , and we can derive from (14)that(15) x ′ i − max (cid:0) log t | K | + max k ∈ K δ F (log t u k ( t ) , u k ) , max ℓ ∈ L δ F (log t v ℓ ( t ) , v ℓ ) (cid:1) log t x i x ′ i + log t ( | K | + | L | ) + max (cid:0) max k ∈ K δ F ( u k , log t u k ( t )) , max ℓ ∈ L δ F ( v ℓ , log t v ℓ ( t )) (cid:1) , for all i ∈ [ d ]. Finally, we deduce that(16) d H (log t P ( t ) , P ) log t ( d + 1) + max (cid:0) max u ∈ U δ F (val u , log t u ( t )) , max v ∈ V δ F (val v , log t v ( t )) (cid:1) + max (cid:0) log t ( d + 1) + max u ∈ U δ F (log t u ( t ) , val u ) , max v ∈ V δ F (log t v ( t ) , val v ) (cid:1) , which tends to 0 when t → + ∞ . (cid:3) Remark . For the sake of brevity, we only stated and proved here the one sided metricestimates which we will use in the proof of our main results, leaving it to the interested reader toderive the symmetrical metric estimates. For instance, the inequality d ∞ (log t S, S trop ) log t d H (val( P ) , log t P ( t )) tends to zero aswell as t → ∞ .Next we refine the convergence result just obtained by providing a metric estimate in thespecial case where P is a polyhedron given by constraints with monomial coefficients. Here aPuiseux series of the form ± t α is called monomial , with the convention t −∞ = 0. Further, avector or a matrix is monomial if all its entries are. For a matrix M of size d × d we introducethe quantity η ( M ) > η ( M ) := min n η : σ, τ ∈ Sym d , η = d X i =1 α iσ ( i ) − d X i =1 α iτ ( i ) > o , where Sym d stands for the symmetric group over [ d ]. We use the convention min ∅ = + ∞ .Phrased differently, the determinant of M is a Puiseux series with finitely many terms withdecreasing exponents, and η ( M ) provides a lower bound on the gap between any two successiveexponents (if any). This allows us to obtain explicit upper and lower bounds for log t | det M ( t ) | n terms of val(det M ). Note that these bounds hold without any assumption on the genericityof the matrix M . Lemma 11.
Let M ∈ K d × d be a monomial matrix. Then, for all t > , we have log t | det M ( t ) | val(det M ) + log t d ! , and, if even t > ( d !) /η ( M ) , then we get val(det M ) log t | det M ( t ) | + log t d ! . Proof.
First note that the statement is trivial when det M = 0, since in this case, the deter-minant of M ( t ) vanishes for all t >
0. Now suppose that det M = 0. Since M is monomial,every entry of M is of the form ǫ ij t α ij where ǫ ij ∈ {± } . Therefore, we obtaindet M = p X k =1 c k t β k , where the following conditions are met: (i) each c k is a non-null integer, and P k | c k | d !,(ii) every β k is of the form P di =1 α iσ ( i ) for a certain permutation σ ∈ Sym d , (iii) β > · · · >β p > −∞ . With this notation, we have val(det M ) = β , and β i − β i +1 > η ( M ). Similarly, forall t >
0, we have det M ( t ) = P pk =1 c k t β k . This leads tolog t | det M ( t ) | β + log t (cid:16) p X k =1 | c k | t β k − β (cid:17) val(det M ) + log t d ! . Further, provided that t > ( d !) /η ( M ) , we have p X k =2 | c k | t β k − β ( d ! − t β − β ( d ! − t − η ( M ) − /d ! , and so log t | det M ( t ) | > β + log t (cid:16) − p X k =2 | c k | t β k − β (cid:17) > val(det M ) − log t d ! . (cid:3) Recall that we write e for the all-1-vector of an appropriate size. Theorem 12.
Let P ⊂ K d + be a polyhedron of the form { x ∈ K d : Ax b } where A and b aremonomial. Let η be the minimum of the quantities η ( M ) where M is a square submatrix of (cid:18) A b e ⊤ (cid:19) of order d . Then, for all t > ( d !) /η , we have: d H (log t P ( t ) , val( P )) log t (cid:0) ( d + 1) ( d !) (cid:1) . Proof.
We employ the notation introduced in the proof of Theorem 9. Note that the inequalitygiven in (16) holds for any sets U , V generating the polyhedron P . In particular, we can set U to the set of vertices of P , and V to a set consisting of precisely one representative of everyextreme ray of the recession cone of P .Let u ∈ U . Since u is a vertex, there exists a subset I of cardinality d such that A I u = b I ,where A I and b I consist of the rows of A and b , respectively, which are indexed by i ∈ I , and A I is invertible. Therefore, by Cramer’s rule, every coordinate u i can be expressed as a fractionof the form ± det M / det A I where M is a submatrix of (cid:0) A b (cid:1) of size d × d . Recall that u i is non-negative, and hence u i = | det M | / | det A I | . By definition, η ( M ) and η ( A I ) are greaterthan or equal to η . From Lemma 11 we derive thatval( u i ) − t d ! log t u i ( t ) val( u i ) + 2 log t d ! , for all t > ( d !) /η . Since these inequalities hold for all i ∈ [ d ], we deduce that the two quantities δ F (val u , log t u ( t )) and δ F (log t u ( t ) , val u ) differ by at most 2 log t d !.The recession cone of P is the set { z ∈ K d : Az } . Since P is contained in the positiveorthant, so does the recession cone. Therefore, without loss of generality, we can assume that very v ∈ V satisfies P i v i = 1. In this way, the elements of V precisely correspond to thevertices of the polyhedron { z ∈ K d : Az , e ⊤ z = 1 } . Using the same arguments as above,we infer that, for all v ∈ V , the two values δ F (val v , log t v ( t )) and δ F (log t v ( t ) , val v ) differ byat most 2 log t d !, as soon as t > ( d !) /η . Now the claim follows from (16). (cid:3) The Tropical Central Path
Our core idea is to introduce the tropical central path of a linear program over Puiseux series.This is defined as the image of the classical primal-dual central path under the valuation map.By (3) the classical central path is a segment of a real algebraic curve, and so its tropicalizationis a segment of a (real) tropical curve and thus piecewise linear. It will turn out that forcertain Puiseux linear programs the tropical central path carries a substantial amount of metricinformation. In Sections 5 and 6 below we will see that this applies to the linear programs LW r ( t ). As its key advantage the tropical central path turns out to be much easier to analyzethan its classical counterpart.4.1. A geometric characterization of the tropical central path.
As in Section 2, weconsider a dual pair of linear programs, except that now the coefficents lie in the field K ofabsolutely convergent real Puiseux series from Section 3.1:minimize h c , x i subject to Ax + w = b , ( x , w ) ∈ K N + , LP ( A , b , c ) maximize h b , y i subject to s − A ⊤ y = c , ( s , y ) ∈ K N + . DualLP ( A , b , c )where A ∈ K m × n , b ∈ K m , c ∈ K n and N := n + m . Here, the Euclidean scalar product h· , ·i is extended by setting h u , v i := P i u i v i for any vectors u , v with entries over K . Further, wedefine P := { ( x , w ) ∈ K N + : Ax + w = b } and Q := { ( s , y ) ∈ K N + : s − A ⊤ y = c } , which correspond to the feasible regions of the two Puiseux linear programs. Moreover, we let F := P × Q be the set of primal-dual feasible points, and F ◦ := { z ∈ F : z > } is the strictlyfeasible subset. Throughout we will make the following assumption: Assumption 13.
