arXiv:2102.02162v1 [math.OC] 3 Feb 2021 01AscainfrCmuigMachinery. Computing for Association 2021 © C utb ooe.Asrcigwt rdti emte.T permitted. is credit with Abstracting honored. be must ACM C SN978-x-xxxx-xxxx-x/YY/MM...$15.00 ISBN ACM C eeec Format: Reference ACM ABSTRACT https://doi.org/10.1145/nnnnnnn.nnnnnnn SA’1 uy22,SitPtrbr,Russia Petersburg, Saint 2021, [email protected] July from ISSAC’21, permissions requ Request lists, fee. to a redistribute to and/or or servers on post to work publish, this of components n for this Copyrights bear page. copies first that the and on thi tion advantage of commercial ar part copies or that or profit provided all for fee of without granted copies is hard use or classroom digital make to Permission ytm onm e 1] n en bet ffiinl ov P solve efficiently importance. to great able of being and is dyn [13], of few a analysis name and to ap systems control many optimization, are global There general. in in cations science and mathematics, of eas INTRODUCTION 1 (ISSAC’21). Conference ACM In The of Optimization. 2021. Noncommutative Magron. in Victor Property and Trace Bhardwaj, Constant Abhishek Mai, Anh Hoang Ngoc ming p semidefinite property, trace constant Lagrangian, mented gradient- conditional optimization, trace and eigenvalue hermitia of sums optimization, polynomial noncommutative KEYWORDS noncommutativ lem. the sol of to relaxations methods semidefinite numerical the p order ficiently first trace via constant exploited this be can how erty demonstrate trace then constant We a variable. with program trix semidefinite a as cast be can lem optimizatio polynomial noncommutative ball-constrained a nlge oteeapoce nten etn,cmn f coming setting, nc the geometry algebraic in free approaches a these There to positivity. analogues global of ral certificates provide to aim ods problem solutio moment Schmüdgen’s by Inspired polynomials. positive for from results representation powerful using grams rl[6 n unu nomto 6 9 21]. 19, [6, information quantum and [26] i trol applications several has NCPOP variables. non-commuting problems mization h osatTaePoet nNnomttv Optimizatio Noncommutative in Property Trace Constant The nti ril,w hwta ahsmdfiierlxto of relaxation semidefinite each that show we article, this In nti ril efcson focus we article this In oyoilotmzto problems optimization Polynomial ic h deto neirpitmtosfor methods point interior of advent the Since SP 2,teehv enmn prahst ovn POP, solving to approaches many been have there [2], (SDP) gcHagAhMai Anh Hoang Ngoc olue France Toulouse, [email protected] ncmatsmagbacst 2] hs meth- these [25], sets semialgebraic compact on NPP,ta s oyoilotmzto with optimization polynomial is, that (NCPOP), LAAS 9,adthe and [9], C,NwYr,N,UA pages. 8 USA, NY, York, New ACM, ocmuaie(c oyoilopti- polynomial (nc) noncommutative rca oetproblems moment tracial PP r rsn nmn ar- many in present are (POP) elagbacgeometry algebraic real rspirseicpermission specific prior ires o aeo distributed or made not e g. oyohrie rre- or otherwise, copy o okfrproa or personal for work s tc n h ulcita- full the and otice we yohr than others by owned eient pro- semidefinite [email protected] ae aug- based Proceedings bihkBhardwaj Abhishek rogram- othe to n enatu- re prob- n prob- e amical squares, n olue France Toulouse, con- n [4]. eef- ve rom rop- ma- OP pli- a LAAS hsheacyadisn xeso oegnau/rc opt eigenvalue/trace to extension nc its and hierarchy This hc r h nlg ftecmuaiewrsaotC [27], CS about [31]. works CS-TS commutative and 30] the [29, of TS analogs the are which aibe,wt h iuto en os nten setting. nc the in worse being situation the with variables, aie(c polynomials (nc) tative l in als ments aua osrit codn oPtnrsPositivstelle Putinar’s so to under according convergence constraints guaranteed natural with POPs, for values timal Hierarchy in[,2] nov ovn Dsoe h pc fmultivar of space respectively. the matrices, over Hankel nc SDPs and solving moment involve 23], [5, tion i.e., fe etitn hi s oplnmaso o ere,o degrees, s low “moderate” of polynomials of to are use their matrices restricting Hankel) Often nc the when (or applied moment be only tivariate can hierarchies these 20], [1, Mosek priy(S n[1 n emsast T) ST n[8 al [28] in CS-TS (TS), sparsity term corre and include [11] They in POPs. (CS) of sparsity structures sparsity the exploit hwdhwt xli the exploit to how showed . ocmuaiepolynomials Noncommutative 2.1 [5]. to referred is topics reader the the to section, introduction this detailed in more duced a For sequel. the in PRELIMINARIES & DEFINITIONS 2 cor- with NCPOPs and NCPOPs dense sparsity. relative of classes sample app some this demon on of also efficiency We computational CTP. and with numerical SDPs the strate solving for us methods makes order package first The CTP. of with NCPOPs solving for package lia provid We Secondly, sparsity. CTP-framewor correlative this with extend NCPOPs also some We holds. CTP cons then polydisc) present, nc cla (or is broad ball a nc for if particular, CTP In the NCPOPs. ensure dense which 18], [17, to obta results we ogous First, following: the are contributions main two Our