Exact algorithms for linear matrix inequalities
aa r X i v : . [ m a t h . O C ] S e p Exact algorithms for linear matrix inequalities
Didier Henrion ∗ Simone Naldi † Mohab Safey El Din ‡ September 20, 2016
Abstract
Let A ( x ) = A + x A + · · · + x n A n be a linear matrix, or pencil, generated bygiven symmetric matrices A , A , . . . , A n of size m with rational entries. The setof real vectors x such that the pencil is positive semidefinite is a convex semi-algebraic set called spectrahedron, described by a linear matrix inequality (LMI).We design an exact algorithm that, up to genericity assumptions on the inputmatrices, computes an exact algebraic representation of at least one point in thespectrahedron, or decides that it is empty. The algorithm does not assume theexistence of an interior point, and the computed point minimizes the rank of thepencil on the spectrahedron. The degree d of the algebraic representation of thepoint coincides experimentally with the algebraic degree of a generic semidefiniteprogram associated to the pencil. We provide explicit bounds for the complexityof our algorithm, proving that the maximum number of arithmetic operations thatare performed is essentially quadratic in a multilinear B´ezout bound of d . When m (resp. n ) is fixed, such a bound, and hence the complexity, is polynomial in n (resp. m ). We conclude by providing results of experiments showing practicalimprovements with respect to state-of-the-art computer algebra algorithms. Keywords : linear matrix inequalities, semidefinite programming, computer algebra al-gorithms, symbolic computation, polynomial optimization.
Let S m ( Q ) be the vector space of m × m symmetric matrices with entries in Q . Let A , A , . . . , A n ∈ S m ( Q ) and A ( x ) = A + x A + · · · + x n A n be the associated linear matrix , x = ( x , . . . , x n ) ∈ R n . We also denote the tuple ( A , A , . . . , A n ) by A ∈ S n +1 m ( Q ). For x ∈ R n , A ( x ) is symmetric, with real entries, and hence its eigenvalues are real numbers. ∗ LAAS-CNRS, Universit´e de Toulouse, CNRS, Toulouse, France; Faculty of Electrical Engineering,Czech Technical University in Prague, Czech Republic. † LAAS-CNRS, Universit´e de Toulouse, CNRS, Toulouse, France. ‡ Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, INRIA Paris Center, LIP6, Equipe PolSys,F-75005, Paris, France. x ∈ R n such that the eigenvalues of A ( x )are all nonnegative, that is the spectrahedron S = { x ∈ R n : A ( x ) (cid:23) } . Here (cid:23) A ( x ) (cid:23) linear matrix inequality (LMI). The set S is convex closed basic semi-algebraic. This paper addresses the followingdecision problem for the spectrahedron S : Problem 1 (Feasibility of semidefinite programming)
Compute an exact algebraicrepresentation of at least one point in S , or decide that S is empty. We present a probabilistic exact algorithm for solving Problem (1). The algorithm de-pends on some assumptions on input data that are specified later. If S is not empty,the expected output is a rational parametrization (see e.g. [55]) of a finite set Z ⊂ C n meeting S in at least one point x ∗ . This is given by a vector ( q , . . . , q n +1 ) ⊂ Z [ t ] n +2 anda linear form λ = λ x + · · · + λ n x n ∈ Q [ t ] such that deg( q n +1 ) = ♯ Z , deg( q i ) < deg( q n +1 )for 0 ≤ i ≤ n , gcd( q n +1 , q ) = 1 and Z coincides with the set { ( x , . . . , x n ) ∈ C n | t = λ x + · · · + λ n x n , q n +1 ( t ) = 0 , x i = q i ( t ) /q ( t ) } (1)A few remarks on this representation are in order. Usually, q is taken as ∂q n +1 ∂t for abetter control on the size of the coefficients [14, Theorem 1] – see also the introductorydiscussion of that theorem in [14]. More precisely, if D bounds the degrees of a finite familyof polynomials in Z [ x , . . . , x n ] defining Z and h bounds the bit size of their coefficientsand those of λ , then the coefficients of a rational parametrization encoding Z have bitsize bounded by hD n .This is to be compared with polynomial parametrizations where the rational fractions q i /q are replaced by polynomials p i ; they are obtained by inverting q w.r.t. q n +1 usingthe extended Euclidean algorithm. That leads to polynomials p i with bit size boundedby hD n .In order to compute such representations, the usual and efficient strategy is to computefirst the image of such representations in a prime field and next use a Newton-Hensellifting to recover the integers. According to [23, Lemma 4] and the above bounds, thecost of lifting integers is log-linear in the output size. Since in the case of polynomialparametrizations, the output size may be D n times larger than in the case of rationalparametrizations, rational parametrizations are easier to compute. In addition, observefrom [45, Lemma 3.4 and Theorem 3.12] that isolating boxes for the real points in Z fromrational or polynomial representations have the same bit complexity (e.g. cubic in thedegree of q and log-linear in the maximum bit size of the coefficients in the parametriza-tion).As an outcome of designing our algorithm, we also compute the minimum rank attainedby the pencil on the spectrahedron. Moreover, since the points in Z are in one-to-one correspondence with the roots of q n +1 , from this representation the coordinates ofthe feasible point x ∗ ∈ S can be computed with arbitrary precision by isolating thecorresponding solution t ∗ of the univariate equation q n +1 ( t ) = 0. If S is empty, theexpected output is the empty list. 2 .1 Motivations Semidefinite programming models a large number of problems in the applications [53,10, 8]. This includes one of the most important questions in computational algebraicgeometry, that is the general polynomial optimization problem. Indeed, Lasserre [42]proved that the problem of minimizing a polynomial function over a semi-algebraic set canbe relaxed to a sequence of primal-dual semidefinite programs called LMI relaxations, andthat under mild assumptions the sequence of solutions converge to the original minimum.Generically, solving a non-convex polynomial optimization problem amounts to solving afinite-dimensional convex semidefinite programming problem [47]. Numerical algorithmsfollowing this approach are available and, typically, guarantees of their convergence arerelated to the feasibility (or strict feasibility) of the LMI relaxations. It is, in general, achallenge to obtain exact algorithms for deciding whether the feasible set of a semidefiniteprogramming (SDP) problem min x ∈ R n n X i =1 c i x i s.t. A ( x ) (cid:23) Q − definable semidefinite program as in (2)(that is, we suppose that the coefficients of A ( x ) have rational entries), decide whether thefeasible set S = { x ∈ R n : A ( x ) (cid:23) } is empty or not, and compute exactly at least onefeasible point. We would like to emphasize the fact that we do not assume the existenceof an interior point in S . Quite the opposite, we are especially interested in degeneratecases for which the maximal rank achieved by the pencil A ( x ) in S is small.This work is a first step towards an exact approach to semidefinite programming. Inparticular, a natural perspective of this work is to design exact algorithms for decidingwhether the minimum in (2) is attained or not, and for computing such a minimum in theaffirmative case. While the number of iterations performed by the ellipsoid algorithm [27]to compute the approximation of a solution of (2) is polynomial in the number of vari-ables, once the accuracy is fixed, no analogous results for exact algorithms are available.Moreover, since the intrinsic complexity of the optimization problem (2) is related to itsalgebraic degree δ as computed in [48, 24], the paramount goal is to design algorithmswhose runtime is polynomial in δ . The algorithm of this paper shows experimentally suchan optimal behavior with respect to δ .We finally recall that solving LMIs is a basic subroutine of computer algorithms in systemscontrol and optimization, especially in linear systems robust control [9, 33], but also forthe analysis or synthesis of nonlinear dynamical systems [67], or in nonlinear optimalcontrol with polynomial data [37, 11]. We design a computer algebra algorithm for solving the feasibility problem of semidefiniteprogramming, that is Problem (1), in exact arithmetic. Let us clarify that we do not3laim that an exact algorithm can be competitive with a numerical algorithm in terms ofadmissible size of input problems: indeed, SDP solvers based on interior-point methods[7, 46] can nowadays handle inputs with a high number of variables that are out of reachfor our algorithms. Our contribution can be summarized as follows:1. we show that the geometry of spectrahedra understood as semi-algebraic sets withdeterminantal structure can be exploited to design dedicated computer algebra al-gorithms;2. we give explicit complexity and output-degree upper bounds for computer algebraalgorithms solving exactly the feasibility problem of semidefinite programming; ouralgorithm is probabilistic and works under assumptions on the input, which aregenerically satisfied;3. we provide results of practical experiments showing the gain in terms of computa-tional timings of our contribution with respect to the state of the art in computeralgebra;4. remarkably, our algorithm does not assume that the input spectrahedron S = { x ∈ R n : A ( x ) (cid:23) } is full-dimensional, and hence it can tackle also examples with emptyinterior.The main idea is to exploit the relation between the geometry of spectrahedra, and thatof the determinantal varieties associated to the input symmetric pencil A ( x ). Let usintroduce, for r = 0 , . . . , m −
1, the algebraic sets D r = { x ∈ C n : rank A ( x ) ≤ r } . These define a nested sequence D ⊂ D ⊂ · · · ⊂ D m − . The dimension of D r for genericlinear matrices A is known, and equals n − (cid:0) m − r +12 (cid:1) (see Lemma 4). The Euclideanboundary ∂ S of S is included in the real trace of the last algebraic set of the sequence: ∂ S ⊂ D m − ∩ R n . In particular, for x ∈ ∂ S , the matrix A ( x ) is singular and one couldask which elements of the real nested sequence D ∩ R n ⊂ · · · ⊂ D m − ∩ R n intersect ∂ S . Notation 1 If S = { x ∈ R n : A ( x ) (cid:23) } is not empty, we define the integer r ( A ) =min { rank A ( x ) : x ∈ S } . When S is not empty, r ( A ) equals the minimum integer r such that D r ∩ R n intersects S . We present our first main result, which states that S contains at least one of theconnected components of the real algebraic set D r ( A ) ∩ R n . We denote by S n +1 m ( Q ) = S m ( Q ) × · · · × S m ( Q ) the ( n + 1) − fold Cartesian product of S m ( Q ). Theorem 2 (Smallest rank on a spectrahedron)
Suppose that S = ∅ . Let C be aconnected component of D r ( A ) ∩ R n such that C ∩ S = ∅ . Then C ⊂ S and hence C ⊂ ( D r ( A ) \ D r ( A ) − ) ∩ R n .