The set F ◦ is non-empty. This allows us to define the central path of the Puiseux linear programs LP ( A , b , c ) and DualLP ( A , b , c ). Indeed, applying Tarski’s principle to the real-closed field K shows that, forall µ ∈ K such that µ >
0, the system(17) Ax + w = bs − A ⊤ y = c (cid:18) xswy (cid:19) = µ e x , w , y , s > K N . We denote this solution as C ( µ ) = ( x µ , w µ , s µ , y µ ) and refer toit as point of the central path with parameter µ . Similarly, by Tarski’s principle, F = ∅ ensuresthat the two linear programs have the same optimal value ν ∈ K . We let ( x ∗ , w ∗ ) ∈ P and( s ∗ , y ∗ ) ∈ Q be a pair of primal and dual optimal solutions.Let P ⊂ T N , Q ⊂ T N and F ⊂ T N the images under the valuation map of the primaland dual feasible polyhedra P , Q and F , respectively. Similarly we write ( x ∗ , w ∗ ) ∈ T N and( s ∗ , y ∗ ) ∈ T N for the coordinate-wise valuations of the optimal solutions ( x ∗ , w ∗ ) and ( s ∗ , y ∗ ).Given a primal-dual feasible point z = ( x , w , s , y ) ∈ F , the duality gap , denoted by gap ( z ),is defined as the difference between the values of the primal and dual objective functions, i.e., h c , x i + h b , y i . We recall that gap ( z ) is equivalently given by the complementarity gap definedas the sum of the pairwise product of primal/dual variables: gap ( z ) = h x , s i + h w , y i . bserve that the right-hand side of the latter equality consists of sums of non-negatives terms(of the form x j s j and w i y i ). Consequently, we can apply the valuation map term-wise anddefine, for all z = ( x, w, s, y ) ∈ F , the tropical duality gap as tgap ( z ) := h x, s i T ⊕ h w, y i T , where h· , ·i T stands for the tropical analogue of the scalar product, i.e., h u, v i T := L i ( u i ⊙ v i ).Then the quantity tgap ( z ) equals the valuation of the duality gap of any primal-dual feasible z with val( z ) = z . The study of the tropical central path will require the following tropicalsublevel set induced by the tropical duality gap F λ := { z ∈ F : tgap ( z ) λ } , which is defined for any λ ∈ R . We collect a few basic facts about this collection of sublevelsets. Proposition 14.
Let µ ∈ K such that µ > , and λ = val( µ ) . Then(i) the set F λ is a bounded tropical polyhedron given by P λ × Q λ where P λ := { ( x, w ) ∈ P : h s ∗ , x i T ⊕ h y ∗ , w i T λ } , Q λ := { ( s, y ) ∈ Q : h x ∗ , s i T ⊕ h w ∗ , y i T λ } , (ii) and the image under val of the point C ( µ ) lies in F λ .Proof. We start with the proof of (ii). By definition, C ( µ ) ∈ F so that val( C ( µ )) ∈ F .Moreover, we have tgap (cid:0) val( C ( µ )) (cid:1) = val( gap ( C ( µ )). Since gap ( C ( µ )) = N µ by (2), we deducethat the previous quantity is equal to the valuation of N µ , which is λ . This forces that val( C ( µ ))lies in F λ .We need to show that F λ is a tropical polyhedron which is bounded. To this end we consider z = ( x , w , s , y ) ∈ F . Recall that ν is the common optimal value of the primal and the dualPuiseux linear programs. Then we obtain gap ( z ) = (cid:0) h c , x i − ν (cid:1) + (cid:0) h b , y i + ν (cid:1) = gap ( x , w , s ∗ , y ∗ ) + gap ( x ∗ , w ∗ , s , y ) . Since the right-hand side of this identity is a sum of two non-negative terms, applying thevaluation map yields tgap ( z ) = tgap ( x, w, s ∗ , y ∗ ) ⊕ tgap ( x ∗ , w ∗ , s, y ) for all z = ( x, w, s, y ) ∈ F .Thus we can express the tropical sublevel set F λ as F λ = { ( x, w, s, y ) ∈ F : tgap ( x, w, s ∗ , y ∗ ) ⊕ tgap ( x ∗ , w ∗ , s, y ) λ } = P λ × Q λ . In particular, the set F λ is a tropical polyhedron.To complete our proof we still have to show that F λ is bounded. To this end, pick some point z ◦ = ( x ◦ , w ◦ , s ◦ , y ◦ ) in F λ which has finite coordinates. For instance, we can take z ◦ = val( C ( µ ))by making use of (ii). Now consider an arbitrary point ( x, w, s, y ) ∈ F λ . As F λ = P λ × Q λ , weknow that ( x, w, s ◦ , y ◦ ) ∈ F λ . In particular, tgap ( x, w, s ◦ , y ◦ ) λ , or, equivalently, ( x j ⊙ s ◦ j λ for all j ∈ [ n ] ,w i ⊙ y ◦ i λ for all i ∈ [ m ] . Since s ◦ j , y ◦ i > −∞ , this entails that x j λ ⊙ ( s ◦ j ) ⊙ ( − and w i λ ⊙ ( y ◦ i ) ⊙ ( − for all i ∈ [ m ] and j ∈ [ n ]. Similarly, the entries of ( s, y ) are bounded, too. This completes the proof of (i). (cid:3) As a bounded tropical polyhedron the set F λ admits a (tropical) barycenter. Recall that thelatter was defined as the coordinate-wise maximum of that set. The following theorem relatesthis barycenter with the valuation of the central path, and gives rise to the definition of thetropical central path: Theorem 15.
The image under the valuation map of the central path of the pair of primal-duallinear programs LP ( A , b , c ) and DualLP ( A , b , c ) can be described by: (18) val( C ( µ )) = barycenter of { z ∈ val( F ) : tgap ( z ) val( µ ) } , for any µ ∈ K such that µ > . roof. Let λ := val( µ ), and denote by ¯ z = (¯ x, ¯ w, ¯ s, ¯ y ) the barycenter of the tropical polyhedron F λ . By construction we have val( C ( µ )) ¯ z . Moreover, since tgap (cid:0) ¯ z (cid:1) λ , we also have¯ x j ⊙ ¯ s j λ and ¯ w i ⊙ ¯ y i λ for all i ∈ [ m ] and j ∈ [ n ]. It follows that:(19) ( λ = val( x µ j s µ j ) = val( x µ j ) ⊙ val( s µ j ) ¯ x j ⊙ ¯ s j λ ,λ = val( w µ j y µ j ) = val( w µ i ) ⊙ val( y µ i ) ¯ w i ⊙ ¯ y i λ . As a consequence, the inequality val( C ( µ )) ¯ z is necessarily an equality. (cid:3) The quantity (18), which depends only on the valuation of µ , is called the tropical centralpath at λ = val( µ ) and is denoted by C trop ( λ ) = ( x λ , w λ , s λ , y λ ) . Analogously, the primal and the dual tropical central paths are defined by projecting to ( x λ , w λ )and ( s λ , y λ ), respectively. As shown in (19), the primal and dual components of the tropicalcentral path are characterized by(20) x λj ⊙ s λj = λ = w λi ⊙ y λi for all i ∈ [ m ] and j ∈ [ n ]. The next statement shows that the tropical central path is apolygonal curve with a particularly simple structure. Proposition 16.