ipedmis .. pee al nuu,bxadsimple and box annulus, ball, sphere, e.g., constrai domains, POPs relaxations, simple for SDP gains firs required computational utilizing the significant solve By attained to trace. methods constant spectral have der relaxations inv SDP matrices the the when in satisfied is which POPs, of relaxations h - 1 tnadapoc ntecmuaiestig is setting, commutative the in approach standard A u otecretcpct fitro-on D ovr su solvers SDP interior-point of capacity current the to Due taeyfrrdcn h ieo h D irrhe sto is hierarchies SDP the of size the reducing for strategy A norgdb 7 3,i 1,1]tefis n hr authors third and first the 18] [17, in 33], [7, by Encouraged nti ril,w xedteepotto fteCPt NCPO to CTP the of exploitation the extend we article, this In ednt by denote We needed knowledge preliminary basic some introduce we Here - , . . . , F ∗ - words = ihcecet in coefficients with - = 1] hc rvdsasqec oe onso h op- the on bounds lower sequence a provides which [12], i 8 A ethe be in - . . . - - ie word a Given . 8 h ocmuigletters noncommuting the 1 osdrtefe algebra free the Consider . remonoid free Endow . osatTaeProperty Trace Constant R [email protected] t lmnsaecalled are elements Its . R itrMagron Victor olue France Toulouse, F h eeae by generated - i = ihteinvolution the with - LAAS 8 1 - . . . - 8 1 A R - - , . . . , , h n alisele- its call and , F - ∗ i CP o SDP for (CTP) sisreverse, its is fpolynomi- of = noncommu- Let . st [24]. nsatz Lasserre’s 5 traint(s) x. nfew in r nanal- in e on ned roach, → imiza- h Ju- a e lative olved intro- - has ch sof ss they mul- or- t n to k iate of l i me ize. Ps. 5 = e ∗ - ISSAC’21, July 2021, Saint Petersburg, Russia Mai, Bhardwaj and Magron which fixes R - pointwise. The length of the longest word We say that & g h is Archimedean if for all @ R - , there is ∪ ( ) + ( ) ∈ h i in a polynomial 5 R - is called the degree of 5 and is de- a positive ' N such that ' @ @ & g h . ∈ h i ∈ − ∗ ∈ ( ) + ( ) noted deg 5 . We write R - for all nc polynomials of degree ( ) h i3 2.2.2 Semialgebraic sets. We define the semialgebraic set associ- at most 3. The set of symmetric elements of R - is defined as h i ated to g as SymR - = 5 R - : 5 ∗ = 5 . We employ the graded lexico- h i ∈ h i = graphic ordering on all structures and objects we consider. g = S : 6 g, 6 0 . D ∈ ∀ ∈ ( ) We write - for the set of all words in - of degree at most h i3 h i We can naturally extend this notion from matrix tuples of the same 3, and we let W - W be the column vector of words in - , 3 ( )≡ 3 h i3 order, to bounded self-adjoint operators on some Hilbert space , and V - V the column vector of words of degree 3. We also H 3 ( )≡ 3 which make 6 psd for all 6 g. This extension is called the denote by W3 (resp. V3 ) the set of all entries of W3 - (resp. ( ) ∈ ( ) operator semialgebraic set associated to g, and we denote it as g∞. V - ). The length of W is equal to s 3,= := 3 =8, which D 3 ( ) 3 ( ) 8=0 Similarly we define the variety associated to h as we write as s 3 , when contextually appropriate. Given a poly- = ( ) s 3 Í h = S : ℎ h,ℎ = 0 , nomial 5 R - 3 , let f = 5F F W R ( ) be its vector of ∈ h i ( ) ∈ 3 ∈ D ∈ ∀ ∈ ( ) coefficients. It is clear that every polynomial 5 R - 3 is of and the natural extension to the operator variety . ∈ h i h∞ the form 5 = 5 F = f W = W f. For 5 R - let D F W3 F ∗ 3 3∗ ∈ ∈ h i 2.2.3 Hankel matrices and the Riesz functional. Suppose we have 5 = deg 5 2 , and given some : N, we define :5 := : 5 , ⌈ ⌉ ⌈ ( )/Í⌉ ∈ < −⌈ ⌉ e.g., W = W . We use standard notations on R , i.e., given a truncated real valued sequence y = ~F F W23 . For each such : 5 :5 ( ) ∈ R< −⌈ ⌉ sequence, we define the Riesz functional, !y : R - R as a , a 2 denotes the usual 2-norm of a. h i23 → ∈ SAk k !y @ := @F~F for @ = @FF R - . Let denote the space of real symmetric matrices of size A, we ( ) F F ∈ h i23 Suppose further that y satisfies~F = ~F for allF W23 . We as- will normally omit the subscript A when we discuss matrices of Í Í ∗ ∈ S sociate to such y the nc Hankel matrix of order 3, M y , defined as arbitrary size, or if the size is clear from context. Given A , 3 ( ) ∈ M y D,E = !y D E , where D,E W . Given @ SymR - , we A is positive semidefinite (psd) (resp. positive definite (pd)), if all ( 3 ( )) ( ∗ ) ∈ 3 ∈ h i define the localizing matrix M @y as M @y D,E = !y D @E , eigenvalues of A are non-negative (resp. positive), and we write 3@ ( ) ( 3@ ( )) ( ∗ ) A where now D,E W . A 0 (resp. A 0). We denote by Tr A the trace ( 8,8) of 3@ ≻ ( ) 8=1 ∈ the matrix A SA and tr A = 1 Tr A is the normalized trace. Let ∈ ( ) A ( ) Í S (resp. S ) be the cone of psd (resp. pd) matrices. For a subset 2.3 Eigenvalue minimization + S, we++ define := S and := S . We write Given 5 SymR - , g, h SymR - , the minimal eigenvalue S ⊆ S+= S∩ + S++ S∩ ++ ∈ h i ⊂ h i = A1,..., A= S , and given @ R - , by @ we mean the of 5 over g∞ ∞ is given by: ( ) ∈ ∈ h i ( ) D ∩ Dh evaluation of @ - on , i.e., replacement of the nc letters -8 with ( ) the matrices A8 . We write diag B1,..., BA for the block diagonal _min 5 , g, h = inf v∗ 5 v : g∞ h∞, v 2 = 1 . (2.1) ( ) ( ) ( ) ∈ D ∩ D k k matrix with diagonal blocks being B . 