4e give a proof of Theorem 2 in Section 2. From Theorem 2, we deduce the followingmutually exclusive conditions on the input symmetric linear pencil A : either S = ∅ , or S contains one connected component C of D r ( A ) ∩ R n . Consequently, an exact algorithmwhose output is one point in the component C ⊂ S ∩ D r ( A ) would be sufficient for ourgoal. Motivated by this fact, we design in Section 3.2 an exact algorithm computing onepoint in each connected component of D r ∩ R n , for r ∈ { , . . . , m − } .The strategy to compute sample points in D r ∩ R n is to build an algebraic set V r ⊂ C n + m ( m − r ) whose projection on the first n variables is contained in D r . This set is definedby the incidence bilinear relation A ( x ) Y ( y ) = 0 where Y ( y ) is a full-rank m × ( m − r )linear matrix whose columns generate the kernel of A ( x ) ( cf. Section 3.1). Unlike D r ,the incidence variety V r , up to genericity conditions on the matrices A , A , . . . , A n , turnsto be generically smooth and equidimensional. The next theorem presents a complexityresult for an exact algorithm solving Problem (1) under these genericity assumptions. Theorem 3 (Exact algorithm for LMI)
Suppose that for ≤ r ≤ m − , the inci-dence variety V r is smooth and equidimensional and that its defining polynomial systemgenerates a radical ideal. Suppose that D r has the expected dimension n − (cid:0) m − r +12 (cid:1) . Thereis a probabilistic algorithm that takes A as input and returns:1. either the empty list, if and only if S = ∅ , or2. a vector x ∗ such that A ( x ∗ ) = 0 , if and only if the linear system A ( x ) = 0 has asolution, or3. a rational parametrization q = ( q , . . . , q n +1 ) ⊂ Z [ t ] such that there exists t ∗ ∈ R with q n +1 ( t ∗ ) = 0 and: • A ( q ( t ∗ ) /q ( t ∗ ) , . . . , q n ( t ∗ ) /q ( t ∗ )) (cid:23) and • rank A ( q ( t ∗ ) /q ( t ∗ ) , . . . , q n ( t ∗ ) /q ( t ∗ )) = r ( A ) .The number of arithmetic operations performed are in O ˜ n (cid:18) m + m + nn (cid:19) X r ≤ m − (cid:18) mr (cid:19) ( n + ( m − r )( m + 3 r )) ! . If S = ∅ , the degree of q is in O (cid:18)(cid:0) m m + nn (cid:1) (cid:19) . An important aspect of our contribution can be read from the complexity and degreebounds in Theorem 3: indeed, remark that when m is fixed, both the output degreeand the complexity of the algorithm are polynomial functions of n . Viceversa, by theconstraint n ≥ (cid:0) m − r +12 (cid:1) (given by Lemma 4), one also easily deduces that when n is fixed,the complexity is polynomial in m .The algorithm of Theorem 3 is described in Section 3. Its probabilistic nature comes fromrandom changes of variables performed during the procedure, allowing to put the sets D r
5n generic position. We prove that for generic choices of parameters the output of thealgorithm is correct.A complexity analysis is performed in Section 5. Bounds in Theorem 3 are explicit ex-pressions involving m and n . These are computed by exploiting the multilinearity ofintermediate polynomial systems generated during the procedure, and are not sharp ingeneral. By experiments on randomly generated symmetric pencils, reported in Section 6,we observe that the output degree coincides with the algebraic degree of generic semidef-inite programs, that is with data given in [48, Table 2]: this evidences the optimality ofour approach. We did not succeed in proving exact formulas for such degrees. On input A , S can be defined by m polynomial inequalities in Q [ x , . . . , x n ] of degree ≤ m (see e.g. [51]). As far as we know, the state-of-the-art for designing algorithmsdeciding the emptiness of S consists only of algorithms for deciding the emptiness ofgeneral semi-algebraic sets; our contribution being the first attempt to exploit structuralproperties of the problem, e.g. through the smallest rank property (Theorem 2).A first algorithmic solution to deciding the emptiness of general semi-algebraic sets is givenby Cylindrical Algebraic Decomposition algorithm [12]; however its runtime is doublyexponential in the number n of variables. The first singly exponential algorithm is givenin [26], and has led to a series of works (see e.g. [54, 31]) culminating with algorithmsdesigned in [5] based on the so-called critical points method. This method is based on thegeneral idea which consists in computing minimizers/maximizers of a well-chosen functionreaching its extrema in each connected component of the set under study. Applying[5] to problem (1) requires m O ( n ) bit operations. Note that our technique for dealingwith sets V r is based on the idea underlying the critical point method. Also, in thearithmetic complexity model, our complexity estimates are more precise (the complexityconstant in the exponent is known) and better. This technique is also related to algorithmsbased on polar varieties for grabbing sample points in semi-algebraic sets; see for example[2, 3, 58, 59] and its application to polynomial optimization [25].To get a purely algebraic certificate of emptiness for S , one could use the classical ap-proach by Positivstellensatz [43, 52, 64]. As a snake biting its tail, this would lead to afamily, or hierarchy, of semidefinite programs [42]. Bounds for the degree of Positivstel-lensatz certificates are exponential in the number of variables and have been computedin [65] for Schmudgen’s, and in [49] for Putinar’s formulation. In the recent remarkableresult [44], a uniform 5 − fold exponential bound for the degree of the Hilbert 17th prob-lem is provided. In [41], an emptiness certificate dedicated to the spectrahedral case, bymeans of special quadratic modules associated to these sets, is obtained.All the above algorithms do not exploit the particular structure of spectrahedra under-stood as determinantal semi-algebraic sets. In [40], the authors showed that decidingemptiness of S can be done in time O ( nm ) + m O (min( n,m )) , that is in polynomial timein m (resp. linear time in n ) if n (resp. m ) is fixed. The main drawback of this algorithmis that it is based on general procedures for quantifier elimination, and hence it does notlead to efficient practical implementations. Note also that the complexity constant in the6xponent is still unknown.Also, in [28], a version of [57] dedicated to spectrahedra exploiting some of their structuralproperties, decides whether a linear matrix inequality A ( x ) (cid:23) S contains a point with coordinates in Q . Remark that such an algorithmis not sufficient to solve our problem, since, in some degenerate but interesting cases, S is not empty but does not contain rational points: in Section 6.2 we will illustrate theapplication of our algorithm to one of these examples.As suggested by the smallest rank property, determinantal structures play an importantrole in our algorithm. This structure has been recently exploited in [18] and [22] for the fastcomputation of Gr¨obner bases of zero-dimensional determinantal ideals and computingzero-dimensional critical loci of maps restricted to varieties in the generic case.Exploiting determinantal structures for determinantal situations remained challenging fora long time. In [34] we designed a dedicated algorithm for computing sample points inthe real solution set of the determinant of a square linear matrix. This has been extendedin [36] to real algebraic sets defined by rank constraints on a linear matrix. Observe thatthis problem looks similar to the ones we consider thanks to the smallest rank property.As in this paper, the traditional strategy consists in studying incidence varieties for whichsmoothness and regularity properties are proved under some genericity assumptions onthe input linear matrix.Hence, in the case of symmetric matrices, these results cannot be used anymore. Becauseof the structure of the matrix, the system defining the incidence variety involves toomany equations; some of them being redundant. Hence, these redundancies need to beeliminated to characterize critical points on incidence varieties in a convenient way. Inthe case of Hankel matrices, the special structure of their kernel provides an efficientway to do that. This case study is done in [35]. Yet, the problem of eliminating theseredundancies remained unsolved in the general symmetric case and this is what we do inSection 3 which is the starting point of the design of our dedicated algorithm. We refer to [6, 13, 30, 16] for the algebraic-geometric background of this paper. We recallbelow some basic definitions and notation. We denote by M p,q ( Q ) the space of p × q rational matrices, and GL n ( C ) the set of n × n non-singular matrices. The transpose of M ∈ M p,q ( Q ) is M T . The cardinality of a finite set T (resp. the number of entries of avector v ) are denoted by ♯T (resp. ♯v ).Let x = ( x , . . . , x n ). A vector f = ( f , . . . , f s ) ⊂ Q [ x ] is a polynomial system, h f i ⊂ Q [ x ]its ideal and Z ( h f i ) = { x ∈ C n : f i ( x ) = 0 , i = 1 , . . . , s } the associated algebraic set. Sets Z ( h f i ) define the collection of closed sets of the Zariski topology of C n . The intersectionof a Zariski closed and a Zariski open set is called a locally closed set. For M ∈ GL n ( C )and Z ⊂ C n , let M − Z = { x ∈ C n : M x ∈ Z} . With I ( S ) we denote the ideal ofpolynomials vanshing on S ⊂ C n .Let f = ( f , . . . , f s ) ⊂ Q [ x ]. Its Jacobian matrix is denoted by Df = ( ∂f i /∂x j ) i,j . Analgebraic set Z ⊂ C n is irreducible if Z = Z ∪ Z where Z , Z are algebraic sets,7mplies that either Z = Z or Z = Z . Any algebraic set is the finite union of irreduciblealgebraic sets, called its irreducible components. The codimension c of an irreduciblealgebraic set Z ⊂ C n is the maximum rank of Df on Z , where I ( Z ) = h f i . Its dimensionis n − c . If all the irreducible components of Z have the same dimension, we say that Z isequidimensional. The dimension of an algebraic set Z is the maximum of the dimensions ofits irreducible components, and it is denoted by dim Z . The degree of an equidimensionalalgebraic set Z of codimension c is the maximum cardinality of finite intersections Z ∩ L where L is a linear space of dimension c . The degree of an algebraic set is the sum of thedegrees of its equidimensional components.Let Z ⊂ C n be equidimensional of codimension c , and let I ( Z ) = h f , . . . , f s i . Thesingular locus of Z , denoted by sing ( Z ), is the algebraic set defined by f = ( f , . . . , f s )and by all c × c minors of Df . If sing ( Z ) = ∅ we say that Z is smooth, otherwisesingular. The points in sing ( Z ) are called singular, while points in reg ( Z ) = Z \ sing ( Z )are called regular. Let Z ⊂ C n be smooth and equidimensional of codimension c , andlet I ( Z ) = h f , . . . , f s i . Let g : C n → C m be an algebraic map. The set of criticalpoints of the restriction of g to Z is the algebraic set denote by crit ( g, Z ) and defined by f = ( f , . . . , f s ) and by all c + m minors of the Jacobian matrix D ( f, g ) . The points in g (crit ( g, Z )) are called critical values, while points in C m \ g (crit ( g, Z )) are called theregular values, of the restriction of g to Z . We prove Theorem 2, which relates the geometry of linear matrix inequalities to therank stratification of the defining symmetric pencil. We believe that the statement of thistheorem is known to the community of researchers working on real algebraic geometry andsemidefinite optimization; however, we did not find any explicit reference in the literature.