The tropical central path λ
7→ C trop ( λ ) is a monotone piecewise linear func-tion, whose derivative at each smooth point is a vector of the form ( e K , e [ N ] \ K ) , for some K ⊂ [ N ] .Proof. Let us denote by g : T N → T the function which sends ( x, w ) to h s ∗ , x i T ⊕ h y ∗ , w i T . Pickfinite generating sets U ⊂ T N and V ⊂ T N \ { ( −∞ , . . . , −∞ ) } for the tropical polyhedron P .Since ( x λ , w λ ) lies in P it can be expressed as( x λ , w λ ) = (cid:16)M u ∈ U α u ⊙ u (cid:17) ⊕ (cid:16)M v ∈ V β v ⊙ v (cid:17) , where L u ∈ U α u = 0. The inequality g ( x λ , w λ ) λ now amounts to α u ⊙ g ( u ) λ and β v ⊙ g ( v ) λ for all u ∈ U and v ∈ V . As ( x λ , w λ ) is the barycenter of P λ , the coefficients α u and β v can be chosen to be maximal. This enforces α u = min (cid:0) , λ ⊙ ( g ( u )) ⊙ ( − (cid:1) and β v = λ ⊙ ( g ( v )) ⊙ ( − , using the convention ( −∞ ) ⊙ ( − = + ∞ . Note that g ( v ) = −∞ forall v ∈ V . Indeed, if there were a ray v ∈ V with g ( v ) = −∞ , then any point of the form( x ∗ , w ∗ ) ⊕ ( β ⊙ v ) would belong to P λ . The latter conclusion would contradict the boundednessof P λ . Therefore, all α u and β v belong to R . Observe also that α u and β v , thought of asfunctions of λ , are monotone, piecewise linear, and that their derivatives at any smooth pointtake value in { , } . It follows that λ ( x λ , s λ ) is piecewise linear and monotone, and that itsderivative at any smooth point is of the form e K .A similar argument reveals that the map λ ( s λ , y λ ) is also piecewise linear and monotone,with a derivative at any smooth point of the form e K ′ for some set K ′ ⊂ [ N ]. In consequence,the derivative of the map C trop at any smooth point is of the form ( e K , e K ′ ) and, from (20), weget that K ′ = [ N ] \ K . This proves our claim. (cid:3) We end this section with the following direct consequence of Proposition 16.
Corollary 17. If λ λ ′ , then C trop ( λ ) C trop ( λ ′ ) C trop ( λ ) + ( λ ′ − λ ) e . A uniform metric estimate on the convergence of the central path.
For now, wehave related the tropical central path with the central path of the linear programs LP ( A , b , c )and DualLP ( A , b , c ) over Puiseux series. These give rise to a parametric family of dual linearprograms LP( A ( t ) , b ( t ) , c ( t )) and DualLP( A ( t ) , b ( t ) , c ( t )) over the reals. The purpose of thissection is to relate the resulting family of central paths with the tropical central path. More recisely, we will show that the logarithmic deformation of these central paths uniformly con-verge to the tropical curve. In fact, we will even show that the logarithmic deformation of thewide neighborhoods N −∞ θ of these central paths collapses onto the tropical central path.Let P ( t ) , Q ( t ) ⊂ R N be the feasible sets of LP( A ( t ) , b ( t ) , c ( t )) and DualLP( A ( t ) , b ( t ) , c ( t )),respectively. (Note that this notation is compatible with the one introduced in Section 3.1thanks to Proposition 3.) Then F ( t ) := P ( t ) × Q ( t ) is the primal-dual feasible set, while F ◦ ( t ) := { z ∈ F ( t ) : z > } comprises only those primal-dual points which are strictly feasible.The following lemma relates the optimal solutions of LP ( A , b , c ) and DualLP ( A , b , c ) withthe ones of LP( A ( t ) , b ( t ) , c ( t )) and DualLP( A ( t ) , b ( t ) , c ( t )). Lemma 18.
There exists a positive real number t such that for all t > t the following threeproperties hold:(i) the set F ◦ ( t ) is non-empty;(ii) the number ν ( t ) is the optimal value of LP ( A ( t ) , b ( t ) , c ( t )) and DualLP ( A ( t ) , b ( t ) , c ( t )) ;(iii) ( x ∗ ( t ) , w ∗ ( t )) and ( s ∗ ( t ) , y ∗ ( t )) constitute optimal solutions.Proof. For two series u , v ∈ K the equality u = v forces u ( t ) = v ( t ) for all t large enough. Asimilar statement holds for inequalities like u v and u < v . We infer that the set F ◦ ( t ) isnot empty if t ≫
1. Moreover, ( x ∗ ( t ) , w ∗ ( t )) ∈ P ( t ), ( s ∗ ( t ) , y ∗ ( t )) ∈ Q ( t ) and h c ( t ) , x ∗ ( t ) i = h b ( t ) , y ∗ ( t ) i = ν ( t ). Since LP( A ( t ) , b ( t ) , c ( t )) and DualLP( A ( t ) , b ( t ) , c ( t )) are dual to oneanother, we conclude that ( x ∗ ( t ) , w ∗ ( t )) and ( s ∗ ( t ) , y ∗ ( t )) form a pair of optimal solutions. (cid:3) Throughout the following we will keep that value t from Lemma 18. When t > t , we knowfrom Lemma 18(i) that the primal-dual central path of the linear programs LP( A ( t ) , b ( t ) , c ( t ))and DualLP( A ( t ) , b ( t ) , c ( t )) is well defined. In this case we denote by C t ( µ ) the point of thiscentral path with parameter µ , where µ ∈ R and µ >
0. Let us fix the real precision parameter θ in the open interval from 0 to 1. Then the set N −∞ θ,t ( µ ) := n z = ( x, w, s, y ) ∈ F ◦ ( t ) : ¯ µ ( z ) = µ and (cid:18) xswy (cid:19) > (1 − θ ) µe o is a neighborhood of the point C t ( µ ). A direct inspection shows that the union of the sets N −∞ θ,t ( µ ) for µ > N −∞ θ of the entire central path of thelinear program LP( A ( t ) , b ( t ) , c ( t )) over R ; see (5). In order to stress the dependence on t , wedenote this neighborhood by N −∞ θ,t . With this notation, we have N −∞ θ,t = [ µ> N −∞ θ,t ( µ ) . Further let(21) δ ( t ) := 2 d H (log t F ( t ) , F ) , which, by Theorem 9, tends to 0 when t goes to + ∞ . The following result states that we canuniformly bound the distance from the image of N −∞ θ,t ( µ ) under log t to the point C trop (log t µ )of the tropical central path, independently of µ : Theorem 19.
For all t > t and µ > we have d ∞ (cid:0) log t N −∞ θ,t ( µ ) , C trop (log t µ ) (cid:1) log t (cid:16) N − θ (cid:17) + δ ( t ) . Proof.