8 We will assume that then eigenvalue minimization problem (EG)o (2.1) Finally, given a positive < N, we write N < = <,< 1,... , ∈ ≥ { + } has at least one global minimizer. We can approximate the solution < = 1,...,< , and we use to denote the cardinality of a set. [ ] { } |·| of EG (2.1) from below with a hierarchy of converging SOHS relax- ations [23], indexed by : N: 2.2 Algebraic and geometric structures ∈ d: := sup b R : 5 b &: g : h . Let g = 60,...,6< and h = ℎ1,...,ℎℓ be subsets of SymR - , { ∈ − ∈ ( ) + ( )} { } { } h i with the requirement that 60 = 1, unless otherwise stated. Each relaxation gives rise to the following SDP < 2.2.1 adratic modules. The quadratic module generated by g is = 5 b Tr G8 W:6 68 W∗ − 8 :68 the set 8=0 Õ ℓ < A 9 d: = sup b . (2.2) 9 9 9 N 1 R b,G8 ,H9 Tr H9 W:ℎ ℎ 9 W∗ , & g := ? ( )∗68? ( ) : A 9 ≥ ,? ( ) - . + 9 :ℎ 9 ( ) 8 8 ∈ 8 ∈ h i 9=1 8=0 9=1 Õ Õ Õ H9 S, and G8 0 ∈ The ideal generated by the set h is the set h := & ℎ1,..., ℎℓ . ( ) ({± ± }) Our primary interest is in the dual formulation of this SDP, which The quadratic module associated to g = 60 , is the set of sums of { } can be stated as Hermitian squares (SOHS). ~1 = 1, M y 0, Given : N, the :th-order truncation of & g (resp. h ), de- : ( ) ∈ ( ) ( ) M 68 y 0,8 < , noted by &: g (resp. : h ), is the set of all polynomials in & g g: := inf !y 5 :68 . (2.3) Rs 2: ( ) ∈ [ ] ( ) ( ) ( ) y ( ) ( ) (resp. h ) with degree at most 2:. Moreover, one has ∈ M: ℎ 9 y = 0, 9 ℓ ( ) ℎ 9 ( ) ∈ [ ] < Let g = &: Tr G8W:6 68 W∗ : G8 0 , 8 :68 :min := max 5 , 68 , ℎ 9 : 8 < , 9 ℓ . ( ) ( = ( ) ) {⌈ ⌉ ⌈ ⌉ ⌈ ⌉ ∈ [ ] ∈ [ ]} Õ8 0 When & g h is Archimedean, both d: N :min and g: N :min ℓ ( )+ ( ) ( ): ≥ ( ): ≥ converge to _ 5 , g, h due to an nc analog∈ of Putinar’s Posi-∈ h = Tr H8W ℎ8W∗ : H8 S . min : :ℎ8 :ℎ ( ) ( ) ( = ( 8 ) ∈ ) tivstellensatz [8]. Õ8 1 The Constant Trace Property in Noncommutative Optimization ISSAC’21, July 2021, Saint Petersburg, Russia
2.4 Trace minimization Letting D: y := diag M: y , M: 61y ,..., M: 6 Let us start first, with cyclic equivalence. Given two polynomials and we can similarly reformulate SDP (2.7) to ?,@ R - , we say that ? is cyclically equivalent to @ if ? @ is cyc ∈ h i : − ~1 = 1, and ~D = ~E if D E, a sum of commutators, i.e., ? @ = 8=1 D8E8 E8D8 for some ∼ − cyc ( − ) tr : N R g: := inf !y 5 D: y ( ), . (3.2) : and D8, E8 - , and we write ? @. One can now define y Rs 2: ( ) ( )∈S ∈ ∈ h i cyc Í ∼ ∈ ( ) + the cyclic quadratic module & g , as the set of all polynomials M: ℎ 9 y = 0, 9 ℓ ( ) ℎ 9 ( ) ∈ [ ] 5 SymR - which are cyclically equivalent to some element of ∈ h i & g (see [5, Definition 1.56]). Definition 3.1 (CTP) . We say that an NCPOP has CTP if for every ( ) : : We cannot in general work with the sets , , since the al- : N min , there exists 0 > 0 and P ( ) such that for all Dg∞ Dh∞ ∈ ≥ : : ∈ S gebra of bounded operators over a Hilbert space does not admit y Rs 2: , ++ H ∈ ( ) a trace if is infinite dimensional. Instead we restrict to a certain H M ℎ 9 y = 0, 9 ℓ , subset of finite von Neumann algebras of type I and II, a subset :ℎ 9 ( ) ∈ [ ] Tr P:∗ D: y P: = 0: . of the algebra of bounded operators on , and we denote this by ~1 = 1 ) ⇒ ( ( ) ) II1 H g . Then we consider the following relaxation of (2.4): D In other words, we say that NCPOP (2.1) or (2.5) has CTP if each II1 II1 II1 dual SDP relaxation (3.1) or (3.2) has an equivalent form involving tr 5 , g, h = inf tr 5 : g . (2.5) min ( ) ( ( )) ∈ D ∩ Dh a psd matrix whose trace is constant. In this case, 0 is the constant n o : A discussion of von Neumann algebras is beyond the scope of this trace and P: is the change of basis matrix. The next proposition is article, and we refer the reader to [5, Definition 1.59] for more de- an example of an NCPOP which has CTP. tails. An SOHS relaxation hierarchy, indexed by : N, for (2.5) ∈ can be written as Proposition 3.2 (nc polydisc eality). Let < = 0, ℓ = and 2 ≥ ℎ 9 = - 9 1, for 9 = . Then dtr := sup b R : 5 b &cyc g cyc h (2.6) − ∈ [ ] : ∈ − ∈ : ( ) + : ( ) M ℎ 9 y = 0, 9 ℓ , n o :ℎ 9 which once again, can be written and solved as an SDP. The dual ( ) ∈ [ ] Tr D: y = s : . (3.3) formulation of this SDP, which is our primary interest, is ~1 = 1 ) ⇒ ( ( )) ( ) cyc ~1 = 1, and ~D = ~E if D E, Proof. Note that D y = M y since g = 1 . Suppose that ∼ : ( ) : ( ) { } M y 0, M: ℎ 9 y = 0, 9 ℓ , and ~1 = 1. This implies that for every 9 tr : ℎ 9 ( ) ∈ [ ] ∈ g := inf !y 5 ( ) . (2.7) 2 : s 2: = , the diagonal of M ℎ y is zeros, i.e., ! D - 1 D = 0, y R ( ) M: 68 y 0,8 < , :ℎ 9 y ∗ 9 ∈ ( ) 68 ( ) ∈ [ ] [ ] 9 ( ) ( ( − ) ) = M ℎ y = 0, 9 ℓ for all D W: 1. This now implies, for every F -81 ...-8A W: :ℎ 9 ∈ − ∈ 9 ( ) ∈ [ ] = = ~F∗F !y F∗F !y -8A ...-81-81 ...-8A Compared to the relaxation (2.3) for EG, (2.7) has several addi- ( ) ( ) 2 tional linear constraints arising from cyclic equivalences. Under = !y -8A ...-82 -8 1 -82 ...-8A tr ( ( 1 − ) ) Archimedeanity of & g , d : is monotonically increas- : : N≥ min ! - ...- - ...- ( ) ( ) ∈ y 8A 82 82 8A ing, and converges to trII1 5 , g, h , see [5, Corollary 5.5]. + ( ) min = !y -8 ...