Strengthened SDP Relaxation for an Extended Trust Region Subproblem with an Application to Optimal Power Flow
aa r X i v : . [ m a t h . O C ] S e p Strengthened SDP Relaxation for anExtended Trust Region Subproblemwith an Application to Optimal Power Flow
Anders Eltved ∗ Samuel Burer † September 26, 2020
Abstract
We study an extended trust region subproblem minimizing a nonconvex functionover the hollow ball r ≤ k x k ≤ R intersected with a full-dimensional second ordercone (SOC) constraint of the form k x − c k ≤ b T x − a . In particular, we present aclass of valid cuts that improve existing semidefinite programming (SDP) relaxationsand are separable in polynomial time. We connect our cuts to the literature on theoptimal power flow (OPF) problem by demonstrating that previously derived cutscapturing a convex hull important for OPF are actually just special cases of our cuts.In addition, we apply our methodology to derive a new class of closed-form, locallyvalid, SOC cuts for nonconvex quadratic programs over the mixed polyhedral-conic set { x ≥ k x k ≤ } . Finally, we show computationally on randomly generated instancesthat our cuts are effective in further closing the gap of the strongest SDP relaxationsin the literature, especially in low dimensions. The classical trust region subproblem (TRS) minimizes an arbitrary quadratic function overthe unit Euclidean ball defined by k x k ≤ R and is solvable in polynomial-time [10]. Manyauthors have studied variants of TRS that incorporate additional constraints. For example,[20] also imposes the lower bound r ≤ k x k . We collectively refer to variants of TRS that ∗ Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800Kgs. Lyngby, Denmark. Email: [email protected] . † Department of Business Analytics, University of Iowa, Iowa City, IA, 52242-1994, USA. Email: [email protected] . extended TRS . In this paper, we study thefollowing specific form of the extended TRS, which incorporates the lower bound r as wellas an additional SOC (second-order cone) constraint, whose “geometry” matches the ball inthe sense that its Hessian is also the identity matrix:min x T Hx + 2 g T x (1a)s.t. r ≤ k x k ≤ R (1b) k x − c k ≤ b T x − a (1c)where x ∈ R n , H = H T ∈ R n × n , g, c, b ∈ R n , a ∈ R , and r, R ∈ R + . Note that H issymmetric without loss of generality and that we have not scaled the problem to the unitball (i.e., we do not assume R = 1) as is common in the TRS literature. The general upperbound R will be convenient for our presentation, especially in Section 3. The algorithm ofBienstock [3] solves (1) in polynomial time since it can be written as a nonconvex quadraticprogram with a fixed number of quadratic/linear constraints (in this case, four), one ofwhich is strictly convex. However, in this paper, we are interested in developing tight convexrelaxations of (1). In particular, as far as we are aware, (1) has no known tight convexrelaxation.Problem (1) includes, for example, the two trust region subproblem —also called the Celis-Dennis-Tapia subproblem [8]—in which a second ball (or ellipsoidal) constraint is added toTRS. In this case, r = 0, b = 0, and a <
0. Here, however, we are interested in the moregeneral structure represented by (1c), which arises, for example, in the optimal power flowproblem (OPF) as discussed in Section 3. More generally, the study of (1) sheds light onany nonconvex quadratically constrained quadratic program that includes a ball constraintand a second SOC constraint with identity Hessian. In Section 3, we will also show howthis structure is relevant for the mixed polyhedral-SOC set { x ≥ k x k ≤ R } . (In theconcluding Section 6, we briefly mention an extension for handling different Hessians.)Since (1) is a nonconvex problem, a standard approach is to approximate (1) by its so-called Shor semidefinite programming (SDP) relaxation [19], which is solvable in polynomial2ime: min H • X + 2 g T x (2a)s.t. r ≤ tr( X ) ≤ R (2b)tr( X ) − c T x + c T c ≤ bb T • X − a b T x + a (2c)0 ≤ b T x − a (2d) Y ( x, X ) (cid:23) M • X := tr( M T X ) is the trace inner product for conformal matrices and Y ( x, X ) := x T x X (3)is symmetric of size ( n + 1) × ( n + 1). Note that (1c) is represented as the two constraints k x − c k ≤ ( b T x − a ) and 0 ≤ b T x − a before lifting to (2c)–(2d). We also define R shor := { ( x, X ) : ( x, X ) satisfies (2b)–(2e) } to be the feasible set of the Shor relaxation. Then (2) can be alternatively expressed asminimizing H • X + 2 g T x over ( x, X ) ∈ R shor .Various valid inequalities can be added to (2) in order to strengthen the Shor relaxation.For example, if v T x ≥ u and v T x ≥ u are any two valid linear inequalities for the feasibleset of (1), then the redundant quadratic constraint ( v T x − u )( v T x − u ) ≥ RLT constraint [18]: v v T • X − u v T x − u v T x + u u ≥ . However, since (1) does not contain explicit linear constraints, in practice one would needto separate over valid v T x ≥ u and v T x ≥ u to generate violated RLT constraints, butthis separation is a bilinear subproblem, which does not appear to be solvable in polynomialtime.The difficulty of separating the RLT constraints when no linear constraints are explicitlygiven can be circumvented in the case of (1) as follows. By multiplying a valid v T x ≥ u with the ball constraint k x k ≤ R , we have the redundant quadratic SOC constraint k ( v T x − u ) x k ≤ R ( v T x − u ), which in turn yields the valid SOC constraint k Xv − u x k ≤ R ( v T x − u ) (4)3n the lifted ( x, X ) space. In a similar manner, v T x ≥ u can be combined with k x − c k ≤ b T x − a . These are known as SOCRLT constraints [21, 5]. In fact, each SOCRLT constraintis a compact encoding of an entire collection of RLT constraints. For example, (4) capturesall of the RLT constraints corresponding to v T x ≥ u fixed and v T x ≥ u varying over thesupporting hyperplanes of k x k ≤ R . Consequently, the collections of SOCRLT and RLTconstraints for (1) are equivalent, but in contrast to the RLT constraints, the SOCRLTconstraints can be separated in polynomial-time based on the fact that TRS is polynomial-time solvable [5].Anstreicher [1] introduced a further generalization of the SOCRLT constraints, called a KSOC constraint , which is based on relaxing a valid quadratic Kronecker-product matrixinequality. Specifically, the KSOC constraint is constructed from the following observations:first, defining SOC := { ( v , v ) : k v k ≤ v } to be the second-order cone, it is well-known that v v ! ∈ SOC ⇐⇒ v v T v v I (cid:23) R x T x R I ⊗ b T x − a x T − c T x − c ( b T x − a ) I (cid:23) . After relaxing this inequality in the space ( x, X ), we obtain the convex KSOC constraint,which captures all SOCRLT constraints (and hence all RLT constraints) and is generallystronger [1], assuming the Shor constraints remain enforced.Summarizing, defining R rlt and R socrlt to be the set of ( x, X ) satisfying all possible RLTand SOCRLT constraints, respectively, we have R shor ∩ R ksoc ⊆ R shor ∩ R socrlt = R shor ∩ R rlt where R ksoc is the set of all ( x, X ) satisfying the KSOC constraint. Moreover, the firstcontainment is proper in general. Hence, in this paper, we focus on improving the relaxation R shor ∩ R ksoc . The paper [13] provides further insight into the strength of R shor ∩ R ksoc relative to other techniques in the literature.Let F denote the feasible set of (1), i.e., the set of all x ∈ R n satisfying (1b)–(1c). This differs from other papers, which often define RLT constraints only for explicitly given valid linearconstraints, of which (1) has none. So, for the sake of generality, we have defined the RLT constraintsallowing for implicit valid linear constraints. G := conv n ( x, xx T ) : x ∈ F o . (5)Note that G is compact because F is. Moreover, because linear optimization over a compactset is guaranteed to attain its optimal value at an extreme point, solving (1) amounts tooptimizing the linear function H • X + 2 g T x over G . While an exact representation of G is unknown, there are several closely related cases in which G can be described exactly; see[7, 2].In this paper, we propose a new class of valid linear inequalities for (1) in the space ( x, X ),which in general strengthen R shor ∩ R ksoc towards G . Each inequality is derived from severalingredients that exploit the structure of F : the self-duality of SOC; the RLT-type validinequality ( R − k x k )( k x k − r ) ≥
0; and knowledge of a quadratic function q ( x ) and a linearfunction l ( x ), each of which is nonnegative over all x ∈ F . We combine these ingredients toderive a valid quartic inequality, which is then relaxed to a valid quadratic inequality, whichin turn yields a new valid linear inequality in ( x, X ).As a small illustrative example, consider when c = 0 and r = 0, in which case F is definedby k x k ≤ R and k x k ≤ b T x − a . For the specific choices q ( x ) = 0 and l ( x ) = 1, our newinequality can also be derived from the following direct argument: the chain of inequalities k x k ≤ R k x k ≤ R ( b T x − a ) linearizes totr( X ) ≤ R ( b T x − a ) . (6)The following example shows that (6) is not captured by R shor ∩ R ksoc : Example 1.