Choose t > t and µ >
0, and z = ( x, w, s, y ) ∈ N −∞ θ,t ( µ ). Letting λ := log t µ we claimthat it suffices to prove that(22) log t z C trop ( λ ) + (log t N + δ ( t )) e . Indeed, by definition of the wide neighborhood N −∞ θ,t ( µ ), we have log t ( x, w ) > − log t ( s, y ) + (cid:0) λ + log t (1 − θ ) (cid:1) e . Using (22), we obtainlog t ( x, w ) > − ( s λ , y λ ) + (cid:18) λ − log t (cid:16) N − θ (cid:17) − δ ( t ) (cid:19) e = ( x λ , w λ ) − (cid:18) log t (cid:16) N − θ (cid:17) + δ ( t ) (cid:19) e , here the last equality is due to (20). Analogously, we can prove that log t ( s, y ) > ( s λ , y λ ) − (cid:16) log t (cid:0) N − θ (cid:1) + δ ( t ) (cid:17) e .Now let us show that (22) holds. By definition of the duality measure ¯ µ ( z ), we have gap ( z ) = N ¯ µ ( z ) = N µ . Applying the map log t yields(23) tgap (log t z ) log t gap ( z ) = λ + log t N , where the inequality is a consequence of the first inequality in (13).Let z ′ ∈ F such that d H (log t z, z ′ ) < + ∞ . Recall that(24) z ′ − δ F (log t z, z ′ ) e log t z z ′ + δ F ( z ′ , log t z ) e . The first inequality in (24) gives tgap ( z ′ ) tgap (log t z ) + 2 δ F (log t z, z ′ ). In combinationwith (23) this shows that z ′ lies in F λ ′ for λ ′ := λ + log t N + 2 δ F (log t z, z ′ ). The secondinequality in (24) now yieldslog t z z ′ + δ F ( z ′ , log t z ) e C trop ( λ ′ ) + δ F ( z ′ , log t z ) e C trop ( λ ) + (cid:0) log t N + 2 d H (log t z, z ′ ) (cid:1) e , where the second inequality follows from C trop ( λ ′ ) being the barycenter of F λ ′ , and the lastinequality is a consequence of Corollary 17. As this argument is valid for all z ′ ∈ F within afinite distance from log t z we obtain that log t z C trop ( λ ) + (cid:0) log t N + δ ( t ) (cid:1) e . (cid:3) Main example.
The family LW r ( t ) of linear programs over the reals from the introductionmay also be read as a linear program over the field K , thinking of t as a formal parameter. Wedenote this linear program by LW r . The goal of this section is to obtain a complete descriptionof the corresponding tropical central path.Introducing slack variables w , . . . , w r − in the first 3 r − LW r , and addingthe redundant inequalities x i > i < r −
1, gives rise to a linear program LW = r , whichis of the form LP ( A , b , c ) in dimension N = 5 r −
1. (Note that the last two inequalities of LW r are non-negativity constraints, which is why we do not need slack variables for them.) Thedual Puiseux linear program (with slacks) is referred to as DualLW = r . We retain the notationintroduced in Section 4.1; for instance, we denote by P and Q the primal and dual feasible setsrespectively.To begin with, we verify that Assumption 13 is satisfied. Due to the lower triangular natureof the system of inequalities in LW r , we can easily find a vector x satisfying every inequalityof this system in a strict manner. In other words, we can find ( x , w ) ∈ P such that x > w >
0. Moreover, since the feasible set of LW r is bounded, the set P is bounded as well. Thisimplies that the dual feasible set Q contains a point ( s , y ) satisfying s > y >
0. As a result,the set F ◦ is non-empty.We focus on the description of the primal part λ ( x λ , w λ ) of the tropical central path,since the dual part can be readily obtained by using the relations (20). It can be checked thatthe optimal value of LW = r , and subsequently of DualLW = r , is equal to 0. Since in our case, theprimal objective vector c is given by the nonnegative vector (1 , , . . . , ∈ K n , we deduce thatwe can choose the dual optimal solution ( s ∗ , y ∗ ) as (1 , , . . . , ∈ K n + m . As a consequence ofTheorem 15 and Proposition 14(i), the point ( x λ , w λ ) on the primal tropical central path agreeswith the barycenter of the tropical sublevel set(25) P λ = { ( x, w ) ∈ P : x λ } . Recall that P stands for val( P ).We first restrict our attention to the x -component of the tropical central path. To this end,let P ′ be the projection of the primal feasible set P onto the coordinates x , . . . , x r . Thisis precisely the feasible set of the Puiseux linear program LW r . Further, let P ′ be the imageunder val of P ′ . Equivalently, this is the projection of P onto x , . . . , x r . We claim that P ′ is iven by the 3 r + 1 tropical linear inequalities(26) x , x x j +1 x j − , x j +1 x j x j +2 (1 − / j ) + max( x j − , x j ) 1 j < r ,which are obtained by applying the valuation map to the inequalities in LW r coefficient-wise.While this can be checked by hand, we can also apply [AGS16, Corollary 14], as P ′ is a regularset in T r , i.e., it coincides with the closure of its interior.By (25) we deduce that the point x λ is the barycenter of the tropical polyhedron { x ∈P ′ : x λ } . We arrive at the following explicit description of x λ . Proposition 20.
For all λ ∈ R , the point x λ is given by the recursion x λ = min( λ, x λ = 1 x λ j +1 = 1 + min( x λ j − , x λ j ) x λ j +2 = (1 − / j ) + max( x λ j − , x λ j ) 1 j < r .Proof. We introduce the family of maps F j : ( a, b ) (1 + min( a, b ) , − / j + max( a, b )) where1 j < r . With this notation, the point x lies in P ′ and also satisfies x λ if and only if(27) x min( λ, , x , ( x λ j +1 , x λ j +2 ) F j ( x λ j − , x λ j ) , for every 1 j < r . Since the maps F j are order preserving, the barycenter of the tropicalpolyhedron defined by (27) is the point which attains equality in (27). (cid:3) Observe that the map λ x λ is constant on the interval [2 , ∞ [, while it is linear on ] −∞ , , w , . . . , w r − into our analysis. The points ( x , w )in the primal feasible set P is defined by the following constraints:(28) x + w = t x + w = t x j +1 + w j = t x j − x j +1 + w j +1 = t x j x j +2 + w j +2 = t − / j ( x j − + x j )( x , w ) ∈ K N + j < r .This entails that the points ( x, w ) in P = val( P ) satisfy the inequalities(29) w , w w j x j − , w j +1 x j w j +2 (1 − / j ) + max( x j − , x j ) 1 j < r .The following result states that all these inequalities are tight for all points ( x λ , w λ ) on theprimal tropical central path. Proposition 21.
For all λ ∈ R , the point w λ is described by the following relations: (30) w λ = 2 , w λ = 1 w λ j = 1 + x λ j − w λ j +1 = 1 + x λ j w λ j +2 = (1 − / j ) + max( x λ j − , x λ j ) = x λ j +2 j < r λx λ x λ x λ x λ x λ x λ x λ x λ x λ x λ x r − x r r − r − r − r − ( r − r − ( r − r − ( r − r − ( r − r − ( r − r − ( r − r − λ = 0 λ = r − λ = r − λ = r − λ = r − λ = r − λ = r − Figure 2.