-8 -8 ...-8 ( ) ( A 2 2 A ) 2 = = !y -8 -8 = !y - 1 !y 1 = ~1 = 1. 3 CTP FOR NC OPTIMIZATION ··· ( A A ) ( 8A − ) + ( ) In this section we develop a framework which exploits CTP for = = This yields Tr M: y F W: ~F∗F s : . NCPOPs. Our results below hold for both eigenvalue (2.1) and trace ( ( )) ∈ ( ) Í (2.5) minimization hierarchies (2.3) and (2.7) respectively. We pro- A general solution method for solving NCPOPs which satisfy vide sufficient conditions under which CTP is guaranteed, as well CTP can be described as follows. We first convert the :-th order re- as simple linear programming methods to check these conditions. laxation (3.1) or (3.2) to a standard (primal) SDP with CTP and then We conclude by examining some special cases. leverage appropriate first-order algorithms, such as CGAL [32] or spectral method (SM) [17, Appendix A.3], which exploit CTP to 3.1 CTP for Dual Hierarchies solve the SDP. We first give a precise definition of CTP for NCPOP. Recall the For a detailed exposition on how the SDP (3.1) or (3.2) can be : < sets g and h from §2. For every : N min , define s := s :6 , converted to a standard (primal) form, the reader is invited to con- ∈ ≥ : 8=0 ( 8 ) : s: sult [18]. There one will also find explanations of how the pri- and the set ( ) S as S ⊆ Í mal and dual forms of the SDP are related, and their use with : s s : := Y S : : Y = diag Y0,..., Y< , and each Y8 S 68 . S ( ) ∈ ( ) ∈ ( ) CGAL/SM. n o ISSAC’21, July 2021, Saint Petersburg, Russia Mai, Bhardwaj and Magron 3.2 Sufficient condition to have CTP Proof. Let : N. From Lemma 3.4, we obtain that ∈ We now provide a sufficient condition for NCPOP to satisfy CTP. F F = 1 F F : ∗ ∗ For : N≥ min , let &◦ g be the interior of the truncated quadratic + ∈ : ( ) F W: A : F VA module & g , i.e., Õ∈ Õ∈[ ] Õ∈ : ( ) < A 1 2 = 1 : 2D( − )D∗ - 9 1 D , &◦ g := Tr G8W:6 68 W∗ : G8 0 . + + − : ( ) ( 8 :68 ) ≻ A : D WA 1 9 = (8=0 ) Õ∈[ ] ∈Õ− © Õ∈[ ] ª Õ : 1 A 1 ® Theorem 3.3. Suppose that for every : N, the following inclu- yielding the selection3D( − ) = A deg D 1 2D( − ) ,forD W: 1 ∈ ∈[ ( )+ ] « ∈¬ − sion holds: in (3.8). Hence the desired result follows. >0 Í R &◦ g . (3.4) ⊂ : ( ) The next result shows that CTP is satisfied whenever an NCPOP Then NCPOP (2.1) and (2.4) satisfy CTP. involves a ball constraint. For a real symmetric matrix A, denote : Proof. Let : N min and 0 > 0 such that 0 & g . Then the largest eigenvalue of A by _max A . ∈ ≥ : : ∈ :◦ ( ) ( ) we can write < 2 Theorem 3.6. If 1 9 = - 9 g then the inclusions (3.4) hold = − ∈[ ] ∈ 0: Tr G8 W:6 68 W:∗ , (3.5) and therefore NCPOP (2.1) and (2.4) have CTP. 8 68 Í Õ8=0 1 2 2 with each G S . We denote by G / the square root of G . Set Proof. Without loss of generality, set 6< := 1 9 = - 9 and 8 ++ 8 8 − ∈[ ] 1 ∈2 1 2 N :min P = diag G / ,..., G / . From this and (3.5), let : ≥ be fixed. By Lemma 3.5, Í : ( 0 < ) ∈ = = < = < 0: Tr W: W:∗ Tr G Remark 3.8. Combining the proof of Theorem 3.6 with the proof 3.4 Universal algorithm of Theorem 3.3, one can obtain explicit expressions for 0: and P: Algorithm 1 below solves EG (2.1) where CTP can be verified by in Definition 3.1. Namely, 0 = 1 : and : + LP. A similar algorithm solves NCPOP (2.4). 1 2 1 2 P = diag G / , √XI ,..., √XI , G / . : 0 :61 :6< 1 < − Algorithm 1 SpecialEP-CTP However, in our experience this choice leads to poor numerical : Input: EG (2.1) and a relaxation order : N≥ min properties. In the next section we provide a hierarchy of linear ∈ Output: The optimal value g: of SDP (3.1) programs (LPs) inspired from the inclusions (3.4), to obtain the con- 1: Solve LP (3.9) to obtain an (optimal) solution b: , G8,: , H9,: ; stant trace 0 and the change of basis matrix P which achieve a 1 2 1 2 ( ) : : 2: Let 0 = b and P = diag G / ,..., G / ; better numerical performance. : : : ( 0,: <,: ) 3: Compute the optimal value g: of SDP (3.1) by running an algorithm based on first-order methods, and which exploits CTP. 3.3 Verifying CTP via linear programming For any : N :min , let : be the set of real diagonal matrices ∈ ≥ S ( ) Two examples of algorithms based on first-order methods and of size s : and consider the following linear program (LP) ( ) which exploit CTP are CGAL [32] or SM [17, Appendix A.3]. b = Tr G8 W:6 68 W∗ 8 :68 4 CTP WITH CORRELATIVE SPARSITY 8=0 Õ In this section, we show that the CTP can analogously be ex- ℓ inf b , (3.9) ploited for EG (2.1) with sparse nc polynomials. For brevity, we b,G8 ,H9 Tr H9 W:ℎ ℎ 9 W:∗ , + 9 ℎ 9 focus on EG, and show only the framework for correlative sparsity 9=1 Õ (CS) [11], however the trace minimization setting, as well as the :6 G I ( 8 ),8 0 < 8 8 frameworks for term sparsity (TS) as well as correlative-term spar- − ∈ S+ ∈ { }∪[ ] where I is the identity matrix of size B : for 8 0 < . sity (CS-TS) [28] are very similar. We note that the proofs are very 8 b 68 ( ) ∈ { }∪[ ] similar to those presented in §3, and so we omit them for brevity. Lemma 3.9. If LP (3.9) has a feasible solution b:, G8,:, H9,: for To begin with, we define CS and present the associated approx- : ( ) every : N≥ min ,then NCPOP (2.1) and (2.4) have CTP with 0: = b: imation hierarchies for EG (2.1) satisfying CS, which were initially ∈ 1 2 1 2 and P = diag G / ,..., G / . proposed in [11, 27]. : ( 0,: <,: ) The proof of Lemma 3.9 is similar to that of Theorem 3.3 with 4.1 EPs with CS 0: = b: and G8 = G8,: , 8 0 < . = = ∈ { }∪[ ] For F -81 ...