The Constant Trace Property in Noncommutative Optimization Ngoc Hoang Anh Mai Abhishek Bhardwaj Victor Magron LAAS LAAS LAAS Toulouse, France Toulouse, France Toulouse, France [email protected] [email protected] [email protected] ABSTRACT A standard approach in the commutative setting, is Lasserre’s In this article, we show that each semidefinite relaxation of a Hierarchy [12], which provides a sequence lower bounds on the op- ball-constrained noncommutative polynomial optimization prob- timal values for POPs, with guaranteed convergence under some lem can be cast as a semidefinite program with a constant trace ma- natural constraints according to Putinar’s Positivstellensatz [24]. trix variable. We then demonstrate how this constant trace prop- This hierarchy and its nc extension to eigenvalue/trace optimiza- erty can be exploited via first order numerical methods to solve ef- tion [5, 23], involve solving SDPs over the space of multivariate ficiently the semidefinite relaxations of the noncommutative prob- moment and nc Hankel matrices, respectively. lem. Due to the current capacity of interior-point SDP solvers such as Mosek [1, 20], these hierarchies can only be applied when the mul- KEYWORDS tivariate moment (or nc Hankel) matrices are of “moderate” size. Often restricting their use to polynomials of low degrees, or in few noncommutative polynomial optimization, sums of hermitian squares, variables, with the situation being worse in the nc setting. eigenvalue and trace optimization, conditional gradient-based aug- A strategy for reducing the size of the SDP hierarchies is to mented Lagrangian, constant trace property, semidefinite program- exploit the sparsity structures of POPs. They include correlative ming sparsity (CS) in [11] and term sparsity (TS), CS-TS in [28] all of which are the analogs of the commutative works about CS [27], ACM Reference Format: Ngoc Hoang Anh Mai, Abhishek Bhardwaj, and Victor Magron. 2021. The TS [29, 30] and CS-TS [31]. Constant Trace Property in Noncommutative Optimization. In Proceedings Encouraged by [7, 33], in [17, 18] the first and third authors of ACM Conference (ISSAC’21). ACM, New York, NY, USA, 8 pages. showed how to exploit the Constant Trace Property (CTP) for SDP relaxations of POPs, which is satisfied when the matrices involved in the SDP relaxations have constant trace. By utilizing first or- 1 INTRODUCTION der spectral methods to solve the required SDP relaxations, they Polynomial optimization problems (POP) are present in many ar- attained significant computational gains for POPs constrained on eas of mathematics, and science in general. There are many appli- simple domains, e.g., sphere, ball, annulus, box and simplex. cations in global optimization, control and analysis of dynamical In this article, we extend the exploitation of the CTP to NCPOPs. systems to name a few [13], and being able to efficiently solve POP Our two main contributions are the following: First, we obtain anal- is of great importance. ogous results to [17, 18], which ensure the CTP for a broad class of In this article we focus on noncommutative (nc) polynomial opti- dense NCPOPs. In particular, if nc ball (or nc polydisc) constraint(s) mization problems (NCPOP), that is, polynomial optimization with is present, then CTP holds. We also extend this CTP-framework to non-commuting variables. NCPOP has several applications in con- some NCPOPs with correlative sparsity. Secondly, We provide a Ju- trol [26] and quantum information [6, 19, 21]. lia package for solving NCPOPs with CTP. The package makes use Since the advent of interior point methods for semidefinite pro- of first order methods for solving SDPs with CTP. We also demon- grams (SDP) [2], there have been many approaches to solving POP, strate the numerical and computational efficiency of this approach, using powerful representation results from real algebraic geometry on some sample classes of dense NCPOPs and NCPOPs with cor- arXiv:2102.02162v1 [math.OC] 3 Feb 2021 for positive polynomials. Inspired by Schmüdgen’s solution to the relative sparsity. moment problem on compact semialgebraic sets [25], these meth- ods aim to provide certificates of global positivity. There are natu- 2 DEFINITIONS & PRELIMINARIES ral analogues to these approaches in the nc setting, coming from Here we introduce some basic preliminary knowledge needed free algebraic geometry [9], and the tracial moment problems [4]. in the sequel. For a more detailed introduction to the topics intro- duced in this section, the reader is referred to [5]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full cita- 2.1 Noncommutative polynomials tion on the first page. Copyrights for components of this work owned by others than We denote by - the noncommuting letters -1,...,-=. Let - = ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re- h i -1,...,-= be the free monoid generated by -, and call its ele- publish, to post on servers or to redistribute to lists, requires prior specific permission h i and/or a fee. Request permissions from [email protected]. ments words in -. Given a word F = -81 ...-8A , F∗ is its reverse, ISSAC’21, July 2021, Saint Petersburg, Russia R i.e., F∗ = -8A ...-81 . Consider the free algebra - of polynomi- © 2021 Association for Computing Machinery. R h i ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 als in - with coefficients in . Its elements are called noncommu- https://doi.org/10.1145/nnnnnnn.nnnnnnn tative (nc) polynomials. Endow R - with the involution 5 5 h i → ∗ 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 .