Left: the x -components of the primal tropical central path of LW r for r > λ
2. Right: the projection of the tropical central path of LW r onto the ( x r − , x r )-plane. Proof.
Let ¯ w be the element defined by the relations in (30). We want to prove that w λ = ¯ w .By Theorem 15 and Proposition 14(i), it suffices to show that the point ( x λ , ¯ w ) is the barycenterof the tropical polyhedron P λ . Given ( x, w ) ∈ P , we have x x λ as x belongs to P λ and x λ isthe barycenter of the latter set. Moreover, w satisfies the inequalities given in (29). We deducethat w ¯ w .It now remains to show that ( x λ , ¯ w ) belongs to P , since this immediately leads to ( x λ , ¯ w ) ∈P λ . In other terms, we want to find a point ( x , w ) ∈ P such that val( x , w ) = ( x λ , ¯ w ). Let usfix a sequence of positive numbers α = > α > · · · > α r − >
0. We claim that letting(31) x j +1 := α j t x λ j +1 , x j +2 := α j t x λ j +2 , (0 j < r )and defining w in terms of the equalities in (28), yields such an admissible lift.First, observe that x > x ) = x λ . Second, we have w = t − α t x λ and w = α t .Recall that x λ
2. Thus, w and w are non-negative, and they satisfy val w = 2 andval w = 1. Now, let us consider j for 1 j < r . We have w j = t x j − − x j +1 = α j − t x λ j − − α j t x λ j +1 = ( α j − − α j ) t ¯ w j + α j ( t ¯ w j − t x λ j +1 ) . As ¯ w j = 1 + x λ j − > x λ j +1 , we have 0 t ¯ w j − t x λ j +1 t ¯ w j , and this gives us w j > w j = ¯ w j . A similar argument shows that w j +1 > w j +1 = ¯ w j +1 . Finally, wecan write w j +2 = t − / j ( x j − + x j ) − x j +2 = ( (2 α j − − α j ) t x λ j +2 if x λ j − = x λ j , ( α j − − α j ) t x λ j +2 + o ( t x λ j +2 ) otherwise.Since 2 α j − > α j − > α j , we obtain that w j +2 >
0, and val w j +2 = x λ j +2 = ¯ w j +2 . (cid:3) able 1. Coordinates of points on the primal tropical central path of LW r forsome specific values of λ , where 1 j < r and k = 0 , , . . . , j − − λ k j k +22 j k +42 j k +62 j k +82 j x j +1 j + k j j + k +22 j j + k +22 j j + k +42 j j + k +42 j x j +2 j + k +12 j j + k +12 j j + k +32 j j + k +32 j j + k +52 j w j j + k j j + k +22 j j + k +42 j j + k +42 j j + k +42 j w j +1 j + k +22 j j + k +22 j j + k +22 j j + k +42 j j + k +62 j w j +2 j + k +12 j j + k +12 j j + k +32 j j + k +32 j j + k +52 j Table 1 gives a summary of the values of the coordinates of the primal tropical central pathfor specific values of λ which we shall use below.5. Curvature Analysis
The purpose of this section is to show how the combinatorial analysis of the tropical centralpath translates into lower bounds on the total curvature of the central path of a parametricfamily of linear programs over the reals. Our main application will be a detailed version ofTheorem A from the introduction, and a proof of this result.Let us recall some basic facts concerning total curvature. For two non-null vectors x, y ∈ R d we denote by ∠ xy the measure α ∈ [0 , π ] of the angle of the vectors x and y , so thatcos α = h x, y ik x kk y k , where k·k refers to the Euclidean norm. Given three points U, V, W ∈ R d such that U = V and V = W , we extend this notation to write ∠ U V W for the angle formed by the vectors
U V and
V W . If τ is a polygonal curve in R d parameterized over an interval [ a, b ], the total curvature κ ( τ, [ a, b ]) is defined as the sum of angles between the consecutive segments of the curve. Moregenerally, the total curvature κ ( σ, [ a, b ]) can be defined for an arbitrary curve σ , parameterizedover the same interval, as the supremum of κ ( τ, [ a, b ]) over all polygonal curves τ inscribed in σ . If σ is twice continuously differentiable, this coincides with the standard definition of thetotal curvature R ba k κ ′′ ( s ) k ds , when κ is parameterized by arc length; see [AR89, Chapter V] formore background.Our approach is based on estimating the curvature of the central path using approximationsby polygonal curves. Our first observation is concerned with limits of angles between familiesof vectors arising from vectors over K . Lemma 22.
Let x , y be two non-null vectors in K d , and let x := val( x ) and y := val( y ) . Thenthe limit of ∠ x ( t ) y ( t ) for t → + ∞ exists. Moreover, if the sets arg max i ∈ [ d ] x i and arg max i ∈ [ d ] y i are disjoint, then we have lim t → + ∞ ∠ x ( t ) y ( t ) = π . Proof.
Since the field K is real closed, the Euclidean norm k·k can be extended to a function from K d to K by k u k := qP i u i for all u ∈ K d . As a consequence, the quotient h x , y i / (cid:0) k x kk y k (cid:1) isan element of K . Let α be its valuation. We obtain α max i ∈ [ d ] ( x i + y i ) − (cid:0) max i ∈ [ d ] x i + max i ∈ [ d ] y i (cid:1) . uppose without loss of generality that h x , y i 6 = 0. Then, there exists a non-zero number c ∈ R such that h x ( t ) , y ( t ) i / (cid:0) k x ( t ) kk y ( t ) k (cid:1) = ct α + o ( t α )when t → + ∞ . If arg max i ∈ [ d ] x i ∩ arg max i ∈ [ d ] y i = ∅ , then α <
0, implying that the limit ofcos ∠ x ( t ) y ( t ) = h x ( t ) , y ( t ) i / (cid:0) k x ( t ) kk y ( t ) k (cid:1) as t → + ∞ is equal to 0. (cid:3) We will use Lemma 22 in order to estimate the limit when t → + ∞ of the angle betweensegments formed by triplets of successive points of the tropical central path. One remarkableproperty is that the tropical central path of any Puiseux linear program is monotone; seeProposition 16. We refine Lemma 22 to fit this setting. Lemma 23.
Let U , V , W ∈ K d , and U := val( U ) , V := val( V ) and W := val( W ) . If max i ∈ [ d ] U i < max i ∈ [ d ] V i < max i ∈ [ d ] W i , and the sets arg max i ∈ [ d ] V i and arg max i ∈ [ d ] W i aredisjoint, we have lim t → + ∞ ∠ U ( t ) V ( t ) W ( t ) = π . Proof.
Let us remark that for all i ∈ [ d ], we have val( V i − U i ) max( U i , V i ), and this inequalityis an equality if U i = V i . Since max i ∈ [ d ] U i < max i ∈ [ d ] V i , we deduce that max i ∈ [ d ] val( V i − U i ) =max i ∈ [ d ] V i , and that the argument of the two maxima are equal. The same applies to thecoordinates of the vector val( W − V ). We infer from Lemma 22 that ∠ U ( t ) V ( t ) W ( t ) tendsto π/ i ∈ [ d ] V i ∩ arg max i ∈ [ d ] W i = ∅ . (cid:3) This motivates us to introduce a (weak) tropical angle ∠ ∗ U V W := ( π if U, V, W satisfy the conditions of Lemma 23 , U, V, W ∈ T d . We now consider a Puiseux linear program of the form LP ( A , b , c ), together with the associated family of linear programs LP( A ( t ) , b ( t ) , c ( t )) andtheir primal-dual central path C t . We obtain that: Proposition 24.