Proof of Theorem 2:
By assumption, the rank of A ( x ) on S is greater or equalthan r ( A ). We consider the vector function e = ( e , . . . , e m ) : R n −→ R m where e ( x ) ≤ . . . ≤ e m ( x ) are the ordered eigenvalues of A ( x ). Let C be a connected componentof D r ( A ) ∩ R n such that C ∩ S = ∅ , and let x ∈ C ∩ S . One has rank A ( x ) = r ( A ) and e ( x ) = . . . = e m − r ( A ) ( x ) = 0 < e m − r ( A )+1 ( x ) ≤ . . . ≤ e m ( x ) . Suppose ad absurdum thatthere exists y ∈ C such that y / ∈ S . In particular, one eigenvalue of A ( y ) is strictlynegative.Let g : [0 , → C be a continuous semi-algebraic map such that g (0) = x and g (1) = y .This map exists since C is a connected component of a real algebraic set. The image g ([0 , T = { t ∈ [0 ,
1] : g ( t ) ∈ S } = g − ( g ([0 , ∩ S ) . Since g is continuous, T ⊂ [0 ,
1] is closed. So it is a finite union of closed intervals.Since 0 ∈ T ( g (0) = x ∈ S ) there exists t ∈ [0 ,
1] and N ∈ N such that [0 , t ] ∈ T and for all p ≥ N , t + p / ∈ T . One gets that g ( t ) = ˜ x ∈ S and that for all p ≥ N , g ( t + p ) = ˜ x p / ∈ S . By definition, ˜ x, ˜ x p ∈ C ⊂ D r ( A ) ∩ R n for all p ≥ N , and since˜ x ∈ S , we get rank A (˜ x ) = r ( A ) and rank A (˜ x p ) ≤ r ( A ) for all p ≥ N . We also get that8ank A ( g ( t )) = r ( A ) for all t ∈ [0 , t ]. We finally have ˜ x p → ˜ x when p → + ∞ , since g iscontinuous. There exists a map ϕ : n p ∈ N : p ≥ N o → Z which assigns to p the index of eigenvalue-function among e , . . . , e m corresponding to themaximum strictly negative eigenvalue of A (˜ x p ), if it exists, otherwise it assigns 0. Remarkthat since rank A (˜ x p ) ≤ r ( A ) for all p , then ϕ ( p ) ≤ r ( A ) for all p . In other words, theeigenvalues of A (˜ x p ) satisfy e (˜ x p ) ≤ . . . ≤ e ϕ ( p ) (˜ x p ) < e ϕ ( p )+1 (˜ x p ) = . . . = e ϕ ( p )+ m − r ( A ) (˜ x p )0 ≤ e ϕ ( p )+ m − r ( A )+1 (˜ x p ) ≤ . . . ≤ e m (˜ x p ) , for p ≥ N . Since the sequence { ϕ ( p ) } p ≥ N is bounded, up to taking a subsequence, itadmits at least a limit point by the Bolzano-Weierstrass Theorem [4, Th. 3.4.8], this pointis an integer, and j ϕ ( j ) is constant for large j . Suppose that there exists a limit point ℓ >
0, and let { p j } j ∈ N such that ϕ ( p j ) → ℓ and that for j ≥ N ′ , j ϕ ( p j ) is constant.Thus, 0 = e ℓ +1 (˜ x p j ) = . . . = e ℓ + m − r ( A ) (˜ x p j ) for all j ≥ N ′ . Since ˜ x p j → ˜ x , and since e , . . . , e m are continuous functions, we obtain that ℓ = 0 is the unique limit point of ϕ ,hence ϕ converges to 0. Hence ϕ ≡ p . This contradicts the fact that ˜ x p / ∈ S for large p .We conclude that the set C \ S is empty, that is C ⊂ S . By the minimality of the integer r ( A ) in { rank A ( x ) : x ∈ S } , one deduces that C ⊂ ( D r ( A ) \ D r ( A ) − ) ∩ R n . (cid:3) Our algorithm is called
SolveLMI , and it is presented in Section 3.3. Before, we describe inSection 3.2 its main subroutine
LowRankSym , which is of recursive nature and computesone point per connected component of the real algebraic set D r ∩ R n . We start, in thenext section, with some preliminaries. Expected dimension of low rank loci
We first recall a known fact about the dimension of D r , when A is a generic symmetricpencil. Lemma 4
There exists a non-empty Zariski open subset A ⊂ S n +1 m ( C ) such that, if A ∈ A ∩ S n +1 m ( Q ) , for all r = 0 , . . . , m − , the set D r is either empty or it has dimension n − (cid:0) m − r +12 (cid:1) . Proof :
The proof is classical and can be found e.g. in [1, Prop. 3.1]. (cid:3) ncidence varieties Let A ( x ) be a symmetric m × m linear matrix, and let 0 ≤ r ≤ m −
1. Let y =( y i,j ) ≤ i ≤ m, ≤ j ≤ m − r be unknowns. Below, we build an algebraic set whose projection onthe x − space is contained in D r . Let Y ( y ) = y , · · · y ,m − r ... ... y m, · · · y m,m − r , and let ι = { i , . . . , i m − r } ⊂ { , . . . , m } , with ♯ι = m − r . We denote by Y ι the ( m − r ) × ( m − r ) sub-matrix of Y ( y ) obtained by isolating the rows indexed by ι . There are (cid:0) mr (cid:1) such matrices. We define the set V r ( A, ι ) = { ( x, y ) ∈ C n × C m ( m − r ) : A ( x ) Y ( y ) = 0 , Y ι − I m − r = 0 } . We denote by f ( A, ι ), or simply by f , when there is no ambiguity on ι , the polynomialsystem defining V r ( A, ι ). We often consider linear changes of variables x : for M ∈ GL n ( C ), f ( A ◦ M, ι ) denotes the entries of A ( M x ) Y ( y ) and Y ι − I m − r , and by V r ( A ◦ M, ι ) its zeroset. We also denote by U ι ∈ M m − r,m ( Q ) the full rank matrix whose entries are in { , } ,and such that U ι Y ( y ) = Y ι . By simplicity we call U ι the boolean matrix with multi-index ι .By definition, the projection of V r ( A, ι ) on the first n variables is contained in D r . Weremark the similarity between the relation A ( x ) Y ( y ) = 0 and the so-called complemen-tarity condition for a couple of primal-dual semidefinite programs, see for example [48,Th. 3]. The difference in our model is that the special size of Y ( y ) and the affine constraint Y ι = I m − r force a rank condition on Y ( y ) and hence on A ( x ). Eliminating redundancies
The system f ( A, ι ) contains redundancies induced by polynomial relations between itsgenerators. These relations can be explicitly eliminated in order to obtain a minimalpolynomial system defining V r . Lemma 5
Let M ∈ GL n ( C ) . Let ι ⊂ { , . . . , m } , with ♯ι = m − r . Let A ∈ S n +1 m ( Q ) ,and f ∈ Q [ x, y ] m ( m − r )+( m − r ) be the polynomial system defined in Section 3.1. Then wecan explicitly construct a subsystem f red ⊂ f of length m ( m − r ) + (cid:0) m − r +12 (cid:1) such that h f red i = h f i . Proof :
In order to simplify notations and without loss of generality we suppose M = I n and ι = { , . . . , m − r } . We substitute Y ι = I m − r in A ( x ) Y ( y ), and we denote by g i,j the( i, j ) − th entry of the resulting matrix. We denote by f red the following system: f red = ( g i,j for i ≥ j, Y ι − I m − r ) .
10e claim that, for 1 ≤ i = j ≤ m − r , g i,j ≡ g j,i mod h g k,ℓ , k > m − r i , which impliesthat f red verifies the statement. Let a i,j denote the ( i, j ) − th entry of A ( x ). Let i < j andwrite g i,j = a i,j + m X ℓ = m − r +1 a i,ℓ y ℓ,j and g j,i = a j,i + m X ℓ = m − r +1 a j,ℓ y ℓ,i . We deduce that g i,j − g j,i = P mℓ = m − r +1 a i,ℓ y ℓ,j − a j,ℓ y ℓ,i since A is symmetric. Also, modulothe ideal h g k,ℓ , k > m − r i , and for ℓ ≥ m − r + 1, one can explicit a i,ℓ and a j,ℓ , by usingpolynomial relations g ℓ,i = 0 and g ℓ,j = 0, as follows: g i,j − g j,i ≡ m X ℓ = m − r +1 − m X t = m − r +1 a ℓ,t y t,i y ℓ,j + m X t = m − r +1 a ℓ,t y t,j y ℓ,i ! ≡≡ m X ℓ,t = m − r +1 a ℓ,t ( − y t,i y ℓ,j + y t,j y ℓ,i ) ≡ h g k,ℓ , k > m − r i . The previous congruence concludes the proof. (cid:3)
We prove below in Proposition 7 and in Corollary 11 that, up to genericity assumptions,the ideal h f i = h f red i is radical and that ♯f red matches the codimension of V r . In the nextexample, we explicitly write down the redundancies shown in Lemma 5 for a simple case. Example 6
We consider a × symmetric matrix of unknowns, and the kernel corre-sponding to the configuration { , } ⊂ { , , } . Let f f f f f f = x x x x x x x x x y y . We consider the classes of polynomials f , f modulo h f , f i , deducing the followinglinear relation: f − f = y x − y x ≡ y x y − y x y = 0 . Lagrange systems
Let f ( A, ι ) be the polynomial system defining V r ( A, ι ). We set c = m ( m − r ) + (cid:0) m − r +12 (cid:1) and e = (cid:0) m − r (cid:1) , so that V r is defined by c = ♯f red polynomial equations, and e = ♯f − c is the number of redundancies eliminated by Lemma 5. We define, for M ∈ GL n ( C ), thepolynomial system ℓ = ℓ ( A ◦ M, ι ), given by the coordinates of the map ℓ : C n × C m ( m − r ) × C c + e −→ C n + m ( m − r )+ c + e ( x, y, z ) (cid:0) f ( A ◦ M, ι ) , z T Df ( A ◦ M, ι ) − ( e T , (cid:1) , where e ∈ Q n is the first column of the identity matrix I n . We also define Z ( A ◦ M, ι ) = Z ( ℓ ( A ◦ M, ι )). When V r ( A ◦ M, ι ) is smooth and equidimensional, Z ( A ◦ M, ι ) encodesthe critical points of the restriction of Π ( x, y ) = x to V r ( A ◦ M, ι ).11 utput representation
The output of the algorithm is a rational parametrization ( q , . . . , q n +1 ) ⊂ Z [ t ] such thatthe finite set defined in (1) contains at least one point on the spectrahedron S . We describe the main subroutine
LowRankSym , which is a variant for symmetric pencils ofthe algorithms in [34, 35, 36]. Its output is a finite set meeting each connected componentof D r ∩ R n . It takes advantage of the particular properties of the incidence varieties overa symmetric low rank locus, as highlighted by Lemma 5. Properties
We define the following properties for a given A ∈ S n +1 m ( Q ): Property P . We say that A satisfies P if, for all ι ⊂ { , . . . , m } , with ♯ι = m − r , theincidence variety V r ( A, ι ) is either empty or smooth and equidimensional.
Property P . We say that A satisfies P if, for all r , D r has the expected dimension n − (cid:0) m − r +12 (cid:1) . Property P holds generically in S n +1 m ( Q ), as shown by Lemma 4.We also define the following property for a polynomial system f ⊂ Q [ x ] and a Zariskiopen set O ⊂ C n : Property Q . Suppose that f ⊂ Q [ x ] generates a radical ideal and that it defines analgebraic set of codimension c , and let O ⊂ C n be a Zariski open set. We say that f satisfies Q in O , if the rank of Df is c in Z ( h f i ) ∩ O . Formal description of
LowRankSym
The formal description of our algorithm is given next. It consists of a main algo-rithm which checks the genericity hypotheses, and of a recursive sub-algorithm called
LowRankSymRec . 12 owRankSym
Input: A ∈ S n +1 m ( Q ), encoded by the m ( m + 1)( n + 1) / A , A , . . . , A n , and an integer 1 ≤ r ≤ m − Output:
Either the empty list [ ], if and only if D r ∩ R n = ∅ , or an error message stating that the genericity assump-tions are not satisfied, or a rational parametrization q =( q , . . . , q n +1 ) ⊂ Z [ t ], such that for every connected com-ponent C ⊂ D r ∩ R n , with C ∩ D r − = ∅ , there exists t ∗ ∈ Z ( q n +1 ) ∩ R with ( q ( t ∗ ) /q ( t ∗ ) , . . . , q n ( t ∗ ) /q ( t ∗ )) ∈ C . Procedure
LowRankSym ( A, r )1. if n < (cid:0) m − r +12 (cid:1) then • if dim D r = − ι ⊂ { , . . . , m } with ♯ι = m − r do • if IsReg (( A, ι )) = false then return(“the input isnot generic”);3. return(
LowRankSymRec ( A, r )).
Procedure
LowRankSymRec ( A, r )1. choose M ∈ GL n ( Q );2. q ← [ ]; for ι ⊂ { , . . . , m } with ♯ι = m − r do • q ι ← Image ( Project ( RatPar ( ℓ ( A ◦ M, ι ))) , M − ); • q ← Union ( q, q ι );3. choose t ∈ Q ; A ← ( A + tA , A , . . . , A n );4. q ′ ← Lift ( LowRankSymRec ( A, r ) , t );5. return( Union ( q, q ′ )).The routines appearing in the previous algorithm are described next: • IsReg . Input: A ∈ S n +1 m ( Q ) , ι ⊂ { , . . . , m } ; Output: true if V r ( A, ι ) is empty orsmooth and equidimensional of codimension m ( m − r ) + (cid:0) m − r +12 (cid:1) , false otherwise. • Project . Input:
A rational parametrization of ℓ ( A ◦ M, ι ) ⊂ Q [ x, y, z ]; Output: anerror message if the projection of Z ( A ◦ M, ι ) ∩ { ( x, y, z ) : rank A ( M x ) = r } on the x − space is not finite; otherwise a rational parametrization of this projection.13 RatPar . Input: ℓ ( A ◦ M, ι ) ⊂ Q [ x, y, z ]; Output : a rational parametrization of ℓ ( A ◦ M, ι ). • Image . Input: a rational parametrization of a set
Z ⊂ Q [ x , . . . , x N ] and a matrix M ∈ GL N ( Q ); Output: a rational parametrization of M − Z = { x ∈ C N : M x ∈Z} . • Union . Input: two rational parametrizations encoding Z , Z ⊂ Q [ x , . . . , x N ]; Out-put: a rational parametrization of Z ∪ Z . • Lift . Input: a rational parametrization of a set
Z ⊂ Q [ x , . . . , x N ], and t ∈ C ; Output: a rational parametrization of { ( t, x ) : x ∈ Z} . The input of
SolveLMI is a linear matrix A ∈ S n +1 m ( Q ), and the algorithm makes use ofalgorithm LowRankSym described previously, as a subroutine. The formal description isthe following.