Let F = { x ∈ R : k x k ≤ , k x k ≤ − x − x } . Then (6) is tr( X ) ≤ − x − x .Minimizing the objective − x − x − tr( X ) over R shor ∩ R ksoc yields the optimal solution Y ∗ ≈ . . . . . − . . − . . with (approximate) optimal value − . , i.e., the optimal value is negative, which demon-strates that (6) is not valid for R shor ∩ R ksoc . As far as we aware, inequality (6) for this special case has not yet appeared in the literature.We seek in this paper, however, an even more general procedure for deriving valid inequalitiesusing the ingredients described in the previous paragraph.5he paper is organized as follows. In Section 2, we present the derivation of our new validinequalities and discuss several illustrative choices of q ( x ) and l ( x ). We also specialize theresults to c = 0 and a = 0, a case which further enables the derivation of a similar, secondtype of valid linear inequality in ( x, X ). Then, in Section 3, we show that our inequalitiesinclude those introduced in [9] for the study of the OPF problem, and we extend ourapproach to derive a new class of valid SOC constraints for G when F equals the intersectionof the ball k x k ≤ R and the nonnegative orthant. Next, in Section 4, we prove that theseparation problem for our inequalities—which can be viewed as dynamically choosing thenonnegative functions q ( x ) and l ( x )—is polynomial-time based on the availability of anySDP relaxation in the variables ( x, X ), such as the relaxations R shor or R shor ∩ R ksoc . In thissense, we are able to “bootstrap” any existing SDP relaxation for the separation subroutineto generate valid cuts. Finally, in Section 5, we provide computational evidence that our cutsare effective in further closing the gap between (1) and R shor ∩ R ksoc on randomly generatedproblems, especially in low dimensions. We close in Section 6 with a few final thoughts anddirections for future research.This paper is accompanied by the code repository https://github.com/A-Eltved/strengthened_sdr ,which contains full code for the paper’s examples and computational results. In addition, thefirst author’s forthcoming Ph.D. thesis [12] will contain additional discussion and extensions. In the Introduction, we discussed the valid inequality (6) for the specific case c = 0 and r = 0.Now we assume general c and r . Analogous to (6), we use k x k ≤ R and k x − c k ≤ b T x − a along with the self-duality of SOC to obtain the following quadratic inequality: R − x ! T b T x − ax − c ! ≥ ⇒ R ( b T x − a ) ≥ tr( X ) − c T x. (7)Note that this inequality makes use of the equivalent constraint k − x k ≤ R . We seek tostrengthen it further by incorporating two additional ideas.The first idea involves exploiting the lower bound r ≤ k x k and the RLT-type validinequality ( R − k x k )( k x k − r ) ≥
0. Consider the following proposition: Indeed, our initial motivation for this paper was the desire to understand the inequalities in [9] morefully. roposition 1. Suppose r ≤ k x k ≤ R , and define r k x k − := 0 when k x k = r = 0 . Then r + R (1 + rR k x k − ) x ∈ SOC . (8) Proof. If r = 0, then (8) reads ( R, x ) ∈ SOC, which is true by assumption. So suppose0 < r ≤ k x k . Then we wish to prove(1 + rR k x k − ) k x k = k x k + rR k x k − ≤ r + R, which follows by expanding the valid expression ( R − k x k )( k x k − r ) ≥ k x k ≥ r > r + R − (1 + rR k x k − ) x ! T b T x − ax − c ! ≥ ⇐⇒ ( r + R )( b T x − a ) ≥ x T x + rR − c T x − rR k x k − c T x. However, this inequality cannot be directly linearized in ( x, X ) due to the non-quadraticterm k x k − . So we bound the term r k x k − c T x from above by a problem-dependent constant[ c ] max ≥
0, which satisfies r c T x ≤ [ c ] max x T x for all x ∈ F . We then have the valid linearinequality ( r + R )( b T x − a ) ≥ tr( X ) + rR − c T x − [ c ] max R. (9)Such a [ c ] max clearly exists. For example, [ c ] max = k c k works because r c T x ≤ r k c kk x k ≤ k c kk x k , but naturally it is advantageous to take [ c ] max as small as possible. One method for computinga smaller [ c ] max ≤ k c k is binary search on [ c ] max over the interval [0 , k c k ], where at each stepwe check whether the optimal value ofmin x n [ c ] max x T x − r c T x : k x k ≤ R, k x − c k ≤ b T x − a o is nonnegative. The nonconvex lower bound r ≤ k x k has been excluded from this subproblemto ensure convexity and polynomial-time solvability, which also ensures that the binary searchis polynomial-time overall. Note also that, when r = 0 or c = 0, the optimal [ c ] max equals 0.Our second idea to improve (7) and (9) is to replace ( b T x − a, x − c ) ∈ SOC in the7erivation above with another vector—but one that is still in the second-order cone. Inparticular, we consider the nonnegative combination q x Rx ! + l x b T x − ax − c ! ∈ SOC , (10)where q x := q ( x ) is a quadratic function and l x := l ( x ) is a linear function, both of which arenonnegative for all x ∈ F . This approach is similar to polynomial-optimization approachessuch as the one pioneered in [14], which uses polynomial multipliers with limited degree toderive new, albeit redundant, constraints. Then we have the following generalization of (9): r + R − (1 + rR k x k − ) x ! T Rq x + l x ( b T x − a )( q x + l x ) x − l x c ! ≥ r + R ) Rq x + ( r + R ) l x ( b T x − a ) ≥ ( q x + l x ) x T x + rR ( q x + l x ) − l x c T x − [ c ] max R l x . Note that the right-hand side is quartic in x , and hence this inequality cannot be directlylinearized in the space ( x, X ). Hence, we define[ q + l ] min := min { q x + l x : x ∈ F } ≥ . to get the valid quadratic inequality( r + R ) Rq x + ( r + R ) l x ( b T x − a ) ≥ [ q + l ] min x T x + rR ( q x + l x ) − l x c T x − [ c ] max R l x , (11)which can be easily linearized in ( x, X ) as summarized in the following theorem. Note thatthe theorem requires only that [ q + l ] min be a nonnegative lower bound on the value of q ( x ) + l ( x ) over F . Theorem 1.