Let λ, λ ∈ R and λ = λ < λ < · · · < λ p − < λ p = λ . Then lim inf t →∞ κ (cid:16) C t , (cid:2) t λ , t λ (cid:3)(cid:17) > p − X k =1 ∠ ∗ C trop ( λ k − ) C trop ( λ k ) C trop ( λ k +1 ) . Proof.
Let λ ∈ R , and let U be the point of the central path of LP ( A , b , c ) with parameter µ equal to the Puiseux series t λ . If t is substituted by a sufficiently large real number, thepoints C t ( t λ ) and U ( t ) are identical, since both satisfy the constraints given in (1) for A = A ( t ), b = b ( t ), c = c ( t ) and µ = t λ .The monotonicity of the tropical central path shown in Proposition 16 allows to applyLemma 23. From the previous discussion, we get thatlim t →∞ ∠ C t ( t λ k − ) C t ( t λ k ) C t ( t λ k +1 ) > ∠ ∗ C trop ( λ k − ) C trop ( λ k ) C trop ( λ k +1 ) . for all k ∈ [ p − κ (cid:0) C t , [ t λ , t λ ] (cid:1) > P p − k =1 ∠ C t ( t λ k − ) C t ( t λ k ) C t ( t λ k +1 ). (cid:3) We denote by LW = r ( t ) and DualLW = r ( t ) the linear programs over R obtained by substitut-ing the parameter t with a real value in the Puiseux linear programs LW = r and DualLW = r ,respectively. We are now ready to state and prove the following detailed version of Theorem A. Theorem 25.
For all ǫ > , the total curvature of the primal central path of the linear program LW = r ( t ) is greater than (2 r − − π − ǫ , provided that t > is sufficiently large. Moreover, thesame holds for the primal-dual central path. roof. We will use Proposition 24 to provide a lower bound on lim inf t →∞ κ ( C t , [0 , ,
2] by the scalars λ k = k r − for k = 0 , , . . . , r − .Let us first point out that, given λ ∈ [0 , C trop ( λ ) ofthe tropical central path are less than or equal to max(0 , λ − , λ );see Proposition 20 and 21. It follows that the dual components are dominated by the primalones, and thus it suffices to estimate the total curvature of the primal central path.Using Table 1, we deduce that the maximal component of the vector C trop ( λ k ) is equal to r − k +22 r − , and that is uniquely attained by the coordinate w r − ( λ ) when k is odd, andby w r − ( λ ) when k is even. This implies ∠ ∗ C trop ( λ k − ) C trop ( λ k ) C trop ( λ k +1 ) = π , and weobtain the claim from Proposition 24. (cid:3) Remark . One can refine Theorem 25 to additionally obtain a lower bound on the curvatureof the dual central path at the same time. This requires to consider a slightly modified versionof LW = r ( t ). More precisely, it can be shown that it suffices to add the constraints x r +1 + w r = t r x r − and x r +2 + w r +1 = t r x r involving the two extra variables x r +1 , x r +2 and the slackvariables w r and w r +1 . Remark . Let us compare the lower bound of Theorem 25 with the upper bound of Dedieu,Malajovich and Shub [DMS05] obtained from averaging. Given a real m × n matrix A , vectors b ∈ R m and c ∈ R n , and an m × m diagonal matrix E with diagonal entries ±
1, we consider thelinear program P E min c ⊤ x, Ax − s = b, Es > . It is shown there that the sum of the total curvatures of the dual central paths of the 2 m linearprograms P E arising from the various choices of sign matrices E does not exceed(32) 2 πn (cid:18) m − n (cid:19) . It can be verified that the dual linear program
DualLW = r ( t ) is of the form P E for E = − I ,where I is the identity matrix, n = 3 r + 1 and m = 5 r −
1. By applying Stirling’s formulato (32) we see that the sum of the total curvatures of the dual central paths of the 2 m linearprograms P E , arising from varying E , is bounded by2 π (3 r + 1) (cid:18) r − r + 1 (cid:19) = O (cid:18) √ r (cid:16) (cid:17) r (cid:19) . The lower bound of order Ω (2 r ) from Theorem 25 shows that the total curvature of the dualcentral path of at least one of these 2 m linear programs is exponential in r .6. Tropical Lower Bound on the Complexity of Interior Point Methods
In this section, we derive a general lower bound on the number of iterations of interior pointmethods with a log-barrier. That lower bound is given by the smallest number of tropicalsegments needed to describe the tropical central path, see Theorem 29. Applying this result tothe parametric family of linear programs LW = r ( t ) provides a proof of Theorem B.6.1. Approximating the tropical central path by tropical segments.
We return to thegeneral situation from Section 4.1, and consider a dual pair of linear programs LP ( A , b , c )and DualLP ( A , b , c ) over Puiseux series. Lemma 5 and Proposition 16 yield that the tropicalcentral path can be described as a concatenation of finitely many tropical segments. Given λ, λ ∈ R such that λ λ , we let γ (cid:0) [ λ, λ ] (cid:1) the smallest number of tropical segments needed todescribe the section C trop (cid:0) [ λ, λ ] (cid:1) of the tropical central path. r − x r Figure 3.
A tubular neighborhood of the tropical central path (in light blue),containing an approximation by tropical segments (in orange).Let ǫ >
0. For z ∈ R N we denote by B ∞ ( z ; ǫ ) the closed d ∞ -ball centered at z and withradius ǫ . Further, we fix λ, λ ∈ R such that λ λ . The set T ([ λ, λ ]; ǫ ) := [ λ λ λ B ∞ ( C trop ( λ ); ǫ )is the tubular neighborhood of the section C trop ([ λ, λ ]) of the tropical central path; see Figure 3.Let us consider the union S of a finite sequence of consecutive tropical segments S := tsegm ( z , z ) ∪ tsegm ( z , z ) ∪ · · · ∪ tsegm ( z p − , z p ) ( z , . . . , z p ∈ T N )which is contained in the tubular neighborhood T := T ([ λ, λ ]; ǫ ), and which further satisfies z ∈ B ∞ ( C trop ( λ ); ǫ ) and z p ∈ B ∞ ( C trop ( λ ); ǫ ). That is, S approximates the tropical central pathby p tropical segments, starting and ending in small neighborhoods of z and z p , respectively;see Figure 3 for an illustration. Next we will show that, in this situation, the number of tropicalsegments in S is bounded from below by γ (cid:0) [ λ, λ ] (cid:1) , provided that the tubular neighborhood T is tight enough. To this end, we set ǫ > d ∞ -distance betweenany two distinct vertices in the polygonal curve C trop ( R ). Note that ǫ does not depend on thechoices of λ and λ . Proposition 28. If ǫ < ǫ then p > γ (cid:0) [ λ, λ ] (cid:1) .Proof. We abbreviate γ := γ (cid:0) [ λ, λ ] (cid:1) . Let us consider a sequence λ = λ < λ < · · · < λ γ = λ such that C trop ([ λ, λ ]) can be decomposed as the union of γ successive tropical segments, i.e., C trop ([ λ, λ ]) = tsegm ( C trop ( λ ) , C trop ( λ )) ∪ · · · ∪ tsegm ( C trop ( λ γ − , λ γ )) . By definition of γ , these segments are maximal in the sense that none of them is properlycontained in a tropical segment contained in C trop ([ λ, λ ]).Let us look at the shape of the tropical central path in the neighborhood of an intermediatepoint C trop ( λ i ) for 0 < i < γ . Without loss of generality, we assume that C trop ( λ i ) = 0. Sincethe segment tsegm ( C trop ( λ i − ) , C trop ( λ i )) is maximal, the parameter λ i marks a point where thetropical central path is not differentiable. Let K, L ⊂ [2 N ] such that e K and e L are the leftand right derivatives of C trop at λ i , respectively. Since ǫ < ǫ , the point C trop ( λ i ) is the only k z ℓ C trop ( λ i ) H e K e L z k z ℓ C trop ( λ i ) H e K e L Figure 4.