SolveLMI
Input: A ∈ S n +1 m ( Q ), encoded by the m ( m + 1)( n + 1) / A , A , . . . , A n ; Output:
Either the empty list [ ], if and only if { x ∈ R n : A ( x ) (cid:23) } is empty; or an error message stating that thegenericity assumptions are not satisfied, or, otherwise, eithera vector x ∗ = ( x ∗ , . . . , x ∗ n ) such that A ( x ∗ ) = 0, or a rationalparametrization q = ( q , . . . , q n +1 ) ⊂ Z [ t ], such that there ex-ists t ∗ ∈ Z ( q n +1 ) ∩ R with A ( q ( t ∗ ) /q ( t ∗ ) , . . . , q n ( t ∗ ) /q ( t ∗ )) (cid:23) Procedure
SolveLMI ( A )1. x ∗ ← SolveLinear ( A ); if x ∗ = [ ] then return( x ∗ );2. for r from 1 to m − • q ← LowRankSym ( A, r ); • if q = “the input is not generic” then return( q ); • if q = [ ] then b ← CheckLMI ( A, q ); • if b = true then return( q );3. return([ ], “the spectrahedron is empty”).The different subroutines of SolveLMI are described next:14
SolveLinear . Input : A ∈ S n +1 m ( Q ); Output the empty list if A ( x ) = 0 has no solution,otherwise x ∗ such that A ( x ∗ ) = 0; • CheckLMI . Input : A ∈ S n +1 m ( Q ) and a rational parametrization q ⊂ Z [ t ]; Output : true if A ( q ( t ∗ ) /q ( t ∗ ) , . . . , q n ( t ∗ ) /q ( t ∗ )) (cid:23) t ∗ ∈ Z ( q n +1 ) ∩ R , false otherwise. We prove in Theorem 10 that
SolveLMI returns a correct output if genericity propertieson input data and on random parameters chosen during its execution are satisfied; theproof relies on some preliminary results that are described before. The proofs of theseintermediate results share some techniques with [34, 35, 36] and are given in Section 4.
Intermediate results
The first result is a regularity theorem for the incidence varieties V r ( A, ι ). We focus onproperty P for the input matrix A ( cf. Section 3.2).
Proposition 7
Let m, n, r ∈ N , with ≤ r ≤ m − .1. There exists a non-empty Zariski-open set A ⊂ S n +1 m ( C ) such that if A ∈ A ∩ S n +1 m ( Q ) , then A satisfies P ;2. if A satisfies P , there exists a non-empty Zariski open set T ⊂ C such that if t ∈ T ∩ Q , then ( A + tA , A , . . . , A n ) satisfies P . Next, we prove that the projection of Z ( A ◦ M, ι ) ∩ { ( x, y, z ) : rank A ( M x ) = r } over the x − space is finite and that this set meets the critical points of the restriction of the mapΠ : ( x, y ) → x to V r ( A, ι ). Proposition 8
Let A ∈ S n +1 m ( Q ) satisfy P . Then there exists a non-empty Zariski openset M ⊂ GL n ( C ) such that, if M ∈ M ∩ M n,n ( Q ) , for all ι ⊂ { , . . . , m } , with ♯ι = m − r ,the following holds:1. The system ℓ ( A ◦ M, ι ) satisfies Q in { ( x, y, z ) : rank A ( M x ) = r } ;2. the projection of Z ( A ◦ M, ι ) ∩{ ( x, y, z ) : rank A ( M x ) = r } on the x − space is emptyor finite;3. the projection of Z ( A ◦ M, ι ) ∩ { ( x, y, z ) : rank A ( M x ) = r } on ( x, y ) containsthe set of critical points of the restriction of Π : ( x, y ) → x to V r ( A ◦ M, ι ) ∩{ ( x, y ) : rank A ( M x ) = r } . x , closure properties of theprojection maps π i ( x ) = ( x , . . . , x i ) restricted to D r . Also, in order to compute samplepoints on the connected components of D r ∩ R n not meeting D r − , the next propositionshows that to do that it is sufficient to compute critical points on the incidence variety V r . Proposition 9
Let A ∈ S n +1 m ( Q ) satisfy P , and let d = dim D r . There exists a non-empty Zariski open set M ⊂ GL n ( C ) such that if M ∈ M ∩ M n,n ( Q ) , for any connectedcomponent C ⊂ D r ∩ R n , the following holds:1. for i = 1 , . . . , d , π i ( M − C ) is closed; further, for t ∈ R lying on the boundary of π ( M − C ) , then π − ( t ) ∩ M − C is finite;2. let t lie on the boundary of π ( M − C ) : for x ∈ π − ( t ) ∩ M − C , such that rank A ( M x ) = r , there exists ι ⊂ { , . . . , m } , with ♯ι = m − r , and ( x, y ) ∈ V r ( A ◦ M, ι ) , such that Π ( x, y ) = t . Theorem of correctness
Let A ∈ S n +1 m,m ( Q ) be the input of SolveLMI . We say that hypothesis H holds if: • A and all parameters generated by SolveLMI belong to the Zariski open sets definedin Proposition 7, 8 and 9, for all recursive steps of
LowRankSym ; • A satisfies Property P .We can now state the correctness theorem for SolveLMI . Theorem 10 (Correctness of
SolveLMI ) Suppose that H holds. Let S = { x ∈ R n : A ( x ) (cid:23) } be the spectrahedron associated to A . Then two alternatives hold:1. S = ∅ : hence the output of SolveLMI with input A is the empty list;2. S = ∅ : hence the output of SolveLMI with input A is either a vector x ∗ such that A ( x ∗ ) = 0 , if it exists; or a rational parametrization q = ( q , . . . , q n +1 ) ⊂ Z [ t ] suchthat there exists t ∗ ∈ Z ( q n +1 ) ∩ R with: • A ( q ( t ∗ ) /q ( t ∗ ) , . . . , q n ( t ∗ ) /q ( t ∗ )) (cid:23) and • rank A ( q ( t ∗ ) /q ( t ∗ ) , . . . , q n ( t ∗ ) /q ( t ∗ )) = r ( A ) (cf. Notation 1). Proof :
Suppose A ( x ) = 0 has a solution. Hence, at Step 1 of SolveLMI , SolveLinear withinput A returns a vector x ∗ such that A ( x ∗ ) = 0. We deduce that x ∗ ∈ S = ∅ and thatthe rank of A attains its minimum on S at x ∗ . We deduce that, if A ( x ) = 0 has at leastone solution, the algorithm returns a correct output.Suppose now that either S is empty, or r ( A ) ≥
1. We claim that
LowRankSym is correct,in the following sense: with input (
A, r ), with A satisfying P , the output is a rational16arametrization whose solutions meet each connected component C of D r ∩ R n such that C ∩ D r − = ∅ .We assume for the moment this claim, and consider two possible alternatives:1. S = ∅ . Consequently, CheckLMI outputs false at each iteration of Step 2 in
SolveLMI . Thus the output of
SolveLMI is the empty list, and correctness follows.2. S = ∅ . Denote by C a connected component of D r ( A ) ∩ R n such that C ∩ S = ∅ .By Theorem 2, we deduce that C ⊂ S , and that C ∩ D r ( A ) − = ∅ . Let q be theoutput of LowRankSym at Step 2 of
SolveLMI . By our claim, q defines a finite setwhose solutions meet C , hence S . Consequently, CheckLMI returns true at Step 2,and hence the algorithm stops returning the correct output q .We end the proof by showing that LowRankSym is correct. This is straightforwardlyimplied by the correctness of the recursive subroutine
LowRankSymRec , which is provedbelow by using induction on the number of variables n .For n < (cid:0) m − r +12 (cid:1) , since H holds, then A satisfies P r . Hence D r is empty, and LowRankSym returns the correct answer [ ] (the empty list).Let n ≥ (cid:0) m − r +12 (cid:1) , and let ( A, r ) be the input. The induction hypothesis implies that forany ˜ A ∈ S nm ( Q ) satisfying P , then LowRankSymRec with input ( ˜
A, r ) returns a rationalparametrization of a finite set meeting each connected component ˜
C ⊂ ˜ D r ∩ R n − such that˜ C ∩ ˜ D r − = ∅ , with ˜ D r = { x ∈ R n − : rank ˜ A ( x ) ≤ r } . Let C ⊂ D r ∩ R n be a connectedcomponent with C ∩ D r − = ∅ , and let M be the matrix chosen at Step 1. Hence, since H holds, by Proposition 9 the set π ( M − C ) is closed in R . There are two possible scenarios. First case.
Suppose that π ( M − C ) = R , let t ∈ Q be the rational number chosen atStep 3 of LowRankSymRec , and let ˜ A = ( A + tA , A , . . . , A n ) ∈ S nm ( Q ). We deducethat π − ( t ) ∩ M − C 6 = ∅ is the union of some connected components of the algebraicset ˜ D r = { x ∈ R n − : rank ˜ A ( x ) ≤ r } not meeting ˜ D r − . Also, since A satisfies P ,so does A ◦ M ; by Proposition 7, then ˜ A satisfies P . By the induction assumption, LowRankSymRec with input ( ˜
A, r ) returns at least one point in each connected component˜
C ⊂ ˜ D r ∩ R n − not meeting ˜ D r − , hence one point in C by applying the subroutine Lift atStep 4. Correctness follows.
Second case.
Otherwise, π ( M − C ) = R and, since it is a closed set, its boundary isnon-empty. Let t belong to the boundary of π ( M − C ), and suppose w.l.o.g. that π ( M − C ) ⊂ [ t, + ∞ ). Hence t is the minimum of the restriction of the map π to M − C . By Proposition 9, the set π − ( t ) ∩ M − C is non-empty and finite, and for all x ∈ π − ( t ) ∩ M − C , rank A ( M x ) = r (indeed, for x ∈ M − C , then M x ∈ C and hence
M x / ∈ D r − ∩ R n ). Fix x ∈ π − ( t ) ∩ M − C . By Proposition 9, there exists ι and y ∈ C m ( m − r ) such that ( x, y ) ∈ V r ( A ◦ M, ι ). Also, by Proposition 7, the set V r ( A ◦ M, ι ) issmooth and equidimensional. One deduces that ( x, y ) is a critical point of the restrictionof Π : ( x, y ) → x to V r ( A ◦ M, ι ) and that there exists z such that ( x, y, z ) ∈ Z ( A ◦ M, ι ).Hence, at Step 2, the routine
LowRankSymRec outputs a rational parametrization q ι ,among whose solutions the vector x lies. (cid:3) Proof of intermediate results
We prove Assertion 1 and 2 separately.