Let F be the feasible set of (1), and let [ c ] max ∈ [0 , k c k ] be given such that r c T x ≤ [ c ] max x T x for all x ∈ F . In addition, let q ( x ) := x T H q x + 2 g Tq x + f q and l ( x ) :=2 g Tl x + f l be given such that q ( x ) ≥ and l ( x ) ≥ for all x ∈ F . Also, let [ q + l ] min ≥ bea valid lower bound on the sum q ( x ) + l ( x ) over all x ∈ F . Then the linear inequality ( r + R ) R (cid:16) H q • X + 2 g Tq x + f q (cid:17) + ( r + R ) (cid:16) g l b T • X + ( f l b − ag l ) T x − af l (cid:17) ≥ [ q + l ] min tr( X ) + rR (cid:16) H q • X + 2( g q + g l ) T x + ( f q + f l ) (cid:17) − (cid:16) g l c T • X + f l c T x (cid:17) − [ c ] max R (2 g Tl x + f l ) (12)8 s valid for the convex hull G defined by (5). Note that both sides of (11) contain the term rR q x , and so the presentation of both (11)and (12) could be simplified. However, we leave these slightly unsimplified so as to facilitateour discussion in Section 2.2 below.Let ˆ r be any scalar in [0 , r ]. Since ˆ r ≤ k x k is also valid for F , we can replace r by ˆ r in(12) to obtain an alternate inequality based on ˆ r . In fact, considering ˆ r to be variable inthis inequality while all other quantities are fixed, we see that the inequality is linear in ˆ r ,which implies that all such valid inequalities over ˆ r ∈ [0 , r ] are actually dominated by thetwo extremes ˆ r = 0 and ˆ r = r . We summarize this observation in the following corollary. Corollary 1.
Under the assumptions of Theorem 1, the infinite class of inequalities gottenby replacing r with ˆ r ∈ [0 , r ] is dominated by the two inequalities (12) and R (cid:16) H q • X + 2 g Tq x + f q (cid:17) + R (cid:16) g l b T • X + ( f l b − ag l ) T x − af l (cid:17) ≥ [ q + l ] min tr( X ) − (cid:16) g l c T • X + f l c T x (cid:17) − [ c ] max R (2 g Tl x + f l ) . (13) corresponding to the extremes ˆ r = r and ˆ r = 0 , respectively. In this subsection, we introduce a specialization of our inequalities, which we will return toin Section 3.2.Suppose that we have knowledge of s ∈ R n and λ, µ ∈ R such that F ⊆ S := { x : λ ≤ s T x ≤ µ } , (14)i.e., every x ∈ F satisfies λ ≤ s T x ≤ µ . We call S a valid slab and, abusing notation, we referto S by its tuple ( λ, s, µ ). For example, since F is bounded, for any vector s with k s k = 1,choosing λ = − R and µ = R yields a valid slab. Given any slab ( λ, s, µ ), we discuss twochoices of nonnegative q x and l x .First, define q x := µ − s T x ≥ l x := s T x − λ ≥
0. Note that q x is linear in this case,and [ q + l ] min = q x + l x = µ − λ . Then (11) becomes( r + R ) R ( µ − s T x )+( r + R )( s T x − λ )( b T x − a ) ≥ ( µ − λ )( x T x + rR ) − ( s T x − λ ) c T x − [ c ] max R ( s T x − λ ) . (15)Alternatively, we could also take q x := s T x − λ and l x := µ − s T x to obtain another, similarquadratic inequality. 9econd, given the slab ( λ, s, µ ), we may assume without loss of generality that λ + µ ≥ λ ≤ µ . To see this, we consider three cases. First, if both λ, µ ≥
0, then the statementis clear. Second, if both λ, µ ≤
0, we can use instead the equivalent representation of S by − µ ≤ − s T x ≤ − λ . Finally, if λ < µ ≥ λ + µ <
0, then we can likewise use( − µ, − s, − λ ) instead. Now, with λ + µ ≥ λ ≤ µ , we then define q x := µ − ( s T x ) ≥ l x := ( λ + µ )( s T x − λ ) ≥ q x + l x = µ − ( s T x ) + ( λ + µ ) s T x − λµ − λ = µ + ( µ − s T x )( s T x − λ ) − λ ≥ µ + 0 − λ ≥ . Hence, we obtain (11) with [ q + l ] min := µ − λ ≥ c = 0 , a = 0 , and λ ≥ In this subsection, we derive two cuts—see (18) below—that are closely related to the cutsjust discussed in Section 2.1, and these will play a special role in Section 3.1. We assume c = 0 and a = 0, and we will use a slab ( λ, s, µ ) with λ ≥
0. Note that c = 0 implies[ c ] max = 0.For the first cut, consider the inequality (11) with c = 0 and a = 0, which is furtherrelaxed on the right-hand side:( r + R ) Rq x + ( r + R ) l x b T x ≥ [ q + l ] min x T x + rR ( q x + l x ) ≥ [ q + l ] min ( x T x + rR ) . (16)For the second cut, we consider a pair of functions l x := l ( x ) and p x := p ( x ) that satisfy adifferent relationship than the previously considered l x and q x . Specifically, we assume linear l x ≥ p x ≥
0, and we require l x − p x ≥ x ∈ F as well. We also define[ l − p ] min ≥ l x − p x over F . Then we have the following result. Proposition 2.