-8A , let var F : 81,...,8A . For = , let We provide in the following proposition a more detailed descrip- ( ) { } ⊆ [ ] - := - 9 : 9 and W - := F W3 - : var F tion of some feasible solutions to (3.9) in the special cases of the nc ( ) ∈ 3 ( ) { ∈ ( ) ( ) ⊆ - with length s 3, := 3 8. Similarly, we note V - := polydisc and the nc ball. ( )} ( | |) 8=0| | 3 ( ) F V3 - : var F - . Given y = ~F F W , and 2 ∈ ( ) ( ) ⊆ ( )Í ( ) ∈ 23 ⊆ Proposition 3.10. Suppose either g = 1, 1 8 = -8 or that = , the nc Hankel submatrix associated to of order 3 is defined − ∈[ ] [ ] g = 1 - 2 : 8 = 1 . ThenLP (3.9) has a feasible solution as = 8 − ∈: [ ] ∪{ } Í for every : N min , and therefore NCPOP (2.1) and (2.4) satisfy M3 y, := !y D∗E , for D, E W ∈ ≥ ( ( ))D,E ( ) ∈ 3 CTP. and for @ R - , the localizing (sub)matrix is ∈ h ( )i Proof. It is sufficient in both cases to show that (3.9) has a fea- M3 @y, D,E := !y D∗@E , for D,E W . ( @ ( )) ( ) ∈ 3@ sible solution for every : N :min . ∈ ≥ 2 Assume that 9 9 ? (with = 9 := 9 ) are the maximal cliques Let< = 1 and61 = 1 9 = - 9 . By Lemma 3.5, 0: = Tr W: W∗ | | − ∈[ ] ( : )+ of (a chordal extension∈[ ] of) the correlative sparsity pattern (csp) Tr G W 6 W , where G = diag 3 : 1 is pd. 1 : 1 1 :∗ 1 Í 1 D( − ) F W: 1 ( − ) (( ) ∈ − ) graph associated to EG (2.1), as defined in [11, 27]. Let 9 9 ? Denote by I the− identity matrix of size B : . Thus with large ∈[ ] B : ( ) (resp. , ) be a partition of < (resp. ℓ ) such that for all enough A > 0,( ) A0 , AI ,AG , 0 is a feasible solution of (3.9), 9 9 ? : B : 1 ∈[ ] [ ] [ ] ( : ( ( ) ) ) 8 9 , 68 R - 9 (resp. 8 ,9 , ℎ8 R - 9 ), for every for every : N≥ min . ∈ ∈ h ( )i ∈ ∈ h ( )i ∈ 1 2 9 ? . For each 9 ? , let < 9 := 9 , ; 9 := ,9 and g 9 := On the other hand, if < = = and 69 = = - 9 , for 9 = , then ∈ [ ] ∈ [ ] | | | | − ∈ [ ] 68 : 8 9 , h, := ℎ8 : 8 ,9 . Then & g (resp. h, ) is a by Lemma 3.5, 0: = Tr W: W∗ Tr GW: 1 9 = 69 W∗ = ∈ 9 ∈ ( 9 ) ( 9 ) ( : ) + ( ( ) : 1) quadratic module (resp. an ideal) in R - , for each 9 ? . Tr W W Tr GW 6 W , −where∈[ the] diagonal− ma- 9 : :∗ 9 = : 1 9 :∗ 1 : h ( )i ∈ [ ] ( ) + ∈[ ] ( − − ) Í For each : N≥ min , consider the hierarchy of sparse SOHS trix G = diag 3 : 1 is pd. Thus with large enough ∈ Í D( − ) F W: 1 relaxations > (( ) ∈ − ) A 0, A0:, AIB : ,AG , 0 is a feasible solution of (3.9), for every ( ( ( ) ) ) N :min : ≥ . dcs := sup b : 5 b & g h . (4.1) ∈ : − ∈ ( 9 ): + ( ,9 ): 9 ? Since small constant traces are highly desirable for efficiency of Õ∈[ ] first-order algorithms (e.g. CGAL), we search for an optimal solu- This relaxation can be stated as a primal SDP similar to (2.2), but tion of LP (3.9) instead of just a feasible solution. we are mostly interested in the dual SDP, which can be stated as ISSAC’21, July 2021, Saint Petersburg, Russia Mai, Bhardwaj and Magron follows Lemma 4.3. Let EG (2.1) with CS be as described in §4.1. If LP 9 9 9 : ~1 = 1, M: y, 9 0, 9 ? . (4.4) has a feasible solution b ( ), G( ) , H( ) , for every : N min ( ) ∈ [ ] ( : 8,: 8,: ) ∈ ≥ cs M 6 y, 0,8 , 9 ? , 9 g := inf !y 5 :68 8 9 9 . (4.2) and for every 9 ? , then EG (2.1) satisfies CS-CTP with P( ) = : Rs 2: ( ) ∈ ∈ [ ] : y ( ) ( ) ∈ [ ] ∈ = 9 1 2 9 1 2 9 9 : M:ℎ ℎ8y, 9 0,8 ,9 , 9 ? = N min 8 ( ) ∈ ∈ [ ] diag G( ) / , G( ) / 8 8 and 0 ( ) b ( ) , for : ≥ (( 0,: ) (( 8,: ) ) ∈ ) : : ∈ It is shown in [11, Corollary 6.6] that the primal-dual SDP pair and for 9 ? . ∈ [ ] arising from (4.1) are guaranteed to converge to the optimal value Similar to §3, two special cases where CS-CTP can be verified if there are ball constraints present on each clique of variables. through LP (4.4), are the nc polydisc, and the nc ball on each clique 4.2 Exploiting CTP with correlative sparsity of variables. Consider EG (2.1) with CS described in §4.1. Given 9 ? , : Proposition 4.4. Let EG (2.1) with CS be as described in §4.1. ∈ [ ] ∈ N :min , and y Rs 2: , let Suppose either of the following holds ≥ ∈ ( ) = g = 2 D: y, 9 : diag M: y, 9 , M:6 68y, 9 8 9 , Case 1: 9 1 8 9 -8 , 9 ? . ( ) ( ( ) ( 8 ( )) ∈ ) • − ∈ ∈ [ ] with size denoted by B . Then SDP (4.2) can be rewritten as g = n 1 2 o :,9 Case 2: 9 Í -8 : 8 9 , 9 ? . • | 9 | − ∈ ∈ [ ] ~1 = 1, D y, 9 0, 9 ? , n o :min cs : Then LP (4.4) has a feasible solution for every : N≥ , and there- g = inf !y 5 ( ) ∈ [ ] . (4.3) ∈ : Rs 2: M ℎ y, = 0,8 , , 9 ? fore EG (2.1) satisfies CS-CTP. y ( ) ( ( ) :ℎ 8 9 9 ) ∈ 8 ( ) ∈ ∈ [ ] As in the dense case, let us define The proof of the Proposition 4.4 is similar to the dense setting. :,9 s s :,= 9 ( ) := Y S :,9 : Y = diag Y0, Y8 8 , Y0 S ( ) S ∈ ( ( ) ∈ 9 ) ∈ 4.4 Universal algorithm Algorithm 2 below solves EG (2.1) with CS and whose CS-CTP s : ,= and each Y8 S 68 8 . ∈ ( ) can be verified by solving LP (4.4). We define CTP for EP with CS as follows. Algorithm 2 SpecialEP-CS-CTP Definition 4.1. (CS-CTP) We say that EG (2.1) with CS has CTP if N :min : Input: An EG (2.1) with CS and a relaxation order : ≥ for every : N≥ min and for every 9 ? , there exists a positive cs ∈ ∈ ∈ [ ] Output: The optimal value g: of SDP (4.2) 9 9 :,9 Rs 2: number 0:( ) and P:( ) ( ) such that for all y ( ) , 1: for 9 ? do ∈S++ ∈ ∈ [ ] 9 9 9 = 2: Solve LP (4.4) to obtain an optimal solution b:( ) , G8,:( ) , H9,:( ) ; M:ℎ ℎ8y, 9 0,8 ,9 , 9 9 9 ( ) 8 ( ) ∈ Tr P D y, P = 0 . 