Proof of Assertion 1:
Suppose w.l.o.g. that M = I n . For ι ⊂ { , . . . , m } , with ♯ι = m − r , let f red ⊂ Q [ x, y ] be the system defined in Lemma 5. We prove that thereexists a non-empty Zariski open set A ι ⊂ S n +1 m ( C ) such that, if A ∈ A ι ∩ S n +1 m ( Q ), f red generates a radical ideal and Z ( f red ) is empty or equidimensional, of codimension ♯f red = m ( m − r ) + (cid:0) m − r +12 (cid:1) . We deduce that, for A ∈ A ι , A satisfies P , and we concludeby defining A = ∩ ι A ι (non-empty and Zariski open).Suppose w.l.o.g. that ι = { , . . . , m − r } . We consider the map ϕ : C n + m ( m − r ) × S n +1 m ( C ) −→ C m ( m − r )+ ( m − r +12 )( x, y, A ) f red and, for a fixed A ∈ S n +1 m ( C ), its section map ϕ A : C n + m ( m − r ) → C m ( m − r )+ ( m − r +12 ) definedby ϕ A ( x, y ) = ϕ ( x, y, A ). Remark that, for any A , Z ( ϕ A ) = V r ( A, ι ).Suppose ϕ − (0) = ∅ : this implies that, for all A ∈ S n +1 m ( C ), Z ( f red ) = V r ( A, ι ) = ∅ , thatis A satisfies P for A ∈ A ι = S n +1 m ( C ).If ϕ − (0) = ∅ , we prove below that 0 is a regular value of ϕ . We conclude that by Thom’sWeak Transversality Theorem [60, Section 4.2] there exists a non-empty and Zariski openset A ι ⊂ S n +1 m ( C ) such that if A ∈ A ι ∩ S n +1 m ( Q ), 0 is a regular value of ϕ A . Hence,by applying the Jacobian criterion ( cf. [16, Theorem 16.19]) to the polynomial system f red , we deduce that for A ∈ A ι ∩ S n +1 m ( Q ), V r ( A, ι ) is smooth and equidimensional ofcodimension ♯f red .Let Dϕ be the Jacobian matrix of ϕ : it contains the derivatives of polynomials in f red with respect to variables x, y, A . We denote by a ℓ,i,j the variable encoding the ( i, j ) − thentry of the matrix A ℓ , ℓ = 0 , . . . , n . We isolate the columns of Dϕ corresponding to: • the derivatives with respect to variables { a ,i,j : i ≤ m − r or j ≤ m − r } ; • the derivatives with respect to variables y i,j such that i ∈ ι .Let ( x, y, A ) ∈ ϕ − (0), and consider the evaluation of Dϕ at ( x, y, A ). The above columnscontain the following non-singular blocks: • the derivatives w.r.t. { a ,i,j : i ≤ m − r or j ≤ m − r } of the entries of A ( x ) Y ( y )after the substitution Y ι ← I m − r , that is I ( m − r )( m + r +1) / ; • the derivatives w.r.t. { y i,j : i ∈ ι } of polynomials in Y ι − I m − r , that is I ( m − r ) .Hence, the above columns define a maximal non-singular sub-matrix of Dϕ at ( x, y, A ),of size m ( m − r ) + (cid:0) m − r +12 (cid:1) = ♯f red (indeed, remark that the entries of Y ι − I m − r do not18epend on variables a ,i,j ). Since ( x, y, A ) ∈ ϕ − (0) is arbitrary, we deduce that 0 is aregular value of ϕ , and we conclude. (cid:3) Proof of Assertion 2:
Fix ι ⊂ { , . . . , m } with ♯ι = m − r . Since A satisfies P , V r ( A, ι ) is either empty or smooth and equidimensional of codimension m ( m − r )+ (cid:0) m − r +12 (cid:1) .Suppose first that V r = ∅ . Hence for all t ∈ C , V r ∩ { x − t = 0 } = ∅ , and we concludeby defining T = C . Otherwise, consider the restriction of the projection map Π :( x, y ) → x to V r ( A, ι ). By Sard’s Lemma [60, Section 4.2], the set of critical values ofthe restriction of π to V r ( A, ι ) is included in a finite subset
H ⊂ C . We deduce that, for t ∈ T = C \ H , then ( A + tA , A , . . . , A n ) satisfies P . (cid:3) Corollary 11
Let A ⊂ S n +1 m ( Q ) be as in Proposition 7, and let A ∈ A . Then for every ι ⊂ { , . . . , m } with ♯ι = m − r , the ideal h f red i = h f i is radical, and V r ( A, ι ) is a completeintersection of codimension ♯f red . Proof :
We recall from the proof of Assertion 1 of Theorem 7 that, for A ∈ A , therank of the Jacobian matrix of f red is ♯f red = m ( m − r ) + (cid:0) m − r +12 (cid:1) at every point of V r ( A, ι ). By the Jacobian criterion [16, Theorem 16.19], the ideal h f red i is radical andthe algebraic set Z ( f red ) = V r ( A, ι ) is smooth and equidimensional of codimension ♯f red .Hence I ( V r ( A, ι )) = h f red i can be generated by a number of polynomials equal to thecodimension of V r ( A, ι ), and we conclude. (cid:3)
Local equations of V r ( A, ι )Suppose A is a (not necessarily symmetric) linear matrix. Let us give a local descriptionof the algebraic sets D r and V r ( cf. also [34, Section 5]). Consider first the locallyclosed set D r \ D r − = { x ∈ C n : rank A ( x ) = r } . This is given by the union of sets D r ∩ { x ∈ C n : det N ( x ) = 0 } where N runs over all r × r sub-matrices of A ( x ). Fix ι ⊂ { , . . . , m } with ♯ι = m − r . Let N be the upper left r × r sub-matrix of A ( x ), andconsider the corresponding block division of A : A = (cid:18) N QP T R (cid:19) (3)with P, Q ∈ M r,m − r ( Q ) and R ∈ M m − r,m − r ( Q ). Let Q [ x, y ] det N be the local ring obtainedby localizing Q [ x, y ] at h det N i . Let Y (1) (resp. Y (2) ) be the matrix obtained by isolatingthe first r (resp. the last m − r ) rows of Y ( y ). Hence, the local equations of V r in { ( x, y ) : det N ( x ) = 0 } are given by: Y (1) + N − QY (2) = 0 , Σ( N ) Y (2) = 0 , Y ι − I m − r = 0 , (4)where Σ( N ) = R − P T N − Q is the Schur complement of N in A . This follows from thefollowing straightforward equivalence holding in the local ring Q [ x, y ] det N ( cf. also [34,Lemma 13]): A ( x ) Y ( y ) = 0 iff (cid:18) I r − P T I m − r (cid:19) (cid:18) N − I m − r (cid:19) (cid:18) N QP T R (cid:19) Y ( y ) = 0 . ntermediate lemma Let w ∈ C n be a non-zero vector, and consider the projection map induced by w :Π w : ( x , . . . , x n , y ) w x + · · · + w n x n . For A ∈ A (given by Proposition 7), for all ι as above, the critical points of the restrictionof Π w to V r ( A, ι ) are encoded by the polynomial system ( f, g, h ) where f = f ( A, ι ) , ( g, h ) = z T (cid:18) DfD Π w (cid:19) = z T (cid:18) D x f D y fw T (cid:19) , (5)and z = ( z , . . . , z c + e ,
1) is a vector of Lagrange multipliers. Indeed, equations induced by( g, h ) imply that the vector w is normal to the tangent space of V r at ( x, y ).We prove an intermediate lemma towards Proposition 8. Lemma 12
Let A ∈ S n +1 m ( Q ) satisfy P . Then there exists a non-empty Zariski open set W ⊂ C n such that, if w ∈ W ∩ Q n , for all ι ⊂ { , . . . , m } , with ♯ι = m − r , the followingholds:1. the system ( f, g, h ) in (5) satisfies Q in { ( x, y, z ) : rank A ( x ) = r } ;2. the projection of Z ( f, g, h ) ∩ { ( x, y, z ) : rank A ( x ) = r } on the x − space is empty orfinite;3. the projection of Z ( f, g, h ) ∩ { ( x, y, z ) : rank A ( x ) = r } on ( x, y ) contains the set ofcritical points of the restriction of Π w to V r ∩ { ( x, y ) : rank A ( x ) = r } . Proof of Assertion 1:
The strategy relies on applying Thom Weak TransversalityTheorem and the Jacobian criterion, as in the proof of Proposition 7. The similar passageswill be only sketched.We claim (and prove below) that, given a r × r sub-matrix N of A , there exists anon-empty Zariski open set W N ⊂ C n such that, for w ∈ W N , ( f, g, h ) satisfies Q in { ( x, y, z ) : det N = 0 } . We deduce Assertion 1 by defining W = T N W N , where N runsover all r × r sub-matrices of A ( x ).Let U ι ∈ C ( m − r ) × m be the boolean matrix such that U ι Y ( y ) = Y ι , and let U ι = ( U (1) ι | U (2) ι )be the subdivision with U (1) ι ∈ C ( m − r ) × r and U (2) ι ∈ C ( m − r ) × ( m − r ) . The third equation in(4) equals U ι Y ( y ) − I m − r = 0 . From (4) we deduce the equality I m − r = U (1) ι Y (1) + U (2) ι Y (2) = ( U (2) ι − U (1) ι N − Q ) Y (2) and hence that both Y (2) and U (2) ι − U (1) ι N − P are non-singular in Q [ x, y ] det N . We deducethat the above local equations of V r are equivalent to Y (1) + N − QY (2) = 0 , Σ( N ) = 0 , Y (2) − ( U (2) ι − U (1) ι N − P ) − = 0in Q [ x, y ] det N . We collect the above equations in a system ˜ f , of length c + e . Hence, theJacobian matrix of ˜ f is D ˜ f = D x [Σ( N )] i,j ( m − r ) × m ( m − r ) ⋆ I r ( m − r ) ⋆ I ( m − r ) . A satisfies P , the rank of D ˜ f is constant and equal to c if evaluated at ( x, y ) ∈ Z ( ˜ f ) = V r ( A, ι ) ∩ { ( x, y ) : det N = 0 } . Similarly to (5), we define(˜ g, ˜ h ) = z T (cid:18) D ˜ fw T (cid:19) with z = ( z , . . . , z c + e , h i = 0 implies that z ( m − r ) + i = 0,for i = 1 , . . . , m ( m − r ), and hence they can be eliminated, together with the variables z ( m − r ) + i , i = 1 , . . . , m ( m − r ). Hence, one can consider the equivalent equations ( ˜ f , ˜ g, ˜ h )where the last m ( m − r ) variables z do not appear in ˜ g .Let us define the map ϕ : C n + c + e + m ( m − r ) × C n → C n + c + e + m ( m − r ) , ϕ ( x, y, z, w ) = ( ˜ f , ˜ g, ˜ h ),and for w ∈ C n , its section map ϕ w : ( x, y, z ) ϕ ( x, y, z, w ).If ϕ − (0) = ∅ , we define W N = C n and conclude. Otherwise, let ( x, y, z, w ) ∈ ϕ − (0). Re-mark that polynomials in ˜ f just depend on ( x, y ), hence their contribution in Dϕ ( x, y, z, w )is the block D ˜ f , whose rank is c , since ( x, y ) ∈ V r . Hence, we deduce that the row-rank of Dϕ at ( x, y, z, w ) is at most n + c + m ( m − r ). Further, by isolating thecolumns corresponding to the derivatives with respect to x, y , to w , . . . , w n , and to z ( m − r ) + i , i = 1 , . . . , m ( m − r ), one obtains a ( n + c + e + m ( m − r )) × (2 n + 2 m ( m − r ))sub-matrix of Dϕ of rank n + c + m ( m − r ). (cid:3) Proof of Assertion 2:
From Assertion 1 we deduce that the locally closed set E = Z ( f, g, h ) ∩{ ( x, y, z ) : rank A ( x ) = r } is empty or e − equidimensional. If it is empty, we aredone. Suppose that it is e − equidimensional. Consider the projection map π x ( x, y, z ) = x ,and its restriction to E . Let x ∗ ∈ π x ( E ). Then rank A ( x ∗ ) = r and there exists a unique y ∈ C m ( m − r ) such that f ( x ∗ , y ) = 0. Hence the fiber π − x ( x ∗ ) is isomorphic to the linearspace { ( z , . . . , z c + e ) : ( z , . . . , z c + e ) Df = ( w T , } . Since the rank of Df is c , one deducesthat π − x ( x ∗ ) is a linear space of dimension e , and by the Theorem on the Dimension ofFibers [63, Sect. 6.3, Theorem 7] we deduce that π x ( E ) has dimension 0. (cid:3) Proof of Assertion 3:
Since V r ∩ { ( x, y ) : rank A ( x ) = r } is smooth and equidimen-sional, by [60, Lemma 3.2.1], for w = 0, the set crit (Π w , V r ) coincides with the set ofpoints ( x, y ) ∈ V r such that the matrix D ( f, Π w ) = (cid:18) DfD Π w (cid:19) has a rank ≤ c . In particular there exists z = ( z , . . . , z c + e , z c + e +1 ) = 0, such that z T D ( f, Π w ) = 0. One can exclude that z c + e +1 = 0, since this implies that Df has anon-zero vector in the left kernel, which contradicts the fact that A satisfies P . Hencewithout loss of generality we deduce that z c + e +1 = 1, and we conclude. (cid:3) Proof of the proposition
We can finally deduce the proof of Proposition 8.