Suppose c = 0 , a = 0 , and l x := l ( x ) and p x := p ( x ) are nonnegativefunctions on F such that l x − p x is also nonnegative on F . Then l x b T x − rp x ( l x − p x ) x ! ∈ SOC . Proof. ( l x − p x ) k x k = l x k x k − p x k x k ≤ l x b T x − rp x . r + R − (1 + rR k x k − ) x ! T l x b T x − rp x ( l x − p x ) x ! ≥ , which rearranges and relaxes to( r + R ) l x b T x − ( r + R ) rp x ≥ ( l x − p x ) x T x + rR ( l x − p x ) ≥ [ l − p ] min ( x T x + rR ) . (17)Note that (17) simplifies to R l x b T x ≥ [ l − p ] min x T x when r = 0, which is a consequence ofthe simpler inequality R b T x ≥ x T x ; see (6) with a = 0. In other words, (17) appears to beinteresting only when r > q x , l x , and p x for the inequalities (16) and (17)based on the slab 0 ≤ λ ≤ s T x ≤ µ . We choose q x := µ − ( s T x ) , l x := ( λ + µ ) s T x , and p x := ( s T x ) − λ as the nonnegative functions, resulting in q x + l x = µ − ( s T x ) + ( λ + µ ) s T x ≥ µ + λµ =: [ q + l ] min l x − p x = λ − ( s T x ) + ( λ + µ ) s T x ≥ λ + λµ =: [ l − p ] min , where the inequalities follow from the RLT inequality ( µ − s T x )( s T x − λ ) ≥
0. Pluggingthese into (16)–(17), respectively, and linearizing, we obtain( r + R ) R ( µ − ss T • X ) + ( r + R )( λ + µ ) sb T • X ≥ ( µ + λµ )(tr( X ) + rR ) (18a)( r + R )( λ + µ ) sb T • X − ( r + R ) r ( ss T • X − λ ) ≥ ( λ + λµ )(tr( X ) + rR ) . (18b) In this section, we explore two applications of the inequalities developed in Section 2. Thefirst application shows that the valid inequalities for the optimal power flow problem (OPF)derived in [9] are in fact just special cases of our inequalities, whereas the derivation in[9] was specifically tailored to OPF. Our second application investigates the convex hull of G , where—departing from the form of (1)— F equals the intersection of the ball with thenonnegative orthant, i.e., F possesses polyhedral aspects as well. We study this form of F since it is relevant for any bounded feasible set with nonnegative variables, where the boundis given by a Euclidean ball. 11 .1 Optimal power flow problem In this subsection, we consider a result of Chen et al. [9], which provides an exact formulationfor the convex hull of a nonconvex, quadratically constrained set appearing in the study ofthe optimal power flow (OPF) problem. In particular, the authors added two new linearinequalities to the Shor relaxation in order to capture the convex hull. Whereas these twoinequalities were specifically derived for OPF, we will show that they are just special casesof (18) derived in Section 2.2. For additional background on convex relaxations of OPF, werefer the reader to the two-part survey [15, 16].We restate the result of Chen et al. using their notation. Let J C ⊆ R be the convex hullof the following nonconvex quadratic system: L jj ≤ W jj ≤ U jj ∀ j = 1 , L W ≤ T ≤ U W (19b) W ≥ W W = W + T (19d)where the four variables are ( W , W , W , T ) ∈ R and the data L = ( L , L , L ) and U = ( U , U , U ) satisfy L ≤ U and L jj ≥ j = 1 ,
2. Chen et al.’s interest in thisparticular convex hull arose from an analysis of the OPF problem, where (19) appears as arepeated substructure. As explained in [9], J C can alternatively be expressed as the followingconvex hull using two complex variables z , z ∈ C : J C = conv z z ∗ z z ∗ Re( z z ∗ )Im( z z ∗ ) ∈ R : L jj ≤ z j z ∗ j ≤ U jj ∀ j = 1 , L Re( z z ∗ ) ≤ Im( z z ∗ ) ≤ U Re( z z ∗ )Re( z z ∗ ) ≥ . (20)In particular, equation (19d) is the usual “rank-1” condition, capturing the link betweenthe linear variables ( W , W , W , T ) and the quadratic expressions in z , z . The authorsproved that the pair of linear inequalities π + π W + π W + π W + π T ≥ U W + U W − U U (21a) π + π W + π W + π W + π T ≥ L W + L W − L L (21b)12re valid for J C , where π := − q L L U U π := − q L U π := − q L U π := (cid:18)q L + q U (cid:19) (cid:18)q L + q U (cid:19) − f ( L ) f ( U )1 + f ( L ) f ( U ) π := (cid:18)q L + q U (cid:19) (cid:18)q L + q U (cid:19) f ( L ) + f ( U )1 + f ( L ) f ( U )and where f ( x ) := ( √ x − /x when x > f (0) := 0. In fact, they proved that(21), when added to the Shor relaxation, is sufficient to capture J C : J C = ( W , W , W , T ) : (19a)–(19c) W W ≥ W + T (21) . Here, the convex constraint W W ≥ W + T is equivalent to the regular positive-semidefinite condition.We now relate (21) to our inequalities (18). Defining F := x ∈ R : L ≤ x + x ≤ U L ≤ x ≤ U L x x ≤ x x ≤ U x x x x ≥ , x ≥ . and G by (5), the following proposition establishes an equivalence between J C and G . Proposition 3. J C = { ( X + X , X , X , X ) : ( x, X ) ∈ G} .Proof. Consider (20). Because the quadratic terms z z ∗ , z z ∗ , and z z ∗ are unaffected bya rotation of C applied simultaneously to both z and z , we may enforce Re( z ) ≥ z ) = 0 without changing the definition of J C . Then writing z = x + ix and z = x for x ∈ R , we thus have J C = conv { ( x + x , x , x x , x x ) : x ∈ F ⊆ R } , which provesthe proposition.Our next proposition establishes an alternative form for F , which matches the developmentin Section 2 except that the SOCs involve only two scalar variables, even though F is 3-dimensional. However, the results of Section 2 can easily be adapted to this case, the key13oint being that the Hessians of the SOCs are equal. First we need a lemma. Lemma 1.
For n = 2 , let P := { x ∈ R : Ax ≤ } be a polyhedral cone with A ∈ R × .Then P = { x : k (cid:16) x x (cid:17) k ≤ b T x } for some b ∈ R .Proof. First assume that P is contained in the right side of the plane, i.e., P ⊆ { x : x ≥ } and that P is symmetric about the x axis. Then, for some β ≥ P = { x : x ≥ , − βx ≤ x ≤ βx } = { x : x ≥ , x ≤ β x } = { x : x ≥ , x + x ≤ (1 + β ) x } = { x : k (cid:16) x x (cid:17) k ≤ √ β x } , which proves the result in this case. For general P , we may apply an orthogonal rotationto revert to the previous case, which does not affect the norm k (cid:16) x x (cid:17) k (but does change theexact form of b ).We next state and prove the proposition. Note that the assumptions L > U > L in the proposition are realistic for power networks: the first ensures the voltage magnitudeat a bus is positive, and the second allows for a positive voltage-angle difference between theinvolved buses. Proposition 4.
Suppose L > and U > L . Then F = x ∈ R : √ L ≤ (cid:13)(cid:13)(cid:13)(cid:16) x x (cid:17)(cid:13)(cid:13)(cid:13) ≤ √ U (cid:13)(cid:13)(cid:13)(cid:16) x x (cid:17)(cid:13)(cid:13)(cid:13) ≤ b x + b x √ L ≤ x ≤ √ U where b and b uniquely solve the system L U b b = q L q U . Proof.