9 9 9 9 9 :( ) : 9 :( ) :( ) = = 1 2 1 2 ~1 = 1 ⇒ ( ) 3: Let 0:( ) b:( ) and P:( ) diag G0(,:) / ,..., G<,:( ) / ; (( ) ( ) ) 4: cs The following result provides a sufficient condition for an EG Compute the optimal value g: of SDP (4.3) by running an algorithm (2.1) with CS to satisfy CTP. based on first-order methods and which exploits CTP. Theorem 4.2. Assume that there is an nc ball constraint on each 2 g clique of variables, i.e., 1 8 9 -8 9 , for every 9 ? . Then > − ∈ ∈ ∈ [ ] 5 NUMERICAL EXPERIMENTS one has R 0 & g , for all : N :min and for all 9 ? . As a ⊆ :◦ ( 9 ) Í ∈ ≥ ∈ [ ] consequence, EG (2.1) has CTP. In this section we report results of numerical experiments con- ducted on the eigenvalue minimization problem (2.1). These results A proof of Theorem 4.2 can be obtained in a similar fashion to were obtained by executing Algorithm 1 and Algorithm 2, respec- §3.2, by considering each clique of variables. tively for dense and sparse randomly generated instances of nc quadratically constrained quadratic problems (QCQPs) with CTP. 4.3 Verifying CS-CTP via linear programming In the dense case, one computes the first and second order SDP As in the dense case, given an EG (2.1) with CS, we can verify relaxations, namely the optimal values g1 and g2 of SDP (3.1). Sim- if CS-CTP is satisfied via a hierarchy of LPs. For every : N :min ilarly in the sparse case, one computes the optimal values gcs and ∈ ≥ 1 and for every 9 ? , let :,9 be the set of real diagonal matrices gcs of SDP (4.3). The experiments are performed in Julia 1.3.1 with ∈ [ ] S ( ) 2 of size s :,= 9 and consider the following LP the following software packages: ( ) 9 b :6 ,9 NCTSSOS [28] is a modeling library for solving Moment-SOS ( 8 ) G8 I8( ) ,8 9 0 , • − ∈ S+ ∈ ∪{ } relaxations of sparse EPs based on JuMP (with Mosek 9.1 b = Tr G W 9 6 W 9 used as SDP solver). b 8 : 8 : ∗ inf b 68 ( 68 ) , (4.4) Arpack [15] is used to compute the smallest eigenvalues and 8 Õ0 9 • b,G8 ,H8 ∈{ }∪ the corresponding eigenvectors of real symmetric matrices 9 9 Tr H8 W ℎ8 W ∗ of (potentially) large size, which is based on the implicitly + :ℎ8 ( :ℎ8 ) 8 ,9 restarted Arnoldi method [14]. Õ∈ 9 where I( ) is the identity matrix of size B :6 , 9 , for every 8 0 Both implementation of Algorithm 1 and 2 are available online: 8 ( 8 ) ∈ { }∪ 9 . https://github.com/maihoanganh/ctpNCPOP. The Constant Trace Property in Noncommutative Optimization ISSAC’21, July 2021, Saint Petersburg, Russia Table 1: Numerical results for randomly generated dense QCQPs Table 2: Numerical results for randomly generated dense QCQPs with nc ball constraint with nc polydisk constraints EP size: < = 1, ; = = 4 ; SDP size: l = 2, 0max = 3; CGAL EP size: < = =, ; = = 7 ; SDP size: l = = 1, 0max = 3, CGAL • 4 ⌈ / ⌉ • 4 ⌈ / ⌉ + accuracy: Y = 10− . accuracy: Y = 10− . EP size SDP size Mosek CGAL EP size SDP size Mosek CGAL = ; : Bmax Z val time val time = ; : Bmax Z val time val time 1 11 5 -3.2413 1 -3.2411 2 1 11 13 -3.2165 0.009 -3.2154 0.4 10 3 10 2 2 111 815 -3.1110 28 -3.1107 59 2 111 1343 -3.2039 26 -3.2037 229 1 21 7 -3.5534 0.03 -3.5525 1 1 21 24 -4.5773 0.03 -4.5767 2 20 5 20 3 2 421 5587 -3.5026 203 2 421 9514 -4.5147 753 − − − − 1 31 10 -4.6984 0.1 -4.6954 1 1 31 36 -5.1182 0.8 -5.1172 3 30 8 30 3 2 931 18415 -4.6819 1392 2 931 31311 -5.0717 2215 − − − − We use a desktop computer with an Intel(R) Core(TM) i7-8665U approximate optimal value which differs from 1% w.r.t. the one re- CPU @ 1.9GHz 8 and 31.2 GB of RAM. turned by Mosek when = 20. Mosek runs out of memory when × ≤ We use the following notation for the numerical results. The = 20 and : = 2, while CGAL still works well. With our current ≥ number of variables, inequality and equality constraints are de- setting for the CGAL accuracy, the approximate optimal value is noted by =, < and ;, respectively. We denote by : the relaxation correct up to the two first accuracy digits, so we can guarantee that order used to solve the dense SDP (3.1) and the sparse SDP (4.3). the bound improves from : = 1 to : = 2. For NCPOP with CS, let us denote by Dmax the largest size of vari- able cliques and ? the number of variable cliques. We note l, Bmax, 5.2 Randomly generated QCQPs with CS Z and 0max the number of psd blocks, the largest size of psd blocks, Test problems: We construct randomly generated nc QCQPs with the number of affine equality constraints and the largest constant CS and ball constraints on each clique of variables as follows: trace of matrices involved in the SDP relaxations, respectively. Let “val” stand for the approximate optimal value of the SDP relaxation (1) Take a positive integer D, ? := = D 1 and let ⌊ / ⌋ + with desired accuracy Y for CGAL, and let “time” be the correspond- ing running time in seconds. We use “ ” to indicate that the calcu- D , if 9 = 1 , − [ ] lation runs out of space. For all examples tested in this paper, the 9 = D 9 1 ,...,D9 , if 9 2,...,? 