Proof of Proposition 8:
Define M as the set of matrices M ∈ GL n ( C ) such that thefirst row of M − is contained in W (defined in Lemma 12). The proof of all assertions21ollows from Lemma 12 since, for M ∈ M , one gets (cid:18) Df ( A ◦ M, ι ) e T · · · (cid:19) = (cid:18) Df ( A, ι ) ◦ Mw T · · · (cid:19) (cid:18) M I m ( m − r ) (cid:19) , (6)where w T is the first row of M − . Indeed, for z = ( z , . . . , z c + e ), we deduce from theprevious relation that the set of solutions to the equations f ( A, ι ) = 0 , z T Df ( A, ι ) = ( w T ,
0) (7)is the image of the set of solutions of f ( A ◦ M, ι ) = 0 , z T Df ( A ◦ M, ι ) = ( e T ,
0) (8)by the linear map xyz M − I m ( m − r )
00 0 I c + e xyz . This last fact is straightforward since from (6) we deduce that system (8) is equivalent to f ( A ◦ M, ι ) = 0 , z T ( Df ( A, ι ) ◦ M ) = ( w T , . Hence the three assertions of Proposition 8are straightforwardly deduced by those of Lemma 12. (cid:3)
For the proof of Assertion 1 of Proposition 9, we need to recall some notation from [34,Sec. 5]. Let
Z ⊂ C n be an algebraic set of dimension d . Its equidimensional component ofdimension p , for 0 ≤ p ≤ d , is Ω p ( Z ). We define S ( Z ) = Ω ( Z ) ∪· · ·∪ Ω d − ( Z ) ∪ singΩ d Z , where we recall that sing V denotes the singular locus of an algebraic set V , and C ( π i , Z ) = Ω ( Z ) ∪ · · · ∪ Ω i − ( Z ) ∪ d [ r = i crit ( π i , reg Ω r Z ) , Here reg V denotes V \ sing V , π i : x ( x , . . . , x i ) and crit ( g, V ) the set of critical pointsof the restriction of a map g to V . For M ∈ GL n ( C ) we recursively define • O d ( M − Z ) = M − Z ; • O i ( M − Z ) = S ( O i +1 ( M − Z )) ∪ C ( π i +1 , O i +1 ( M − Z )) ∪ C ( π i +1 , M − Z ) for i =0 , . . . , d − M is generic in GL n ( C ) (that is, it lies out ofa proper algebraic set) the algebraic sets O i ( M − Z ) have dimension at most i and arein Noether position with respect to x , . . . , x i ( cf. [63, 16] for a background in Noetherposition). We used this fact in [34, Prop. 18] to prove closure properties of the restrictionof maps π i , i = 1 , . . . , d , to the connected components of Z ∩ R n .22 roof of Assertion 1: We denote by M ⊂ GL n ( C ) the non-empty Zariski open setdefined in [34, Prop. 17], for the algebraic set D r . Hence, for M ∈ M , we deduce by[34, Prop. 18] that for i = 1 , . . . , d , and for any connected component C ⊂ D r ∩ R n , theboundary of π i ( M − C ) is contained in π i ( O i − ( M − D r ) ∩ M − C ) ⊂ π i ( M − C ), and hencethat π i ( M − C ) is closed. Moreover, let C ⊂ D r ∩ R n be a connected component and let t ∈ R be in the boundary of π ( M − C ). Then [34, Lemma 19] implies that π − ( t ) ∩ M − C is finite. (cid:3) Proof of Assertion 2:
Let M ∈ M . Consider the open set O = { ( x, y ) ∈ C n + m ( m − r ) : rank A ( M x ) = r, rank Y ( y ) = m − r } . Let Π x : C n + m ( m − r ) → C n , Π x ( x, y ) = x . Then Π x ( O ) is the locally closed set M − ( D r \D r − ) = { x ∈ C n : rank A ( M x ) = r } . We consider the restriction of polynomial equations A ( M x ) Y ( y ) = 0 to O . By definition of O , we can split the locally closed set O ∩ Z ( A ( M x ) Y ( y )) into the union O ∩ Z ( A ( M x ) Y ( y )) = [ ι ⊂ { , . . . , m } ♯ι = m − r (cid:16) O ι ∩ Z ( A ( M x ) Y ( y )) (cid:17) , where O ι = { ( x, y ) : det Y ι = 0 } .Let C be a connected component of D r ∩ R n . Let t lie in the frontier of π ( M − C ), and x ∈ π − ( t ) ∩ M − C with rank A ( M x ) = r . Hence there exists ι ⊂ { , . . . , m } such that x lies in the projection of V r ( A ◦ M, ι ) on the x − space. Hence there exists y such that( x, y ) ∈ V r ( A ◦ M, ι ) and such that π ( x, y ) = t . (cid:3) Our next goal is to estimate the arithmetic complexity of algorithm
SolveLMI , that isthe number of arithmetic operations performed over Q . This will essentially rely on thecomplexities of state-of-the-art algorithms computing rational parametrizations. We start by computing Multilinear B´ezout bounds ( cf. [60, Ch. 11]) on the output degree.
Proposition 13
Let A ∈ S n +1 m be the input of SolveLMI , and ≤ r ≤ m − . Let p r = ( m − r )( m + r + 1) / . If H holds, for all ι ⊂ { , . . . , m } , with ♯ι = m − r , the degreeof the parametrization q ι computed at Step 2 of LowRankSymRec is at most θ ( m, n, r ) = X k ∈G m,n,r (cid:18) p r n − k (cid:19)(cid:18) n − k + p r − − r ( m − r ) (cid:19)(cid:18) r ( m − r ) k (cid:19) , ith G m,n,r = { k : max { , n − p r } ≤ k ≤ min { n − (cid:0) m − r +12 (cid:1) , r ( m − r ) }} . Moreover, for all m, n, r , θ ( m, n, r ) is bounded above by (cid:0) p r + nn (cid:1) . Proof :
We can simplify the system f = f ( A, ι ) to a system of p r bilinear equations withrespect to variables x = ( x , . . . , x n ) and y = ( y m − r +1 , , . . . , y m,m − r ). Indeed, by Lemma5, V r ( A, ι ) is defined by Y ι − I m − r = 0 and by m ( m − r ) − e = p r entries of A ( x ) Y ( y ),where e = (cid:0) m − r (cid:1) . Hence we can eliminate equations Y ι − I m − r = 0 and the correspondingvariables y i,j . Consequently, the Lagrange system can be also simplified, by admittingonly p r Lagrange multipliers z . We can also eliminate the first Lagrange multiplier z (since z = 0, one can assume z = 1) and impose a rank defect on the truncated Jacobianmatrix obtained by Df by eliminating the first column (that containing the derivativeswith respect to x ).The bound θ ( m, n, r ), by [60, Ch. 11], equals the coefficient of s nx s r ( m − r ) y s p r − z in ( s x + s y ) p r ( s y + s z ) n − ( s x + s z ) r ( m − r ) . This can be easily obtained by writing down such anexpansion and solving the associated linear system forcing the constraints on the expo-nents of the monomials. The result is exactly the claimed closed formula. The estimate θ ( m, n, r ) ≤ (cid:0) p r + nn (cid:1) is obtained by applying the standard formula: (cid:18) a + ba (cid:19) = min( a,b ) X i ,i ,i =0 (cid:18) ai (cid:19)(cid:18) bi (cid:19)(cid:18) ai (cid:19)(cid:18) bi (cid:19)(cid:18) ai (cid:19)(cid:18) bi (cid:19) with a = n and b = p r . (cid:3) Corollary 14
With the hypotheses and notations of Proposition 13, the sum of the de-grees of the rational parametrizations computed during
SolveLMI is bounded above by P r ≤ r ( A ) (cid:0) mr (cid:1) θ ( m, n, r ) . The degree of the rational parametrization whose solutions intersect S is in O (cid:18) m + m + nn (cid:19) ! . Proof :
Remark that the number of subsets ι ⊂ { , . . . , m } , with ♯ι = m − r is (cid:0) mm − r (cid:1) = (cid:0) mr (cid:1) , and that SolveLMI stops when r reaches r ( A ). Hence the first part follows by applyingProposition 13. Finally, remark that p ≥ p ≥ · · · ≥ p r ≥ · · · for all m , and hence, byProposition 13, the degree of the rational parametrization whose solutions intersect S isof the order of (cid:0) p r + nn (cid:1) ≤ (cid:0) p + nn (cid:1) = (cid:0) m m + nn (cid:1) . (cid:3) In the column deg of Table 1 we report the degrees of the rational parametrization q ι returned by LowRankSymRec at Step 2, compared with its bound θ ( m, n, r ) computed inProposition 13. For this table, the input are randomly generated symmetric pencils withrational coefficients. When the algorithm does not compute critical points (that is, whenthe Lagrange system generates the empty set) we put deg = 0.We recall that the routine LowRankSymRec computes points in components of the realalgebraic set D r ∩ R n not meeting the subset D r − ∩ R n , hence of the expected rank r .Moreover, we recall that LowRankSym calls recursively its subroutine
LowRankSymRec ,24liminating at each call the first variable. Hence, the total number of critical pointscomputed by
LowRankSym for a given expected rank r is obtained by summing up theinteger in column deg for every admissible value of n . We remark here that both the degreeand the bound are constant and equal to 0 if n is large enough. Hence, the previous sum isconstant for large values of n . Similar behaviors appear, for example, when computing theEuclidean Distance degree (EDdegree) of determinantal varieties, as in [15] or [50]. In [50,Table 1], the authors report on the EDdegree of determinantal hypersurfaces generatedby linear matrices A ( x ) = A + x A + · · · + x n A n : for generic weights in the distancefunction, and when the codimension of the vector space generated by A , . . . , A n is small(for us, when n is big, since matrices A i are randomly generated, hence independent for n ≤ (cid:0) m +12 (cid:1) = dim S m ( Q )) the EDdegree is constant. Similar comparisons can be donewith data in [50, Example 4] and [50, Corollary 3.5].( m, r, n ) deg θ ( m, n, r ) ( m, r, n ) deg θ ( m, n, r )(3 , ,
2) 6 9 (4 , ,
9) 0 0(3 , ,
3) 4 16 (5 , ,
5) 0 0(3 , ,
4) 0 15 (5 , ,
6) 35 924(3 , ,
5) 0 6 (5 , ,
7) 140 10296(3 , ,
6) 0 0 (5 , ,
3) 20 84(4 , ,
3) 10 35 (5 , ,
4) 90 882(4 , ,
4) 30 245 (5 , ,
2) 20 30(4 , ,
5) 42 896 (5 , ,
3) 40 120(4 , ,
6) 30 2100 (5 , ,
4) 40 325(4 , ,
7) 10 3340 (5 , ,
5) 16 606(4 , ,
8) 0 3619 (6 , ,
3) 0 0(4 , ,
9) 0 2576 (6 , ,
4) 0 0(4 , ,
12) 0 0 (6 , ,
5) 0 0(4 , ,
3) 16 52 (6 , ,
6) 112 5005(4 , ,
4) 8 95 (6 , ,
2) 0 0(4 , ,
7) 0 20 (6 , ,
3) 35 165(4 , ,
8) 0 0 (6 , ,
3) 80 230Table 1: Degrees and bounds for rational parametrizationsThe values in column deg of Table 1 must also be compared with the associated algebraicdegree of semidefinite programming. Given integers k, m, r with r ≤ m −
1, Nie, Ranes-tad, Sturmfels and von Bothmer computed in [48, 24] formulas for the algebraic degree δ ( k, m, r ) of a generic semidefinite program associated to m × m k − variate linear matrices,with expected rank r . Since the values in column deg match exactly the correspondingvalues in [48, Table 2], we conclude this section with the following expected result, whichis a work in progress. Conjecture 15
Let A ∈ S n +1 m ( Q ) be the input of SolveLMI , and suppose that S = { x ∈ R n : A ( x ) (cid:23) } is not empty. Let δ ( k, m, r ) be the algebraic degree of a generic semidefi-nite program with parameters k, m, r as in [48, 24]. If H holds, then the sum of the degrees f the rational parametrizations computed during SolveLMI is r ( A ) X r =1 (cid:18) mr (cid:19) min( n,p r + r ( m − r )) X k = p r − r ( m − r ) δ ( k, m, r ) , where p r = ( m − r )( m + r + 1) / . SolveLMI
Complexity of some subroutines