The assumption L > x >
0, which in turn implies F = x ∈ R : L ≤ x + x ≤ U √ L ≤ x ≤ √ U L x ≤ x ≤ U x x ≥ . U > L makes x ≥ F is equivalent to √ L ≤ k (cid:16) x x (cid:17) k ≤ √ U .To complete the proof, we claim that L x ≤ x ≤ U x is equivalent to the SOCconstraint k (cid:16) x x (cid:17) k ≤ b x + b x . Indeed, it is clear that the set defined by these two linearinequalities is a polyhedral cone with the two extreme rays r = (cid:16) L (cid:17) and r = (cid:16) U (cid:17) . So, bythe lemma, the set is SOC-representable in the form k (cid:16) x x (cid:17) k ≤ b x + b x for some b ∈ R .In particular, the extreme rays r j must satisfy k r j k = b T r j . By plugging in the values of r and r , we get the 2 × b , as desired. Note that the 2 × U − L is positive.Based on Propositions 3 and 4, we now prove that (21) is simply (18) tailored to theOPF case. Theorem 2.
Inequalities (21) are the inequalities (18) tailored to system (19).Proof.
By Proposition 3, we can translate (21a) to the variables ( x, X ). After collectingterms, (21a) becomes( π + U U ) + ( π − U )( X + X ) + ( π − U ) X + π X + π X ≥ . (22)Using Proposition 4, consider (18a) with the following replacements: x ← x x ! , r ← q L , R ← q U , λ ← q L , s T x ← x , µ ← q U . This results in the following valid inequality: (cid:18)q L U + U (cid:19) X + X + √ L U √ L + √ U ≤ (cid:18)q L + q U (cid:19) ( b X + b X ) + ( U − X ) q U . Simple, although tedious, algebraic manipulations establish that this inequality is precisely(22). A similar argument establishes that (21b) corresponds to (18b). We also verified numerically that (21) is not captured by R shor ∩ R ksoc in this case. We provide Matlab code for these manipulations at the website https://github.com/A-Eltved/strengthened_sdr . .2 Intersection of the ball and nonnegative orthant As stated in the Introduction, the critical feature of F studied in this paper is its intersectionof the ball with a second SOC-representable set, which shares the Hessian identity matrix.However, there are of course many other forms of F that can be of interest in practice.For example, when F is the nonnegative orthant, then G is the completely positive cone,which can be used to model many NP-hard problems as linear conic programs [4]. Anothercommon case is when F is a box, e.g., the set [0 , n [6].Let us examine the case in which F is the intersection of the nonnegative orthant andthe unit ball. For general n , define F := { x ≥ k x k ≤ } ⊆ R n . Since x ∈ F ⇒ k x k ≤ k x k = e T x, we have F ⊆ { x : k x k ≤ , k x k ≤ e T x } , (23)and for n = 2, one can actually show that (23) is an equation. Since F is a subset ofthe nonnegative orthant, any inequality, which is valid for the completely positive cone,is also valid for F , but here we focus on the implied structure in (23). Section 2 applieswith r = 0 , R = 1 , c = 0 , b = e , and a = 0. In particular, the constraints tr( X ) ≤ X ) ≤ e T x are valid for G ; see the Introduction and inequality (6).We can strengthen tr( X ) ≤ X ) ≤ e T x using the slab inequalities of Section 2.1.Geometrically, given any s ∈ R n with s ≥ k s k = 1, we have the slab λ := 0 ≤ s T x ≤ µ , which is valid for F : 0 ≤ s T x ≤ k s kk x k = k x k ≤ . After linearization, inequality (15) in this case reads 1 − s T x + s T Xe ≥ tr( X ). Moreover,if we switch the role of q x and l x in (15)—recall that q x is linear for slabs—then we have s T x + e T x − s T Xe ≥ tr( X ). Rearranging, we write these two inequalities astr( X ) ≤ s T ( Xe − x ) (24a)tr( X ) ≤ e T x − s T ( Xe − x ) . (24b)Letting s vary over its constraints k s k = 1 and s ≥
0, we derive a compact SOC-representationof this class of inequalities over various domains of G .16 heorem 3. Let ( I, J ) be a partition of the index set { , . . . , n } , and define the domain D IJ := ( x, X ) : [ Xe − x ] I ≥ Xe − x ] J ≤ . Then the following SOC constraints are locally valid for G on D IJ : tr( X ) ≤ − k [ Xe − x ] J k (25a)tr( X ) ≤ e T x − k [ Xe − x ] I k . (25b) Moreover, (25) imply all valid inequalities (15) derived from slabs of the form ≤ s T x ≤ ,where s is any vector satisfying k s k = 1 and s ≥ .Proof. Consider the constraints (24), and for notational convenience, define y := Xe − x .Because s ≥
0, the quantity s T y on the right-hand side of (24a) breaks into s TI y I ≥ s TJ y J ≤ D IJ . By minimizing the right-hand side of (24a) with respect to s , we achieve thetightest cut corresponding to s = ( s I , s J ) = (0 , − y J / k y J k ), which yields tr( X ) ≤ − k y J k ,as desired. A similar argument for (24b) yields tr( X ) ≤ e T x − k y I k .We remark that, when I is empty, inequality (25b) reduces to the inequality tr( X ) ≤ e T x over D IJ . Similarly, when J is empty, (25a) is tr( X ) ≤ x, X ),solve the relaxation to obtain an optimal solution (¯ x, ¯ X ). Then define the partition ( I, J )and corresponding domain D IJ according to ¯ Xe − ¯ x . Then, if either of the inequalitiesin (25) is violated, we can derive a violated supporting hyperplane of the SOC constraint.After adding the violated linear inequality to the current relaxation, which is globally validbecause it is linear, we can resolve and repeat the process.We close this section with an example showing that the cuts derived above are not impliedby R shor ∩ R ksoc . Example 2.
Let n = 2 , and consider I = { , } and J = ∅ . Then tr( X ) ≤ e T x −k Xe − x k isvalid on the domain D IJ = { ( x, X ) : Xe − x ≥ } . In particular, tr( X ) ≤ e T x − u T ( Xe − x ) for all vectors u satisfying k u k = 1 , and taking u = e , we have tr( X ) ≤ e T x − [ Xe − x ] ,which is globally valid since it is linear. Minimizing e T x − [ Xe − x ] − tr( X ) over R shor ∩R ksoc yields the optimal value − . , indicating that R shor ∩ R ksoc does not capture this validconstraint. Separation
In this section, we argue that the inequalities (12)–(13) given by Theorem 1 and Corollary1 are separable in polynomial time. To state this result precisely, we assume that [ c ] max hasalready been pre-computed and that a fixed convex relaxation of the convex hull G definedby (5) is available. For convenience, we write this fixed convex relaxation R := n ( x, X ) : Y ( x, X ) ∈ c R o ⊇ G , where Y ( x, X ) is given by (3) and c R is a closed, convex cone in the space of ( n + 1) × ( n + 1)symmetric matrices. In particular, R is just the slice of c R with the top-left corner of Y setto 1. Then the relaxation of (1) over R can be stated as min { H • X + 2 g T x : ( x, X ) ∈ R} with dual max y : − y g T g H ∈ c R ∗ where c R ∗ is the dual cone of c R . We state this general form for ease of notation and tomake evident that one can choose different R in computation. For example, one could take R = R shor at one extreme or R = R shor ∩ R ksoc at the other.In fact, to separate (12)–(13) we will use the following observation concerning R , c R , and c R ∗ : Observation.
