Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

Gvozdenović, Nebojša; Laurent, Monique

doi:10.1007/s10107-006-0062-8

Cited by 29 publications

(36 citation statements)

References 22 publications

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“…In recent years, semidefinite programming [37] and sum of squares optimization [33, 22,31] have proven to be powerful techniques for tackling a diverse set of problems in applied and computational mathematics. The reason for this, at a high level, is that several fundamental problems arising in discrete and polynomial optimization [23,17,7] or the theory of dynamical systems [32,19,3] can be cast as linear optimization problems over the cone of nonnegative polynomials. This observation puts forward the need for efficient conditions on the coefficients c α := c α1,...,αn of a multivariate polynomial p(x) = α c α1,...,αn x α1 1 .…”

Section: Introductionmentioning

confidence: 99%

Sum of squares basis pursuit with linear and second order cone programming

Ahmadi¹,

Hall²

2017

Algebraic and Geometric Methods in Discrete Mathematics

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We devise a scheme for solving an iterative sequence of linear programs (LPs) or second order cone programs (SOCPs) to approximate the optimal value of any semidefinite program (SDP) or sum of squares (SOS) program. The first LP and SOCP-based bounds in the sequence come from the recent work of Ahmadi and Majumdar on diagonally dominant sum of squares (DSOS) and scaled diagonally dominant sum of squares (SDSOS) polynomials. We then iteratively improve on these bounds by pursuing better bases in which more relevant SOS polynomials admit a DSOS or SDSOS representation. Different interpretations of the procedure from primal and dual perspectives are given. While the approach is applicable to SDP relaxations of general polynomial programs, we apply it to two problems of discrete optimization: the maximum independent set problem and the partition problem. We further show that some completely trivial instances of the partition problem lead to strictly positive polynomials on the boundary of the sum of squares cone and hence make the SOS relaxation fail.

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Section: Introductionmentioning

confidence: 99%

Sum of squares basis pursuit with linear and second order cone programming

Ahmadi¹,

Hall²

2017

Algebraic and Geometric Methods in Discrete Mathematics

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“…During the completion of this paper, we learned about the related independent work by Gvozdenović and Laurent [6]. Gvozdenović and Laurent studied some properties of the approximations ϑ (r) (G).…”

Section: Introductionmentioning

confidence: 99%

Computing the Stability Number of a Graph Via Linear and Semidefinite Programming

2007

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We study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph. Our work is based on, and refines de Klerk and Pasechnik's approach to approximating the stability number via copositive programming (SIAM J. Optim. 12 (2002), 875-892). We provide a closed-form expression for the values computed by the linear programming approximations. We also show that the exact value of the stability number α(G) is attained by the semidefinite approximation of order α(G) − 1 as long as α(G) ≤ 6. Our results reveal some sharp differences between the linear and the semidefinite approximations. For instance, the value of the linear programming approximation of any order is strictly larger than α(G) whenever α(G) > 1.

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“…An illustration of how sections of these different cones compare on an example is given in Figure 1, taken from [39]. We consider a parametric family of polynomials parameterized by a and b, p a,b (x 1 , x 2 ) = 2x 4 1 + 2x 4 2 + ax 3 1 x 2 + (1 − a)x 2 1 x 2 2 + bx 1 x 3 2 , and plot in the figure the values of (a, b) for which the polynomial is DSOS (innermost set), SDSOS (set containing the DSOS set), and SOS (or equivalentally nonnegative in this case) (outermost set).…”

Section: A Dsos and Sdsos Programsmentioning

confidence: 99%

Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations

Ahmadi

Hall

Papachristodoulou

et al. 2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

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It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.

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Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

Cited by 29 publications

References 22 publications

Sum of squares basis pursuit with linear and second order cone programming

Sum of squares basis pursuit with linear and second order cone programming

Computing the Stability Number of a Graph Via Linear and Semidefinite Programming

Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations

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