We first provide complexity estimates for subroutines
SolveLinear , CheckLMI , Project , Lift , Image and
Union . Complexity of
SolveLinear . This subroutine computes, if it exists, a solution of the A ( x ) =0. This can be essentially performed by Gaussian elimination. The complexity of solving (cid:0) m +12 (cid:1) linear equations in n variables is linear in (cid:0) m +12 (cid:1) and cubic in n . Complexity of
CheckLMI . This subroutine can be performed as follows. Let q ⊂ Z [ t ]be the rational parametrization in the input of CheckLMI . The spectrahedron S = { x ∈ R n : A ( x ) (cid:23) } is the semi-algebraic set defined, e.g. , by sign conditions on the coefficientsof the characteristic polynomial p ( s, x ) = det( A ( x ) + s I m ) = f m ( x ) + f m − ( x ) s + · · · + f ( x ) s m − + s m . That is, S = { x ∈ R n : f i ( x ) ≥ , ∀ i = 1 , . . . , m } . We make the substitution x i ← q i ( t ) /q ( t ) in A ( x ) and compute the coefficients of p ( s, x ( t )), that are rational functionsof the variable t . Hence CheckLMI boils down to deciding on the sign of m univariaterational functions (that is, of 2 m univariate polynomials) over the finite set defined by q n +1 ( t ) = 0. We deduce that the complexity of CheckLMI is polynomial in m and on thedegree of q n +1 (that is, on the degree of q ) see [6, Ch. 13]. Complexity of
Project, Lift, Image and
Union
Estimates for the arithmetic complexitiesof these routines are given in [60, Ch. 10]. In particular, if θ = θ ( m, n, r ) is the boundcomputed in Proposition 13, and ˜ n = n + r ( m − r ) + p r , then: • By [60, Lemma 10.1.5],
Project is performed within O ˜(˜ n θ ) operations; • By [60, Lemma 10.1.6],
Lift is performed within O ˜(˜ nθ ) operations; • By [60, Lemma 10.1.1],
Image is performed within O ˜(˜ n θ + ˜ n ) operations; • By [60, Lemma 10.1.3],
Union is performed within O ˜(˜ nθ ) operations. Complexity of
LowRankSym and
SolveLMI
The complexity of
LowRankSym boils essentially down to the complexity of
LowRankSym-Rec , that is the complexity of
RatPar . This is performed with the symbolic homotopy26lgorithm [39]: we bound its complexity in this section. We just remark that computingthe dimension of D r at Step 1 of LowRankSym and performing the control routine
IsReg can be done by combining the Jacobian criterion and Gr¨obner bases computations. Ourcomplexity analysis omits this step.We recall that the simplified Lagrange system defined in the proof of Proposition 13contains: p r = ( m − r )( m + r + 1) / , , n − , , r ( m − r ) polynomials of multidegreebounded by (1 , , ℓ this system. We denote by∆ xy = { , x i , y j , x i y j : i = 1 , . . . , n, j = 1 , . . . , r ( m − r ) } ∆ yz = { , y j , z k , y j z k : j = 1 , . . . , r ( m − r ) , k = 2 . . . , p r } ∆ xz = { , x i , z k , x i z k : i = 1 , . . . , n, k = 2 , . . . , p r } the supports of the aforementioned three groups of polynomials. Let e ℓ ⊂ Q [ x, y, z ] be apolynomial system such that: • the length of ˜ ℓ equals that of ℓ ; • for i = 1 , . . . , n − m − r , the support of ˜ ℓ i equals that of ℓ i ; • the solutions of ˜ ℓ are known.Remark that e ℓ can be easily built by considering suitable products of linear forms. Webuild the homotopy τ ℓ + (1 − τ )˜ ℓ ⊂ Q [ x, y, z, τ ] , (9)where τ is a new variable. The system (9) defines a curve. From [39, Proposition 6.1], if thesolutions of ˜ ℓ are known, one can compute a rational parametrization of Z ( tℓ + (1 − t )˜ ℓ )within O ((˜ n N log Q + ˜ n ω +1 ) ee ′ ) arithmetic operations over Q , where: ˜ n is the numberof variables in ℓ ; N = p r ♯ ∆ xy + ( n − ♯ ∆ yz + r ( m − r ) ♯ ∆ xz ; Q = max {k q k : q ∈ ∆ xy ∪ ∆ yz ∪ ∆ xz } ; e is the number of isolated solutions of ℓ ; e ′ is the degree of Z ( tℓ + (1 − t )˜ ℓ ); ω is the exponent of matrix multiplication.The technical lemma below gives a bound on the degree of Z ( tℓ + (1 − t )˜ ℓ ). Lemma 16
Let G m,n,r and θ ( m, n, r ) be the set and the bound defined in Proposition13, and suppose that G m,n,r = ∅ . Let e ′ be the degree of Z ( tℓ + (1 − t )˜ ℓ ) . Then e ′ ∈O (( n + p r + r ( m − r )) min { n, p r } θ ( m, n, r )) . Proof :
Similarly to Proposition 13, we exploit the multilinear structure of tℓ + (1 − t )˜ ℓ ,to compute a bound on e ′ . The system is bilinear with respect to x, y, z, τ . We recall ♯x = n, ♯y = r ( m − r ) , ♯z = p r − , ♯τ = 1, with p r = ( m − r )( m + r + 1) /
2. By [60,Ch. 11], e ′ is bounded by the sum of the coefficients of q = ( s x + s y + s τ ) p r ( s y + s z + s τ ) n − ( s x + s z + s τ ) r ( m − r ) modulo I = h s n +1 x , s r ( m − r )+1 y , s p r z , s τ i ⊂ Z [ s x , s y , s z , s τ ]. We see27hat q = q + s τ ( q + q + q ) + g with s τ that divides g and q = ( s x + s y ) p r ( s y + s z ) n − ( s x + s z ) r ( m − r ) q = p r ( s x + s y ) p r − ( s y + s z ) n − ( s x + s z ) r ( m − r ) q = ( n − s x + s y ) p r ( s y + s z ) n − ( s x + s z ) r ( m − r ) q = r ( m − r )( s x + s y ) p r ( s y + s z ) n − ( s x + s z ) r ( m − r ) − . Hence q ≡ q + s τ ( q + q + q ) mod I , and the bound is given by the sum of thecontributions of q , q , q and q (multiplying q , q , q by s τ does not change the sum ofthe coefficients modulo I ). The contribution of q is the sum of the coefficients of itsclass modulo I ′ = h s n +1 x , s r ( m − r )+1 y , s p r z i . This has been computed in Proposition 13, andcoincides with θ ( m, n, r ). We compute the contribution of q . Let q = p r ˜ q . It is sufficientto compute the sum of the coefficients of ˜ q modulo I ′ (defined above), multiplied by p r .Since deg ˜ q = n − p r + r ( m − r ), and since the maximal powers admissible modulo I ′ are s nx , s r ( m − r ) y , and s p r − z , three configurations are possible.(A) The coefficient of s n − x s r ( m − r ) y s p r − z in ˜ q , that isΣ A = X k (cid:18) p r − n − − k (cid:19)(cid:18) n − k − p r − r ( m − r ) (cid:19)(cid:18) r ( m − r ) k (cid:19) ;(B) the coefficient of s nx s r ( m − r ) − y s p r − z in ˜ q , that isΣ B = X k (cid:18) p r − n − k (cid:19)(cid:18) n − k − p r − r ( m − r ) (cid:19)(cid:18) r ( m − r ) k (cid:19) ;(C) the coefficient of s nx s r ( m − r ) y s p r − z in ˜ q , that isΣ C = X k (cid:18) p r − n − k (cid:19)(cid:18) n − k − p r − r ( m − r ) (cid:19)(cid:18) r ( m − r ) k (cid:19) . Hence we need to bound the expression p r (Σ A + Σ B + Σ C ). One can easily check thatΣ A ≤ θ ( m, n, r ) and Σ B ≤ θ ( m, n, r ), while the same inequality is false for Σ C . However,we claim that Σ C ≤ (1 + min { n, p r } ) θ ( m, n, r ) and hence that the contribution of q is p r (Σ A + Σ B + Σ C ) ∈ O ( p r min { n, p r } θ ( m, n, r )). We prove now this claim. Let χ = max { , n − p r } χ = min { n − p r + r ( m − r ) , r ( m − r ) } α = max { , n − p r + 1 } α = min { n − p r + r ( m − r ) + 1 , r ( m − r ) } so that the sum in θ ( m, n, r ) runs over χ ≤ k ≤ χ and that in Σ C over α ≤ k ≤ α .Remark that χ ≤ α and χ ≤ α . Denote by ϕ ( k ) the k − th term in the sum definingΣ C , and by γ ( k ) the k − th term in the sum defining θ ( m, n, r ). Then for all indices k ,admissible both for θ ( m, n, r ) and Σ C , that is for α ≤ k ≤ χ , one gets, by basic propertiesof binomial coefficients, that ϕ ( k ) = Ψ( k ) γ ( k ), with Ψ( k ) = k − p r − r ( m − r ) n − k − p r + r ( m − r ) − . When k k ) is non-decreasing monotone, and its maximum isΨ( χ ) and is bounded by min { n, p r } . We deduce the claimed inequality Σ C ≤ (1 +min { n, p r } ) θ ( m, n, r ), since if χ < α then χ = α − ϕ ( α ) is bounded above by θ ( m, n, r ). Contributions of q and q . As for q , we deduce that the contribution of q is in O ( n min { n, p r } θ ( m, n, r )) and that of q is in O ( r ( m − r ) min { n, p r } θ ( m, n, r )). (cid:3) We use this degree estimate to conclude our complexity analysis of
LowRankSym . Proposition 17
Let A ∈ S n +1 m ( Q ) be the input of SolveLMI and ≤ r ≤ m − . Let θ ( m, n, r ) be the bound defined in Proposition 13. Let p r = ( m − r )( m + r + 1) / . Then RatPar returns a r.p. within O ˜ (cid:0)(cid:0) mr (cid:1) ( n + p r + r ( m − r )) θ ( m, n, r ) (cid:1) arithmetic opera-tions over Q . Proof :
Let ℓ be the simplified Lagrange system as in the proof of Proposition 13. Weconsider the bound on the degree of the homotopy curve given by Lemma 16. We deducethe claimed complexity result by applying [39, Proposition 6.1], and by recalling thatthere are (cid:0) mr (cid:1) many subsets of { , . . . , m } of cardinality m − r . (cid:3) We straightforwardly deduce the following complexity estimate for
SolveLMI . Theorem 18 (Complexity of
SolveLMI ) Let A ∈ S n +1 m ( Q ) be the input symmetric pen-cil and suppose that H holds. Then the number of arithmetic operations performed by SolveLMI are in O ˜ n (cid:18) m + m + nn (cid:19) X r ≤ m − (cid:18) mr (cid:19) ( n + ( m − r )( m + 3 r )) ! . Proof :
From Proposition 17, we deduce that
LowRankSymRec runs essentially within O ˜( (cid:0) mr (cid:1) ( n + p r + r ( m − r )) θ ( m, n, r ) ) arithmetic operations. The inequality θ ( m, n, r ) ≤ (cid:0) n + p r n (cid:1) is proved in Proposition 13. Moreover, there are at most n recursive calls of LowRankSymRec in LowRankSym , and
SolveLMI stops at most when r reaches m −
1. Fi-nally, the cost of subroutines
SolveLinear , CheckLMI , Project, Lift, Image and
Union is negli-gible. We deduce that the complexity of
SolveLMI is in O ˜ (cid:16) n P r ≤ m − (cid:0) mr (cid:1) ( n + p r + r ( m − r )) (cid:0) p r + nn (cid:1) (cid:17) Since p r ≤ p = m + m and p r + r ( m − r ) ≤ ( m − r )( m + 3 r ), we conclude. (cid:3) SolveLMI is implemented in a maple function, and it is part of a library called spectra (Semidefinite Programming solved Exactly with Computational Tools of Real Algebra),released in September 2015. It collects efficient and exact algorithms solving a large classof problems in real algebraic geometry and semidefinite optimization.We present in this section the results of our experiments. These have been performedon a machine with the following characteristics: Intel(R) Xeon(R) CPU [email protected] 256 Gb of RAM. We rely on the maple implementation of the software
FGb [17],for fast computation of Gr¨obner bases. To compute the rational parametrizations we usethe implementation in maple of the change-of-ordering algorithm FGLM [20] and of itsimproved versions [21, 19].