Given a quadratic function q ( x ) := x T H q x + 2 g Tq x + f q , if there exists y ∈ R such that − y + f q g Tq g q H q ∈ c R ∗ , then q ( x ) ≥ y for all x ∈ F . This observation follows by weak duality because y is a lower bound on the optimal relaxationvalue of H q • X + 2 g Tq x + f q over ( x, X ) ∈ R , which is itself a lower bound on the minimumvalue of q ( x ) over x ∈ F . As a result, the following system guarantees that the conditions ofTheorem 1 on q ( x ) and l ( x ) hold, where ( H q , g q , f q ), ( g l , f l ), and [ q + l ] min are the variables: f q g Tq g q H q ∈ c R ∗ , f l g Tl g l ∈ c R ∗ , (26a)[ q + l ] min ≥ , − [ q + l ] min + f q + f l ( g q + g l ) T g q + g l H q ∈ c R ∗ . (26b)Then, separation amounts to optimizing the linear function in (12)—or (13) as the case18ay be—over (26) for fixed values of ( x, X ). However, before we state the exact separationproblem for (12), we require one additional assumption, namely that F is full-dimensional,i.e., there exists ˆ x ∈ F such that k ˆ x k < k ˆ x − c k < b T x − a . In this case, it iswell known that G and hence R are also full-dimensional in ( x, X )-space. In particular,(ˆ x, ˆ x ˆ x T ) ∈ int( G ) ⊆ int( R ), and henceˆ Y := x ! x ! T ∈ int( c R ) . It thus follows by standard duality theory that c R ∗ ∩ { J : ˆ Y • J ≤ } is a bounded truncationof c R ∗ . This truncation is important so that the separation problem below is bounded andthus has a well-defined optimal value.We are now ready to state the separation subproblem for (12) given fixed values (¯ x, ¯ X )of the variables ( x, X ):min ( r + R ) R (cid:16) H q • ¯ X + 2 g Tq ¯ x + f q (cid:17) + ( r + R ) (cid:16) g l b T • ¯ X + ( f l b − ag l ) T ¯ x − af l (cid:17) (27a) − [ q + l ] min tr( ¯ X ) − rR (cid:16) H q • ¯ X + 2( g q + g l ) T ¯ x + ( f q + f l ) (cid:17) + (cid:16) g l c T • ¯ X + f l c T ¯ x (cid:17) + [ c ] max R (2 g Tl ¯ x + f l ) (27b)s.t. (26) (27c)ˆ Y • f q g Tq g q H q ≤ , ˆ Y • f l g Tl g l ≤ . (27d)The subproblem for (13) is similar—just replace r with 0.We remark that system (26) could be simplified in certain cases. For example, if r = 0and hence F is convex, then it is not difficult to see that the second condition of (26a),which ensures that l ( x ) is nonnegative over F , could be replaced by a dual system based on F alone, not on R . One could also simplify by forcing additional structure on q ( x ) and l ( x ).For example, one could separate against the slabs λ ≤ s T x ≤ µ introduced in Section 2.1by forcing ( H q , g q , f q ) = (0 , − s, µ ), ( g l , f l ) = ( s, − λ ), and [ q + l ] min = µ − λ , in which case(26b) is automatically satisfied.The following example demonstrates the separation procedure, whose implementationwill be discussed in the next section: 19 xample 3. Consider the -dimensional problem min − x − x − . x − x s.t. k x k ≤ k x k ≤ − x − x with H = − I , g = ( − . , − . , r = 0 , R = 1 , a = − , b = ( − , − , and c = (0 , in (1) . All values reported here are truncated from the computations and therefore approximate.The optimal value of min { H • X + 2 g T x : ( x, X ) ∈ R shor ∩ R ksoc } is − . with optimalsolution ¯ x = . − . , ¯ X = . − . − . . . Solving the separation subproblem at (¯ x, ¯ X ) , we obtain the cut corresponding to q ( x ) = x T − . − . x + 2 − . − . x + 0 . ,l ( x ) = 2 . . x + 1 , [ q + l ] min = 1 . . We add the corresponding cut, resolve to obtain a new (¯ x, ¯ X ) , and repeat this loop two moretimes, resulting in the cuts q ( x ) = x T − . . . − . x + 2 − . − . x + 1 ,l ( x ) = 2 . . x + 1 , [ q + l ] min = 1 . ,q ( x ) = x T − . . . − . x + 2 − . − . x + 1 ,l ( x ) = 2 . . x + 1 , [ q + l ] min = 1 . . e finally obtain the rank-1, and hence optimal, solution Y ( x ⋆ , X ⋆ ) = . − . . . − . − . − . . with objective value − . . We note that, even though the procedure generates three cuts,the last cut is actually enough to recover the rank-1 solution. Moreover, running this proce-dure starting from R shor instead of R shor ∩ R ksoc , we also get the same optimal ( x ⋆ , X ⋆ ) afteradding 16 cuts. To quantify the practical effect of the cuts proposed in Theorem 1 and Corollary 1, we embedthe separation subproblem described in Section 4 in a straightforward implementation tosolve random instances of the form (1). We consider two relaxations to “bootstrap” theseparation procedure: R shor and R shor ∩ R ksoc . We will denote by R cuts the points ( x, X )satisfying the added cuts, so that our improved relaxations will be expressed as R shor ∩ R cuts and R shor ∩ R ksoc ∩ R cuts .We implement our experiments in Matlab 9.6 (R2019a) using CVX [11] to model therelaxations and MOSEK 9.1 [17] to solve them. We run the problem instances on a singlecore of an Intel Xeon E5-2650v4 processor using a maximum of 2GB memory. We do notreport complete run times because we are most interested in the strength of the addedcuts, but we do report the number of cuts added to measure the overall effort. Recall thatcalculating a single cut requires solving the separation problem (27) described in Section 4,which in essence involves three copies of the current bootstrap relaxation— R shor ∩ R cuts or R shor ∩ R ksoc ∩ R cuts . However, to give the reader a sense of the run times, consider thefollowing: for an instance of our largest dimension, n = 10, solving R shor took approximately0.6 seconds, solving R shor ∩ R ksoc required about 50 seconds, and solving a single separationproblem for R shor ∩ R ksoc took approximately 64 seconds. We note that our implementationis rudimentary and makes no effort to take advantage of, for example, any particular problemstructure or sparsity, so these times can probably be improved significantly.We generate a single random instance by fixing the dimension n and generating randomdata a, b, c, r, R, H, g in such a way that (1) is feasible with a known interior point ˆ x , whichis also randomly generated. In short, we first set R = 1 without loss of generality, generate r uniformly in [0 , R ], generate ˆ x uniformly in { x : r ≤ ˆ x ≤ R } , generate b, c, H, g with entries21.i.d. standard normal, and finally set a := b T ˆ x − k ˆ x − c k − θ , where θ is uniform in [0 ,