1 , (5.1) modeling time for both NCTSSOS and ctpNCPOP is typically negligi- { ( − ) } ∈ { − } D ? 1 ,...,= , if 9 = ? ; ble compared to the solving time of Mosek and CGAL. Hence the { ( − ) } total running time mainly depends on the solvers and we compare (2) Generate a quadratic polynomial objective function 5 = 9 ? 59 ∈[ ] their performances below. ¯ 9 such that for each 9 ? , 59 = 5 ( ) F F ∈ [ ] F W 9 F ( + ∗Í) ∈ ∈ 2 SymR - 9 2, and the coefficients are randomly generated 5.1 Randomly generated dense QCQPs h ( )i Í as in the dense setting; Test problems: We construct randomly generated dense QCQPs 2 (3) Take < = ? and 69 := 1 - , 9 < . with nc ball and nc polydisk constraints as follows: − 8 9 8 ∈ [ ] (4) Take a random point a as in the∈ dense setting; Í (1) Generate a dense quadratic polynomial objective function (5) Let A := ; ? and 1 ⌊ / ⌋ 5 = 5¯F F F SymR - 2 with coefficients 5¯F 2 F W2 ( + ∗) ∈ h i randomly∈ chosen w.r.t. the uniform probability distribution 9 1 A 1,...,9A , if 9 ? 1 , Í ,9 := {( − ) + } ∈ [ − ] (5.2) on 1, 1 . = (− ) ( ? 1 A 1,...,; , if 9 ? . (2) Do one of the following two cases: {( − ) + } 2 nc ball: let < = 1 and 61 := 1 A = -A ; For every 9 ? and every 8 ,9 , generate a quadratic • − ∈[ ] ∈ [ ] ∈ nc polydisk: let < = = and 6 := 1 - 2, 8 = ; 1 ¯ 8 R 8 = 8 polynomial ℎ8 = 9 ℎ ( ) F F∗ Sym - 9 2 • = Í − ∈ [ ] 2 F W F (3) Take a random point a in G R : 68 G 0,8 < w.r.t. 2 ( + ) ∈ h ( )i { ∈ ( ) ≥ ∈ [ ]} as in the dense setting to∈ ensure that a is a feasible solution the uniform distribution; Í of EG (2.1). (4) For every 9 ℓ , generate a dense quadratic polynomial ∈ [ ] = 1 ¯ 9 R The number of variables is fixed as = = 1000. We increase the ℎ 9 2 F W2 ℎF( ) F F∗ Sym - 2: ∈ ( + ) ∈ h i 9 clique size D so that the number of variable cliques ? decreases (i) for each F W2 1 , select a random coefficient ℎ¯( ) Í ∈ \{ } F accordingly. The numerical results are displayed in Table 3. Again in 1, 1 w.r.t. the uniform distribution; (− 9 ) 9 results in Table 3 show that CGAL is slower than Mosek for : = 1 (ii) set ℎ¯( ) := ℎ¯( )F a . 1 − F W2 1 F ( ) but is faster for : = 2 and returns an approximate optimal value Then a is a feasible solution∈ \{ of} EG (2.1). which differs from 1% w.r.t. the one returned by Mosek (for D 10). Í ≤ The numerical results are displayed in Tables 1 and 2. The re- Mosek runs out of memory for : = 2 when D 15, while CGAL is ≥ sults show that CGAL is typically the fastest solver and returns an once again able to obtain improved lower bounds. ISSAC’21, July 2021, Saint Petersburg, Russia Mai, Bhardwaj and Magron Table 3: Numerical results for randomly generated QCQPs with CS [3] T. Barthel and R. Hübener. Solving condensed-matter ground-state problems by and nc ball constraint on each clique of variables semidefinite relaxations. Physical review letters, 108(20):200404, 2012. [4] S. Burgdorf and I. Klep. The truncated tracial moment problem. Journal of Operator Theory, pages 141–163, 2012. EP size: = = 1000, < = ?, ; = 143, Dmax = D 1; SDP size: l = 2?, [5] S. Burgdorf, I. Klep, and J. Povh. Optimization of polynomials in non-commuting • max 3 + variables, volume 2. Springer, 2016. 0 = 3; CGAL accuracy: Y = 10− . [6] S. Gribling, D. de Laat, and M. Laurent. Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization. EP size SDP size Mosek CGAL Mathematical Programming, 170(1):5–42, 2018. D ? : l Bmax Z val time val time [7] C. Helmberg and F. Rendl. A spectral bundle method for semidefinite program- 1 200 12 541 -2.9659 3 -2.9662 206 ming. SIAM Journal on Optimization, 10(3):673–696, 2000. 10 100 [8] J. Helton and S. McCullough. A positivstellensatz for non-commutative poly- 2 200 133 91850 -2.9594 32008 -2.9598 7790 nomials. Transactions of the American Mathematical Society, 356(9):3721–3737, 1 132 27 405 -2.3230 1 -2.3225 38 2004. 15 66 2 132 703 185592 -2.3179 10051 [9] J. W. Helton, I. Klep, and S. McCullough. Free convex algebraic geometry. Semi- − − 1 100 22 341 -2.1517 4 -2.1515 54 definite optimization and convex algebraic geometry, 13:341–405, 2013. 20 50 [10] C. Josz and D. Henrion. Strong duality in lasserre’s hierarchy for polynomial 2 100 463 290908 -2.1260 11791 − − optimization. Optimization Letters, 10(1):3–10, 2016. [11] I. Klep, V. Magron, and J. Povh. Sparse noncommutative polynomial optimiza- tion. arXiv preprint arXiv:1909.00569, 2019. 6 CONCLUSION [12] J. B. Lasserre. Global optimization with polynomials and the problem of mo- ments. SIAM Journal on optimization, 11(3):796–817, 2001. We have provided a constructive proof that the constant trace [13] J.-B. Lasserre. Moments, positive polynomials and their applications, volume 1. property holds for semidefinite relaxations of eigenvalue or trace World Scientific, 2010. optimization problems, whenever an nc ball constraint is present. [14] R. B. Lehoucq and D. C. Sorensen. Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM Journal on Matrix Analysis and Applications, This property can be easily verified by solving a hierarchy of linear 17(4):789–821, 1996. programs, when the only involved inequality constraints are either [15] R. B. Lehoucq, D. C. Sorensen, and C. Yang. ARPACK users’ guide: solution of noncommutative ball or nc polydisk constraints. This allowsoneto large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, 1998. use first order methods exploiting the constant trace property (e.g., [16] V. Magron and M. S. E. Din. On Exact Polya and Putinar’s Representations. In CGAL) to solve the semidefinite relaxations of large-scale eigen- ISSAC’18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation. ACM, New York, NY, USA, 2018. value problems more efficiently than with second order interior- [17] N. H. A. Mai, J.-B. Lasserre, V. Magron, and J. Wang. Exploiting constant trace point solvers (e.g., Mosek). We have experimentally demonstrated property in large-scale polynomial optimization. arXiv preprint arXiv:2012.08873, some of these computational gains on eigenvalue minimization. 2020. [18] N. H. A. Mai, V. Magron, and J.-B. Lasserre. A hierarchy of spectral relaxations Similar gains shall be achievable for trace minimization. For many for polynomial optimization. arXiv preprint arXiv:2007.09027, 2020. testing examples in this paper, the relative optimality gap of CGAL [19] J.MarecekandJ.Vala. Quantumoptimal controlviamagnus expansion: The non- w.r.t. Mosek is always smaller than 1%. commutative polynomial optimization problem. arXiv preprint arXiv:2001.06464, 2020. As a topic of further research, we intend to rely on our frame- [20] A. MOSEK. The MOSEK Optimization Toolbox, Version 8.1, 2017. work to tackle applications arising from quantum information and [21] K. F. Pál and T. Vértesi. Quantum bounds on bell inequalities. Physical Review A, 79(2):022120, 2009. condensed matter, including bounds on maximal violation levels [22] H. Peyrl and P. Parrilo. Computing sum of squares decompositions with rational for Bell inequalities [21] or ground state energies of many body coefficients. Theoretical Computer Science, 409(2):269–281, 2008. Hamiltonians [3]. Preliminary experiments not reported in this pa- [23] S. Pironio, M. Navascués, and A. Acin. Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM Journal on Opti- per show that relying on CGAL improves some existing bounds for mization, 20(5):2157–2180, 2010. Bell inequalities, while Mosek runs out memory. We intend to im- [24] M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana Univ. prove our software implementation to overcome the accuracy is- Math. J., 42(3):969–984, 1993. [25] K. Schmüdgen. The k-moment problem for compact semi-algebraic sets. Math- sues arising when using CGAL. A related investigation track is to ematische Annalen, 289(1):203–206, 1991. design a numerical method for finding the constant trace and the [26] R. E. Skelton, T. Iwasaki, and D. E. Grigoriadis. A unified algebraic approach to control design. CRC Press, 1997. change of basis for noncommutative problems with arbitrary in- [27] H. Waki, S. Kim, M. Kojima, and M. Muramatsu. Sums of squares and semidefi- equality constraints (possibly including nc ball constraint). Ideally, nite program relaxations for polynomial optimization problems with structured first order semidefinite solvers should have rich numerical prop- sparsity. SIAM Journal on Optimization, 17(1):218–242, 2006. [28] J. Wang and V. Magron. Exploiting term sparsity in noncommutative polynomial erties when combined with the constant trace and the change of optimization. arXiv preprint arXiv:2010.06956, 2020. basis matrix obtained in our method. This will allow us to design, [29] J. Wang, V. Magron, and J.-B. Lasserre. Chordal-TSSOS: a moment-SOS hierarchy implement and analyze a hybrid numeric-symbolic scheme as in that exploits term sparsity with chordal extension. SIAM Journal on Optimiza- tion, 31(1):114–141, 2021. [16, 22], to obtain exact nonnegativity certificates of noncommuta- [30] J. Wang, V. Magron, and J.-B. Lasserre. Tssos: A moment-sos hierarchy that tive problems. exploits term sparsity. SIAM Journal on Optimization, 31(1):30–58, 2021. [31] J. Wang, V. Magron, J. B. Lasserre, and N. H. A. Mai. CS-TSSOS: Correla- tive and term sparsity for large-scale polynomial optimization. arXiv preprint REFERENCES arXiv:2005.02828, 2020. [1] E. D. Andersen and K. D. Andersen. The MOSEK interior point optimizer for [32] A. Yurtsever, O. Fercoq, and V. Cevher. A conditional gradient-based augmented linear programming: an implementation of the homogeneous algorithm. In High lagrangian framework. arXiv preprint arXiv:1901.04013, 2019. performance optimization, pages 197–232. Springer, 2000. [33] A. Yurtsever, J. A. Tropp, O. Fercoq, M. Udell, and V. Cevher. Scalable semidefi- [2] M. F. Anjos and J. B. Lasserre,editors. Handbook on semidefinite, conic and poly- nite programming. arXiv preprint arXiv:1912.02949, 2019. nomial optimization, volume 166 of International Series in Operations Research & Management Science. Springer, New York, 2012.