We implemented the function
LowRankSym and tested the running time of the imple-mentation with input generic symmetric linear matrices. We recall that the algorithm
SolveLMI amounts to iterating
LowRankSym by increasing the expected rank r . By genericdata we mean that a natural number N ∈ N large enough is fixed, and numerators anddenominators of the entries of A ℓ , ℓ = 0 , . . . , n are uniformly generated in the interval[ − N, N ]. We report in Table 2 the timings and the degrees of output rational parametriza-tions. ( m, r, n ) PPC LRS totaldeg deg ( m, r, n ) PPC LRS totaldeg deg (3 , ,
2) 0.2 8 9 6 (4 , , ∞
28 40 0(3 , ,
3) 0.3 11 13 4 (4 , , ∞
29 40 0(3 , ,
4) 0.9 13 13 0 (4 , , ∞
30 40 0(3 , ,
5) 5.1 14 13 0 (5 , ,
2) 0.6 0 0 0(3 , ,
6) 15.5 15 13 0 (5 , ,
3) 0.9 0 0 0(3 , ,
7) 31 16 13 0 (5 , ,
4) 1 1 0 0(3 , ,
8) 109 17 13 0 (5 , ,
5) 1.6 1 0 0(3 , ,
9) 230 18 13 0 (5 , , ∞ , ,
2) 0.2 0 0 0 (5 , ,
2) 0.4 1 0 0(4 , ,
3) 0.3 2 10 10 (5 , ,
3) 0.5 3 20 20(4 , ,
4) 2.2 9 40 30 (5 , , ∞ , ,
5) 12.2 29 82 42 (5 , , ∞ , , ∞
71 112 30 (5 , ,
2) 0.5 7 25 20(4 , , ∞
103 122 10 (5 , ,
3) 10 42 65 40(4 , , ∞
106 122 0 (5 , , ∞
42 105 40(4 , , ∞
106 122 0 (5 , , ∞
858 121 16(4 , ,
3) 1 10 32 16 (6 , ,
3) 4 0 0 0(4 , ,
4) 590 21 40 8 (6 , ,
4) 140 1 0 0(4 , , ∞
22 40 0 (6 , , ∞ , , ∞
24 40 0 (6 , , ∞
704 112 112(4 , , ∞
26 40 0 (6 , ,
2) 0.6 1 0 0(4 , , ∞
27 40 0 (6 , , ∞
591 116 80
Table 2: Timings and degrees for dense symmetric linear matricesWe recall that m is the size of the input matrix, n is the number of variables and r isthe expected maximum rank (that is, the index of the algebraic set D r ). We compare ourtimings (reported in LRS ) with those of the function
PointsPerComponents (column
PPC )of the library raglib developed by the third author [56]. The input of
PointsPerCompo-nents are the ( r + 1) × ( r + 1) minors of the linear matrix, and the output is a rationalparametrization of a finite set meeting each connected component of D r ∩ R n . We donot consider the time needed to compute all the minors of A ( x ) in PPC . The symbol ∞ means that we did not succeed in computing the parametrizations after 48 hours. Column deg contains the degree of the parametrization returned by LowRankSymRec at Step 2,30r 0 if the empty list is returned. Column totaldeg contains the sum of the values in deg for k varying between 1 and n . For example, for m = 4 , r = 2, for n ≤ n ≥ , , , ,
10 for n = 3 , , , ,
7; the number 82 in totaldeg for ( m, n, r ) = (4 , ,
5) is obtained as the sum 10 + 30 + 42 of the integers in deg for m = 4 , r = 2 and n = 3 , ,
5. We remark that, as for Table 1, the value in deg for agiven triple m, n, r coincides with the algebraic degree of semidefinite programming, thatis δ ( n, m, r ) as defined in [48].Our algorithm allows to tackle examples that are out of reach for raglib and that, mostof the time, the growth in terms of running time is controlled when m, r are fixed. Thisshows that our dedicated algorithm leads to practical remarkable improvements: indeed,for example, 4 × R defined bythe determinant of 5 × We consider the symmetric pencil A ( x ) = x − / − x x − x / x − − x − x x / x x x x − x + 2 x / − / − x − − x x − x / x − x x / / . which is the Gram matrix of the trivariate polynomial f ( u , u , u ) = u + u u + u − u u u − u u u + 2 u u + u u + u u + u . In other words, f = v T A ( x ) v for all x ∈ R , where v = ( u , u u , u , u u , u u , u )is the monomial basis of the vector space of homogeneous polynomials of degree 2 in u , u , u . Since f is globally nonnegative, by Hilbert’s theorem [38] it is a sum of atmost three squares in R [ u , u , u ], namely there exist f , f , f ∈ R [ u , u , u ] such that f = f + f + f . Moreover, the spectrahedron S = { x ∈ R : A ( x ) (cid:23) } parametrizesall the sum-of-squares decompositions of f (and it is a particular example of a Gramspectrahedron ). Hence S must contain a matrix of rank at most 3.Scheiderer proved in [61] that f does not admit a sum-of-squares decomposition in the ring Q [ u , u , u ], that is, the summands f , f , f cannot be chosen to have rational coefficients,answering a question of Sturmfels. We deduce that S does not contain points withrational coordinates. In particular, it is not full-dimensional (its affine hull has dimension ≤
5) by straightforward density arguments.We first easily check that D ∩ R = D ∩ R = ∅ ). Further, for r = 2, the algorithm31eturns the following rational parametrization of D ∩ R : (cid:18) t − t , − t − t , t + 8 t − t ,
16 + 6 t − t − t , − − t − t , t − t (cid:19) . where t satisfies 8 t − t − D is, indeed, of dimension 0, degree 3, and itcontains only real points. Remark that the technical assumption P is not satisfied here,since the expected dimension of D is −
1. Conversely, the regularity assumptions on theincidence varieties are satisfied. By applying
CheckLMI one gets that two of the threepoints lie on S , that is those with the following floating point approximation up to 9certified digits: − . − . . − . . − . and − . − . − . − . . − . . These correspond to the two distinct decompositions of f as a sum of 2 squares. An ap-proximation of such representations can be computed by factorizing the matrix A ( x ( t ∗ )) = V T V where t ∗ is the corresponding root of 8 t − t − V ∈ M , ( R ) is full rank. Thecorresponding decomposition is f = v T V T V v = || V v || . At the third point of D ∩ R thematrix A ( x ) is indefinite, so it is not a valid Gram matrix.To conclude, SolveLMI allows to design a computer-aided proof of Scheiderer’s results.This example is interesting since the interior of S is empty and, typically, this can leadto numerical problems when using interior-point algorithms. We have presented a probabilistic exact algorithm that computes an algebraic represen-tation of at least one feasible point of a LMI A ( x ) (cid:23)
0, or that detects emptiness of S = { x ∈ R n : A ( x ) (cid:23) } . The algorithm works under assumptions which are provedto be generically satisfied. When these assumptions are not satisfied, the algorithm mayreturn a wrong answer or raises an error (when the dimension of some Lagrange systemis not 0). The main strategy is to reduce the input problem to a sequence of real rootfinding problems for the loci of rank defects of A ( x ): if S is not empty, we have shownthat computing sample points on determinantal varieties is sufficient to sample S , andthat it can be done efficiently. Indeed, the arithmetic complexity is essentially quadraticon a multilinear B´ezout bound on the output degree.This is, to our knowledge, the first exact computer algebra algorithm tailored to linearmatrix inequalities. We conjecture that our algorithm is optimal since the degree of theoutput parametrization matches the algebraic degree of a generic semidefinite program,with expected rank equal to the minimal achievable rank on S . Since deciding theemptiness of S is a particular instance of computing the minimizer of a linear functionover this set (namely, of a constant), our algorithm is able to compute minimal-rank32olutions of special semidefinite programs, which is, in general, a hard computational task.Indeed, numerical interior-point algorithms typically return approximations of feasiblematrices with maximal rank among the solutions (those lying in the relative interior ofthe optimal face). Moreover, the example of Scheiderer’s spectrahedron shows that we canalso tackle degenerate situations with no interior point which are typically numericallytroublesome.To conclude, as highlighted by the discussions in Section 6, our viewpoint includes aneffective aspect, by which it is essential to translate into practice the complexity resultsthat have been obtained. This is the objective of our maple library spectra . It mustbe understood as a starting point towards a systematic exact computer algebra approachto semidefinite programming and related questions. References [1] M. F. Anjos, J. B. Lasserre (editors). Handbook of semidefinite, conic and polynomialoptimization. International Series in Operational Research and Management Science.Vol.166, Springer, New York, 2012.[2] B. Bank, M. Giusti, J. Heintz, M. Safey El Din, ´E. Schost. On the geometry of polarvarieties. Appl. Alg. in Eng., Comm. and Comp. 21(1):33–83, 2010.[3] B. Bank, M. Giusti, J. Heintz, G.-M. Mbakop. Polar varieties and efficient real elimi-nation. Mathematische Zeitschrift, 238(1):115–144, 2001.[4] R. G. Bartle, D. R. Sherbert. Introduction to real analysis. John Wiley & Sons, NewYork, 1992.[5] S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexityof quantifier elimination. Journal of the ACM, 43(6):1002-1046, 1996.[6] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in real algebraic geometry. Algorithmsand Computation in Mathematics, Vol. 10. Springer Verlag, Berlin, 2006.[7] A. Ben-Tal, A. Nemirovski. Lectures on modern convex optimization: analysis, algo-rithms, engineering applications. MPS-SIAM Series on Optimization, SIAM, Philadel-phia, 2001.[8] G. Blekherman, P. A. Parrilo, R. R. Thomas (Editors). Semidefinite optimization andconvex algebraic geometry. SIAM, Philadelphia, 2013.[9] S.P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan. Linear matrix inequalities in sys-tem and control theory. Studies in Applied Mathematics, Vol. 15. SIAM, Philadelphia,1994.[10] S. Boyd, L. Vandenberghe. Semidefinite programming. SIAM Review, 38(1):49–95,1996. 3311] M. Claeys. Mesures d’occupation et relaxations semi-d´efinies pour la commandeoptimale. PhD thesis, LAAS CNRS, Univ. Toulouse, France, Oct. 2013.[12] G. Collins. Quantifier elimination for real closed fields by cylindrical algebraic de-compostion. Automata Theory and Formal Languages, pages 134–183. Springer, Berlin,1975.[13] D. A. Cox, J. Little, D. O’Shea. Ideals, varieties, and algorithms. 3rd edition,Springer, New York, 2007.[14] Dahan, X. and Schost, ´E.. Sharp estimates for triangular sets, Proceedings of the2004 international symposium on Symbolic and algebraic computation, pp. 103–110,2004.[15] J. Draisma, E. Horobet, G. Ottaviani, B. Sturmfels, R.R. Thomas. The Euclideandistance degree of an algebraic variety. Found. of Comp. Math., 16(1):99–149, 2016.[16] D. Eisenbud. Commutative algebra with a view toward algebraic geometry. Springer,New York, 1995.[17] J.-C. Faug`ere.
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