1] sothat F has a nonempty interior. Recall that ˆ x is required for the separation procedure asdiscussed in Section 4. Before running the separation procedure for an instance, we compute[ c ] max by a binary search on [ c ] max over the interval [0 , k c k ] as discussed in Section 2. Then,when running the overall algorithm, we consider the current relaxation’s optimal solution(¯ x, ¯ X ) to be separated if: the objective value of the separation subproblem (27) is less than τ sep = − − ; or the optimal value of the separation subproblem for the inequalities (13)in Corollary 1, i.e., (27) with r = 0, is less than τ sep . If (¯ x, ¯ X ) is indeed separated, we addthe resulting cut represented by the data ( H q , g q , f q , g l , f l , [ q + l ] min ) to the current bootstraprelaxation, optimize for a new point to be separated, and repeat. The overall loop stopswhen the current (¯ x, ¯ X ) is not separated with tolerance τ sep .Regarding a given relaxation and its optimal solution (¯ x, ¯ X ), we say the relaxation is exact if Y (¯ x, ¯ X ) satisfies λ ( Y (¯ x, ¯ X )) λ ( Y (¯ x, ¯ X )) > τ rank , where λ ( M ) denotes the largest eigenvalue of M , λ ( M ) denotes the second largest eigen-value of M , and τ rank > in our implementation,ensuring that Y (¯ x, ¯ X ) is numerically rank-1. We define the gap as the difference betweenthe optimal value of (1) and the relaxation optimal value. Note that an exact relaxationimplies a gap of 0.After running the algorithm on a particular instance, we classify the instance into one oftwo categories: exact initial or inexact initial , when the initial bootstrap relaxation is exactor inexact, respectively. Furthermore, we break all inexact-initial instances into one of threesubcategories: improved , when the initial relaxation gap is improved but not completelyclosed to 0; closed , when the relaxation becomes exact after adding one or more cuts; and noimprovement , when no cuts are successfully added to improve the gap, i.e., the separationroutine does not help. (Actually, in the tables below, we will not directly report informationabout the exact-initial and no-improvement instances, as these details will be implicitlyavailable from the other categories.)We conduct these experiments for several values of n and many randomly generatedinstances. In addition, we also consider special cases where some of the data a, b, c, r, R isfixed to zero in order to assess whether the cuts are more effective in these special cases. Inparticular, we consider the following three cases: the general case, where no data is fixed a priori to zero; the special case with r = a = 0 and c = 0; and the case of the TTRS We refer the reader to our GitHub site ( https://github.com/A-Eltved/strengthened_sdr ) for thefull random-generation procedure. r = 0 and b = 0. For each of these cases, we generate15,000 instances for each dimension 2 ≤ n ≤
10, and we solve each instance twice, oncebootstrapping from R shor and once from R shor ∩ R ksoc .For the improved and closed instances, we report the average number of cuts added. Alsofor the improved instances, we report the average gap closure in percentage terms, i.e., wereport the average relative gap closure. Since we do not actually know the optimal valueof (1) for the improved instances, to approximate the relative gap closure from above, wecalculate a local minimum value, v local , by taking the lowest value of the quadratic objectivefunction gotten by running Matlab’s fmincon with 100 random initial points. The relativegap for the instance is then calculated asrelative gap closure = v relax final − v relax initial v local − v relax initial × , where v relax initial is the optimal value of the initial relaxation and v relax final is the optimalvalue of the final relaxation. We consider 15,000 random instances for each dimension 2 ≤ n ≤
10 and report the resultsseparately for the R shor and R shor ∩R ksoc bootstrap relaxations in Tables 1 and 2, respectively.In Table 1, we see that our cuts improve the R shor relaxation in many instances. For n = 2, it improves more than a third of the inexact instances, and it closes the gap for about9%. As the dimension goes up, these proportions go down, suggesting that our cuts are moreeffective in lower dimensions. n Inexact initial Improved Avg cuts Avg gap closure Closed Avg cuts2 2923 1188 15 51% 264 43 2582 761 17 46% 175 74 2161 422 10 40% 53 75 1801 416 10 36% 46 96 1583 265 12 36% 29 87 1360 186 11 36% 10 118 1091 140 14 39% 15 79 1029 107 12 34% 4 1510 896 86 13 30% 4 11Table 1: Results for the R shor bootstrap relaxation on 15,000 random general instances foreach dimension n . The columns Inexact initial , Improved , and
Closed report the number ofinstances out of 15,000 in each category.Table 2 shows that R shor ∩ R ksoc is generally quite strong for instances of the form (1).23specially for larger n , the number of inexact instances is small, and the ability of our cutsto improve or close the gaps is limited. In particular, for n ≥ n Inexact initial Improved Avg cuts Avg gap closure Closed Avg cuts2 251 40 13 45% 3 33 84 5 36 48% 0 —4 44 0 — — 0 —5 16 0 — — 0 —6 6 0 — — 0 —7 7 0 — — 0 —8 2 0 — — 0 —9 3 0 — — 0 —10 3 0 — — 0 —Table 2: Results for the R shor ∩ R ksoc bootstrap relaxation on the same 15,000 randomgeneral instances as depicted in Table 1 for each dimension n . The columns Inexact initial , Improved , and
Closed report the number of instances out of 15,000 in each category. r = a = 0 and c = 0 We next consider the special case when F equals { x ∈ R n : k x k ≤ , k x k ≤ b T x } with b ∈ R n . Note that, by rotating the feasible space, we may assume without loss of generalitythat b lies in the direction of e , the all ones vector. In particular, we generate instanceswith b = βe , where β ∈ [1 / √ n, / √ n + 2 n ]. The choice of this interval for β is based onthe following observation: for β < / √ n the feasible space F is empty; for β = 1 / √ n thefeasible space F has no interior; for β → ∞ , the constraint k x k ≤ b T x resembles the halfspace 0 ≤ e T x .Similar to Tables 1–2 of the previous subsection, Tables 3–4 contain the results of ourseparation algorithm on 15,000 randomly generated instances for each dimension, whereTable 3 corresponds to R shor and Table 4 to R shor ∩ R ksoc . Contrary to what we saw in thegeneral case in Tables 1–2, there does not seem to be a drop in the proportion of instanceswhere the cuts help as n increases. Overall, our cuts seem to be quite effective in this specialcase.Specifically for n = 2, the results in Table 4 suggest that R shor ∩ R ksoc ∩ R cuts is tight,i.e., it captures the convex hull G . To test this further, we generated an additional 110,000instances with n = 2. The R shor ∩ R ksoc relaxation was exact for 109,938 of these, and ourcuts closed the gap for the remaining 62 instances with an average of 3 cuts added. Our24 Inexact initial Improved Avg cuts Avg gap closure Closed Avg cuts2 7744 2755 22 82% 4988 23 7635 914 23 86% 6495 34 7736 395 13 83% 6966 35 7709 401 4 81% 6596 36 7584 402 5 67% 7182 37 7648 185 5 87% 7463 38 7614 131 8 89% 7483 39 7566 77 7 93% 7489 210 7552 44 7 89% 7508 2Table 3: Results for the R shor bootstrap relaxation on 15,000 random instances with r = a = 0 and c = 0 for each dimension n . The columns Inexact initial , Improved , and
Closed report the number of instances out of 15,000 in each category. n Inexact initial Improved Avg cuts Avg gap closure Closed Avg cuts2 15 0 — — 15 23 50 7 43 37% 30 24 36 4 78 75% 28 25 29 0 — — 27 36 15 3 8 88% 12 37 13 2 4 57% 11 28 12 0 — — 12 29 6 0 — — 5 110 6 0 — — 5 3Table 4: Results for the R shor ∩ R ksoc bootstrap relaxation on the same 15,000 randominstances as depicted in Table 3 with r = a = 0 and c = 0 for each dimension n . Thecolumns Inexact initial , Improved , and
Closed report the number of instances out of 15,000in each category.computational experience thus motivates a conjecture:
Conjecture 1.
