CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization

Abstract

This work proposes a new moment-SOS hierarchy, called CS-TSSOS, for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. [WKKM06] while the latter was initially proposed by Wang et al. [WLX19] and later exploited in the TSSOS hierarchy [WML19, WML20].In doing so we obtain CS-TSSOS -a two-level hierarchy of semidefinite programming relaxations with (i), the crucial property to involve quasi block-diagonal matrices and (ii), the guarantee of convergence to the global optimum. We demonstrate its efficiency on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to several thousands of variables and ten thousands of constraints.

Sparse non-SOS Putinar-type Positivstellensätze

“…In turn, these results imply that these new non-SOS classes of polynomials can be used to construct approximation hierarchies, that are guaranteed to converge to the solution of polynomial optimization problems, with desired characteristics associated with Putinar-type Positivstellensätze, but without the need to incur in the computational expense associated with checking membership in the class of SOS polynomials. Thirdly, our results show that the possibility of simplifying Positivstellensätze expression by exploiting sparsity is by no means limited, as in the current literature, to SOS Positivstellensätze [24,26,30,49,50]. Instead, our results show that even in the case in which no particular sparsity information is available regarding the polynomials involved in the Positivstellensätze, the obtained non-SOS Putinar-type Positivstellensätze can be written with inherent term sparsity (Theorem 4, Corollary 1, and Remark 4).…”

“…Deriving Positivstellensätze that take advantage of both the sparsity of p and the polynomials defining S has been the focus of a wealth of research work. For example, consider the earlier work in [28,37,38,46], and the more recent work in [26,30,49,50]. This latter work focuses on exploiting correlative and term sparsity to derive term and correlative sparse versions of SOS Positivstellensätze such as Putinar's Positivstellensatz (in [26,49,50]); and Reznick's and Putinar-Vasilescu's [41] Positivstellensatz (in [30]).…”

“…For example, consider the earlier work in [28,37,38,46], and the more recent work in [26,30,49,50]. This latter work focuses on exploiting correlative and term sparsity to derive term and correlative sparse versions of SOS Positivstellensätze such as Putinar's Positivstellensatz (in [26,49,50]); and Reznick's and Putinar-Vasilescu's [41] Positivstellensatz (in [30]). The Positivstellensätze (3) guarantees the existence of non-SOS Putinar-type Positivstellensätze which are term sparse, without any assumption on the polynomial p or the polynomials defining S (see Theorem 4 and Corollary 1).…”

“…That is, to investigate how to take advantage of the sparsity characteristics of the the polynomial p, whose non-negativity is being certified, as well as the sparsity of the polynomials g ∈ R[x] m (defining the underlying semialgebraic set S g ⊆ R n ), to obtain a sparse expression that certifies the non-negativity of the polynomial p on S. For example, consider the earlier work in [28,37,38,46, to name a few]. More recently the focus has been on exploiting correlative and term sparsity to derive sparse versions of SOS Positivstellensätze such as Putinar's Positivstellensatz [26,49,50]; and Reznick's and Putinar-Vasilescu's Positivstellensatz [30,41].…”

Bounding the separable rank via polynomial optimization

“…While this permits to eliminate variables (and work with 2(2d − 1) instead of 4d real variables), this reduction does not permit to block-diagonalize the moment matrices as indicated above. We also refer to [23] and the recent paper [57] for more details about exploiting sign symmetries. Block-diagonal reduction example.…”