Approximation Hierarchies for the Copositive Tensor Cone and Their Application to the Polynomial Optimization over the Simplex (original) (raw)
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O C ] 4 J un 2 01 8 Approximation Hierarchies for Copositive Tensor Cone
2018
In this paper we discuss copositive tensors, which are a natural generalization of the copositive matrices. We present an analysis of some basic properties of copositive tensors; as well as the conditions under which class of copositive tensors and the class of positive semidefinite tensors coincides. Moreover, we have describe several hierarchies that approximates the cone of copositive tensors. The hierarchies are predominantly based on different regimes such as; simplicial partition, rational griding and polynomial conditions. The hierarchies approximates the copositive cone either from inside (inner approximation) or from outside (outer approximation). We will also discuss relationship among different hierarchies.
On the accuracy of uniform polyhedral approximations of the copositive cone
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We consider linear optimization problems over the cone of copositive matrices. Such conic optimization problems, called copositive programs, arise from the reformulation of a wide variety of difficult optimization problems. We propose a hierarchy of increasingly better outer polyhedral approximations to the copositive cone. We establish that the sequence of approximations is exact in the limit. By combining our outer polyhedral approximations with the inner polyhedral approximations due to de Klerk and Pasechnik [SIAM J. Optim, 12 (2002), pp. 875-892], we obtain a sequence of increasingly sharper lower and upper bounds on the optimal value of a copositive program. Under primal and dual regularity assumptions, we establish that both sequences converge to the optimal value. For standard quadratic optimization problems, we derive tight bounds on the gap between the upper and lower bounds. We provide closed-form expressions of the bounds for the maximum stable set problem. Our computational results shed light on the quality of the bounds on randomly generated instances.
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arXiv (Cornell University), 2022
We investigate the hierarchy of conic inner approximations K (r) n (r ∈ N) for the copositive cone COP n , introduced by Parrilo (Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD Thesis, California Institute of Technology, 2001). It is known that COP 4 = K (0) 4 and that, while the union of the cones K (r) n covers the interior of COP n , it does not cover the full cone COP n if n ≥ 6. Here we investigate the remaining case n = 5, where all extreme rays have been fully characterized by Hildebrand (The extreme rays of the 5 × 5 copositive cone. Linear Algebra and its Applications, 437(7):1538-1547, 2012). We show that the Horn matrix H and its positive diagonal scalings play an exceptional role among the extreme rays of COP 5. We show that equality COP 5 = r≥0 K (r) 5 holds if and only if any positive diagonal scaling of H belongs to K (r) 5 for some r ∈ N. As a main ingredient for the proof, we introduce new Lasserre-type conic inner approximations for COP n , based on sums of squares of polynomials. We show their links to the cones K (r) n , and we use an optimization approach that permits to exploit finite convergence results on Lasserre hierarchy to show membership in the new cones.
Bilinear optimality constraints for the cone of positive polynomials
Mathematical Programming, 2011
For a proper cone K ⊂ R n and its dual cone K * the complementary slackness condition x T s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K * }. When K is a symmetric cone, this fact translates to a set of n bilinear optimality conditions satisfied by every (x, s) ∈ C(K). This proves to be very useful when optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we examine several well-known cones, in particular the cone of positive polynomials P 2n+1 and its dual, the closure of the moment cone M 2n+1 . We show that there are exactly four linearly independent bilinear identities which hold for all (x, s) ∈ C(P 2n+1 ), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials.
Polyhedral approximations of the semidefinite cone and their application
Computational Optimization and Applications, 2021
We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.
Journal of Optimization Theory and Applications, 2015
We propose the moment cone relaxation for a class of polynomial optimization problems (POPs) to extend the results on the completely positive cone programming relaxation for the quadratic optimization (QOP) model by Arima, Kim and Kojima. The moment cone relaxation is constructed to take advantage of sparsity of the POPs, so that efficient numerical methods can be developed in the future. We establish the equivalence between the optimal value of the POP and that of the moment cone relaxation under conditions similar to the ones assumed in the QOP model. The proposed method is compared with the canonical convexification procedure recently proposed by Peña, Vera and Zuluaga for POPs. The moment cone relaxation is theoretically powerful, but numerically intractable. For tractable numerical methods, the doubly nonnegative cone relaxation is derived from the moment cone relaxation. Exploiting sparsity in the doubly nonnegative cone relaxation and its incorporation into Lasserre's semidefinite relaxation are briefly discussed.