For the 2-dimensional feasible space F := { x ∈ R : k x k ≤ , k x k ≤ b T x } with arbitrary b ∈ R , R shor ∩ R ksoc ∩ R cuts equals the convex hull G defined in (5) . In addition, in Section 3.2, for n = 2 and b = e , we proposed the locally valid cuts (25), whichwere derived from slabs of a particular form. (Note that these cuts would not necessarilybe valid for a different scaling b = βe .) By generating many random objectives, we wereable to find 100 additional instances, which were not solved exactly by R shor ∩ R ksoc , andthen separated just these locally valid cuts—instead of the more general cuts represented by R cuts . All 100 instances were solved exactly, i.e., achieved the tolerance τ rank . We believethis is strong evidence to support the following conjecture as well: Conjecture 2.
For the 2-dimensional feasible space F := { x ∈ R : k x k ≤ , k x k ≤ e T x } =25 x ≥ k x k ≤ } , the constraints defined by R shor ∩ R ksoc intersected with the locally validcuts (25) capture the convex hull G defined in (5) . b = 0 and r = 0 ) Setting b = 0 and r = 0 in (1) with a < ≤ n ≤
10 and bootstrap from the R shor and R shor ∩ R ksoc relaxations. The results areshown in Tables 5 and 6. The trends in these tables are similar to what we saw in the generalcase in Section 5.1. In particular, our cuts are less effective in higher dimensions. n Inexact initial Improved Avg cuts Avg gap closure Closed Avg cuts2 1404 364 16 33% 86 43 1287 172 15 27% 34 44 985 79 12 27% 20 55 745 34 9 22% 7 36 508 14 7 22% 3 27 454 4 5 25% 2 38 347 5 8 58% 0 —9 293 0 — — 1 210 251 1 4 2% 0 —Table 5: Results for the R shor bootstrap relaxation on 15,000 random TTRS instances foreach dimension n . The columns Inexact initial , Improved , and
Closed report the number ofinstances out of 15,000 in each category. n Inexact initial Improved Avg cuts Avg gap closure Closed Avg cuts2 31 4 20 24% 0 —3 78 7 43 29% 1 74 63 3 55 19% 0 —5 34 1 59 6% 0 —6 22 0 — — 0 —7 16 0 — — 0 —8 14 0 — — 0 —9 6 0 — — 0 —10 4 0 — — 0 —Table 6: Results for the R shor ∩ R ksoc bootstrap relaxation on the same 15,000 randomTTRS instances as depicted in Table 5 for each dimension n . The columns Inexact initial , Improved , and
Closed report the number of instances out of 15,000 in each category.We catalog the following example showing an explicit case for n = 2 in which our cutsclose the gap for TTRS compared to just applying R shor ∩ R ksoc .26 xample 4. Consider the instance with n = 2 , r = 0 , R = 1 , a = − . , and b = , c = − . . , H = − .
32 0 . . − . , g = − . . . The (approximate) optimal value of min { H • X + 2 g T x : ( x, X ) ∈ R shor ∩ R ksoc } is − . and the solution is not rank-1. Solving the separation problem starting from this relaxation,we obtain the (approximate) cut corresponding to g l = . − . , f l = 4 . , [ q + l ] min = 1 . ,H q = − . . . − . , g q = − . . , f q = 2 . . Solving the relaxation with this cut, results in the (numerically) rank-1 solution Y ( x ⋆ , X ⋆ ) = . − . . − . . − . . − . . with (approximate) optimal value − . . In this paper, we have derived a new class of valid linear inequalities for SDP relaxations ofproblem (1). These cuts are separable in polynomial time, which, by the equivalence of sep-aration and optimization, ensures that the SDP relaxation enforcing all of these inequalitiesis polynomial-time solvable. We have also shown that a special case of our cuts has been ap-plied by Chen et al. [9] to obtain the convex hull of an important substructure arising in theOPF problem. In addition, we have extended our methodology to derive new, locally valid,second-order-cone cuts for nonconvex quadratric programs over the mixed polyhedral-conicset { x ≥ k x k ≤ } . Using specific examples as well as computational experiments, wehave demonstrated that the new class of valid inequalities strengthens the strongest knownSDP relaxation, R shor ∩ R ksoc , especially in low dimensions.For the specific 2-dimensional feasible set F = { x ∈ R : k x k ≤ , x ≤ b T x } , ourcomputational experiments indicate that our cuts intersected with R shor ∩ R ksoc capture therelevant convex hull G . We leave this as a conjecture requiring further research. Furthermore,27hen b = e , we also conjecture that the locally valid cuts (25), which are derived from slabs,are by themself enough to capture G . For general F , however, our cuts do not close the gapfully, and so there remains room for improvement.One limitation of our approach is the assumption that the SOC constraint (1c) sharesthe identity Hessian with the hollow ball (1b). If instead we are presented with a generalSOC constraint k J x − c k ≤ b T x − a , where J ∈ R n × n is arbitrary, one idea would be tobound b T x − a ≥ k J x − c k≥ k x − c k − k x − J x k = k x − c k − k ( I − J ) x k≥ k x − c k − q λ max [( I − J ) T ( I − J )] R, which yields the valid constraint k x − c k ≤ b T x − (cid:16) a − q λ max [( I − J ) T ( I − J )] R (cid:17) , to whichour methodology can be applied. Additional options for handling arbitrary Hessians can beconsidered by refining the deriviations of Section 2.Further opportunities for future research include streamlining the separation subroutine,investigating the effectiveness of our cuts in higher dimensions, and examining other ap-plications where the structure of (1) appears. Also, the idea of using the self-duality of acone to derive valid linear cuts could be applied to other self-dual cones or possibly evennon-self-dual cones. Acknowledgments
The authors acknowledge the support of their respective universities, which allowed the firstauthor to visit the second author in 2019–20, when this research was initiated.
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