The tropical central path (in blue) and its tubular neighborhood T (in light blue) near a breakpoint C trop ( λ i ), projected onto the plane ( z k , z ℓ ). Thetropical halfspace H defined in (34) is shown in green. The cases ℓ ∈ K and ℓ K are depicted left and right, respectively.breakpoint of C trop ([ λ, λ ]) contained in the ball B ∞ ( C trop ( λ i ); 3 ǫ ). We derive(33) C trop ( λ + λ i ) = ( − λe K if − ǫ λ ,λe L if 0 λ ǫ . Moreover, we have K L , since K ⊂ L would contradict the maximality of the tropical segment tsegm ( C trop ( λ i − ) , C trop ( λ i )), see Lemma 5. Further, by Proposition 16, the set L is non-empty.Thus, let us consider k ∈ K \ L and ℓ ∈ L . We introduce the tropical halfspace(34) H := { z ∈ T N : max(0 , z ℓ − ǫ ) z k + ǫ } and refer to Figure 4 for an illustration of the setting.We claim that at least one point z j belongs to the neighborhood T ∩ H of C trop ( λ i ). Arguingindirectly, we assume that z j
6∈ H for all j . The complement of H is a tropically convex set, i.e.,if z, z ′
6∈ H , then the tropical segment between z and z ′ is contained in the complement of H .It follows that S ⊂ T \ H . As B ∞ ( C trop ( λ i ); ǫ ) ⊂ H the points z and z p are located in the samepath-connected component of T \B ∞ ( C trop ( λ i ); ǫ ). Further, there is a path from z to C trop ( λ ) in B ∞ ( C trop ( λ ); ǫ ) ⊂ T . That path lies in T \B ∞ ( C trop ( λ i ); ǫ ) as B ∞ ( C trop ( λ ); ǫ ) ∩B ∞ ( C trop ( λ i ); ǫ ) = ∅ due to our assumption ǫ < ǫ . The same argument applies to z p and C trop ( λ ). Therefore, C trop ( λ ) and C trop ( λ ) belong to the same path-connected component of T \ B ∞ ( C trop ( λ i ); ǫ ).However, the latter set consists of two components containing C trop ([ λ, λ i − ǫ [) and C trop (] λ i + ǫ, λ ]), respectively. This provides a contradiction.From (33) and the strict monotonicity of the map C trop , we know that the intersection of C trop ([ λ, λ ]) with H reduces to C trop ([ λ i − ǫ, λ i + 2 ǫ ]). Consequently, we have T ∩ H ⊂ B ∞ ( C trop ( λ i ); 3 ǫ ) . Summing up, for all 0 < i < p , there is at least one index j such that the ball B ∞ ( C trop ( λ i ); 3 ǫ )contains z j . This even holds for i = 0 or i = p , by the choices of z and z p . As ǫ < ǫ , all these p + 1 balls are pairwise disjoint. We finally conclude that p > γ . (cid:3) We are now ready to establish a general lower bound on the number of iterations performedby the class of log-barrier interior point methods described in Section 2. The result is stated interms of polygonal curves contained in the wide neighborhood N −∞ θ,t of the central path, sincesuch curves are the trajectories followed by the log-barrier interior point methods. Our proofcombines Theorem 19 with Proposition 28. To this end, we pick a real number t > t such hat(35) log t (cid:16) N − θ (cid:17) + δ ( t ) < ǫ holds for all t > t . Notice that t depends on t , ǫ and, in particular, on the precisionparameter θ . Hilbert’s projective metric d H plays a role through the definition of δ ( t ) in (21).Recall that ¯ µ ( · ) measures the duality gap. Theorem 29.
Suppose that < θ < and t > t . Then, every polygonal curve [ z , z ] ∪ [ z , z ] ∪ · · · ∪ [ z p − , z p ] contained in the neighborhood N −∞ θ,t satisfies p > γ (cid:0) [log t ¯ µ ( z ) , log t ¯ µ ( z p )] (cid:1) . Proof.
We first assume that ¯ µ ( z ) ¯ µ ( z i ) ¯ µ ( z p ) for all i . Consider z ∈ [ z i , z i +1 ] for some i with 0 i < p . By Proposition 1 we have ¯ µ ( z ) ¯ µ ( z ) ¯ µ ( z p ). It follows that z ∈ N −∞ θ,t ( µ )for some µ with ¯ µ ( z ) µ ¯ µ ( z p ).We define S as the union of the tropical segments tsegm (log t z i , log t z i +1 ) for 0 i < p , andset λ := log t ¯ µ ( z ) and λ := log t ¯ µ ( z p ). By Theorem 19 and Lemma 8, we have: d ∞ (cid:0) S , C trop ([ λ, λ ]) (cid:1) log t (cid:16) N − θ (cid:17) + δ ( t ) < ǫ since t > t . By choosing ǫ as d ∞ (cid:0) S , C trop ([ λ, λ ]) (cid:1) , Theorem 19 ensures that z ∈ B ∞ ( C trop ( λ ); ǫ )and z p ∈ B ∞ ( C trop ( λ ); ǫ ). Therefore, we can apply Proposition 28, which yields the claim inthis special case.We need to deal with the general case. Let j and k be indices such that ¯ µ ( z j ) and ¯ µ ( z k )are the minimal and maximal values among the ¯ µ ( z i ), respectively. If j = k then all dualitymeasures ¯ µ ( z i ) agree, and γ (cid:0) [log t ¯ µ ( z ) , log t ¯ µ ( z p )] (cid:1) = 1. If j < k , we can apply the argumentfor the special case to the subsequence [ z j , z j +1 ] , . . . , [ z k − , z k ]. This way we arrive at p > k − j > γ (cid:0) [log t ¯ µ ( z j ) , log t ¯ µ ( z k )] (cid:1) > γ (cid:0) [log t ¯ µ ( z ) , log t ¯ µ ( z p )] (cid:1) , where the latter inequality comes from the fact that [log t ¯ µ ( z ) , log t ¯ µ ( z p )] is contained in[log t ¯ µ ( z j ) , log t ¯ µ ( z k )]. The remaining case j > k is similar. (cid:3) An exponential lower bound on the number of iterations for our main example.
Let us consider the linear programs LW = r ( t ) and DualLW = r ( t ). A direct inspection reveals thatchoosing t = 0 is sufficient to meet the requirements of Lemma 18.We focus on the section C trop ([0 , λ = 0 and λ = 2. As explained in Section 4.3 and illustrated in Figure 2, the projection of C trop (cid:0) [0 , (cid:1) onto the plane ( x r − , x r ) consists of 2 r − ordinary segments, which alternate theirdirections. This projection to two dimensions cannot be expressed as a concatenation of lessthan 2 r − tropical segments in the plane (see Figure 1). Therefore, also the tropical centralpath cannot be written as the union of fewer tropical segments in any higher dimensional space.With our notation from Section 6.1 this means that γ (cid:0) [0 , (cid:1) > r − . Moreover, it can be verified that the minimal d ∞ -distance between any two vertices in C trop equals 1 / r − . Therefore, choosing ǫ = 1 / (3 · r − ) is good enough for being able to applyProposition 28. It remains to find a sufficiently large number t such that (35) holds for all t > t .Every non-null coefficient in the constraint matrix of LW r is a monomial of degree in r − Z ,and thus we may apply Theorem 12, where η > / r − . As a consequence, if t > ((2 N )!) r − ,we have d H (log t F ( t ) , val( F )) log t (cid:0) (2 N + 1) ((2 N )!) (cid:1) . Recall that N = 5 r − heorem 30. Let < θ < , and suppose that (36) t > (cid:18) max (cid:16) (10 r − , (cid:0) (10 r − (cid:1) (1 − θ ) (cid:17)(cid:19) r − . Then, every polygonal curve [ z , z ] ∪ [ z , z ] ∪ · · · ∪ [ z p − , z p ] contained in the neighborhood N −∞ θ,t of the primal-dual central path of LW = r ( t ) , with ¯ µ ( z ) and ¯ µ ( z p ) > t , contains atleast r − segments. Taking into account the discussion at the end of Section 2, we may restate Theorem 30 interms of the complexity of interior point methods and prove Theorem B.
Corollary 31.
Let < θ < , and suppose that t satisfies (36) . Then, any log-barrier interiorpoint method which describes a trajectory contained in the neighborhood N −∞ θ,t of the primal-dualcentral path of LW = r ( t ) , needs to perform at least r − iterations to reduce the duality measurefrom t to .Remark . Corollary 31 requires the size θ of the neighborhood to be fixed independently ofthe parameter t . While this requirement can be relaxed slightly (the lower bound holds as soonas log(1 − θ ) = o (log t )), we point out that it is met by the interior point methods discussed inSection 2. For instance, for predictor-corrector methods, the radius of the outer neighborhoodis usually set to θ = 1 /
2. Although this setting can be refined, one can show that the proof ofthe convergence requires θ to be chosen less than 4 / Remark . It would be interesting to test standard linear programming solvers on the family LW = r ( t ). It is obvious that solvers computing with bounded precision numbers are unable todeal with coefficients as large as in our example, in the light of the condition (36). This alreadyrules out most of the standard solvers. It would be worthwhile to explore how, e.g., SDPA-GMPcan be used to deal with such input [Nak10]. That solver relies on the floating point numberswith arbitrary precision mantissa provided by the GMP library.7. Combinatorial Experiments
We now want to give some hints to the combinatorial properties of the feasible region of thePuiseux linear program LW r , which we denote as R r . These are based on experiments forthe first few values of r , which have been performed with polymake [GJ00]. Notice that sinceversion 3.0 polymake offers linear programming and convex hull computations over the fieldof Puiseux fractions with rational coefficients [JLLS16], and the coefficients of LW r lie in thissubfield. Notice that this is entirely independent of the metric analysis which was necessary forour main results. Throughout we assume that r > R r is a convex polyhedron in the non-negative orthant of K r which containsinterior points, which means that it is full-dimensional. Moreover, it is easy to check that theexterior normal vectors of the defining inequalities positively span the entire space, hence R r is bounded, i.e., a polytope over Puiseux series. None of these 3 r + 1 inequalities is redundant,i.e., each inequality defines a facet. For instance, R is a quadrangle. However, the polytope R r is not simple for r >
2, i.e., there are vertices which are contained in more than 2 r facets.All these non-simple vertices lie in the optimal face, which is given by x = 0. By modifyingthe inequality x r > x r > ǫ for sufficiently small ǫ > R ǫr asthe feasible region of the perturbed linear program LW ǫr minimize x subject to x t x tx j +1 t x j − , x j +1 t x j x j +2 t − / j ( x j − + x j ) x r − > , x r > ǫ j < r or the remainder of this section we refer to our original construction LW r and its feasible regionas the unperturbed case . The unperturbed Puiseux polytope LW r can be seen as the limit ofthe perturbed Puisuex polytopes LW ǫr when ǫ goes to zero. See Figure 5 for a visualizationof R ǫ .The two facets x = t and x = tx play a special role. It can be verified that they do notshare any vertices, neither in the perturbed nor in the unperturbed case. In the unperturbed r = 2 case these two facets cover all the vertices except for one, while four and 16 are uncoveredfor r = 3 and r = 4, respectively.Since the perturbation is only very slight it follows that the dual graphs of the perturbed andthe unperturbed Puiseux polytope are the same. For 2 r D be any dual to 2-neighborlypolytope; any two facets of D share a common ridge , i.e., a face of codimension 2. Now pickany ridge and truncate it to produce a new polytope D ′ . The dual graph of D ′ is a completegraph minus one edge. Note that the polytope D ′ may have very many vertices as it is still veryclose to a polytope which is 2-neighborly.In our experiments, for all r
6, the primal graph of R ǫr has diameter r + 1 = (3 r + 1) − r ,which is precisely the Hirsch bound. This is in stark contrast with the unperturbed case inwhich the diameter equals 3, for 2 r ǫ are small enough. Our experiments suggestthat ǫ = t − works for all r >
2. Employing generalized Puiseux series with valuations of higherrank offers an alternative approach, which will always work: we may introduce a second largeinfinitesimal s ≫ t and set ǫ = s − . (0 , t, , t / ) ( t , t , t , t / + t / ) (0 , t, , t − ) ( t , t, , t / + t / )( t , t, t , t − ) ( t , t, , t − )( t, t, t , t − )(0 , t − / , , t − ) ( t , , , t / )( t , , , t − ) ( t − / , , , t − )( t − / , t − / , t − / , t − ) ( t , t , t , t / ) Figure 5.
Schlegel diagram of perturbed polytope R ǫ (for ǫ = t − ) projectedonto the facet x = 0 8. Concluding Remarks
In the present work, we obtained a family of counter-examples showing that standard polynomial-time interior points method exhibit a non strongly polynomial time behavior. To do so, weconsidered nonarchimedean instances with a degenerate tropical limit that we characterized bycombinatorial means. This strategy is likely to be applicable to other problems in computationalcomplexity: tropicalization generally permits to test the sensitivity of classical algorithms tothe bitlength of the input.Moreover, the present approach may also extend to other interior point methods (e.g. infeasi-ble ones) or other barrier or penalty functions. Indeed, as should be clear from [Ale13, ABGJ14] hat “really” matters is to work with a Hardy field of functions definable in a o-minimal struc-ture. This allows for other fields than the absolutely convergent generalized real Puiseux seriesconsidered here.The weak tropical angle ∠ ∗ U V W used in Proposition 24 yields a bound on the total curvatureof non-decreasing paths. A similar approach allows one, more generally, to define a notion oftropical curvature for arbitrary paths. This should also be compared with the notion of curvaturefor tropical hypersurfaces introduced in [BdMR13]. We leave this for future work.
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Institut f¨ur Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany [email protected]@math.tu-berlin.de