Completely positive reformulations for polynomial optimization

Peña, Javier; Vera, Juan C.; Zuluaga, Luis F.

doi:10.1007/s10107-014-0822-9

Cited by 50 publications

(58 citation statements)

References 46 publications

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“…Furthermore, we emphasize that this result is not straightforward. Contrary to the case of quadratic objective and linear constrained problems with some binary variables, where one can apply Burer's results (Burer, 2009, Theorem 2.6); the above formulation is quadratic in the objective, in the constraints and also includes binary variables; and in our case the most recent, known sufficient conditions for obtaining conic reformulations do not directly apply (Burer, 2012;Burer and Dong, 2012;Peña et al, 2015;Bai et al, 2016). In all cases, those results require some nonnegativity condition of the considered quadratic constraints on the feasible region of the problem.…”

Section: A Completely Positive Reformulation Of Dompmentioning

confidence: 95%

An exact completely positive programming formulation for the discrete ordered median problem: an extended version

Puerto

2019

J Glob Optim

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This paper presents a first continuous, linear, conic formulation for the Discrete Ordered Median Problem (DOMP). Starting from a binary, quadratic formulation in the original space of location and allocation variables that are common in Location Analysis (L.A.), we prove that there exists a transformation of that formulation, using the same space of variables, that allows us to cast DOMP as a continuous linear problem in the space of completely positive matrices. This is the first positive result that states equivalence between the family of continuous convex problems and some hard problems in L.A. The result is of theoretical interest because it allows us to share the tools from continuous optimization to shed new light into the difficult combinatorial structure of the class of ordered median problems.

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Section: A Completely Positive Reformulation Of Dompmentioning

confidence: 95%

An exact completely positive programming formulation for the discrete ordered median problem: an extended version

Puerto

2019

J Glob Optim

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“…The problem will be formulated as a polynomial minimization problem over the intersection of the multi-sphere and the nonnegative orthant. With this formulation, the study can also be applied to the problem of testing the copositivity for a homogeneous polynomial, which is important in the completely positive programming [45].…”

Section: Introductionmentioning

confidence: 99%

Best Nonnegative Rank-One Approximations of Tensors

Sun

Toh

2019

SIAM J. Matrix Anal. Appl.

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In this paper, we study the polynomial optimization problem of multi-forms over the intersection of the multi-spheres and the nonnegative orthants. This class of problems is NP-hard in general, and includes the problem of finding the best nonnegative rank-one approximation of a given tensor. A Positivstellensatz is given for this class of polynomial optimization problems, based on which a globally convergent hierarchy of doubly nonnegative (DNN) relaxations is proposed. A (zero-th order) DNN relaxation method is applied to solve these problems, resulting in linear matrix optimization problems under both the positive semidefinite and nonnegative conic constraints. A worst case approximation bound is given for this relaxation method. Then, the recent solver SDPNAL+ is adopted to solve this class of matrix optimization problems. Typically, the DNN relaxations are tight, and hence the best nonnegative rank-one approximation of a tensor can be revealed frequently. Extensive numerical experiments show that this approach is quite promising.2010 Mathematics Subject Classification. 15A18; 15A42; 15A69; 90C22. Key words and phrases. Tensor, nonnegative rank-1 approximation, polynomial, multi-forms, doubly nonnegative semidefinite program, doubly nonnegative relaxation method.

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“…It is known that the cone of completely positive tensors has a natural associated dual cone of copositive tensors [29]. In light of this relationship, any completely positive program stated in [24] has a natural dual conic program over the cone of copositive tensors. As a consequence of this characterization, it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems.…”

Section: Introductionmentioning

confidence: 99%

Copositive tensor detection and its applications in physics and hypergraphs

Chen

Huang

2017

Comput Optim Appl

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Copositivity of tensors plays an important role in vacuum stability of a general scalar potential, polynomial optimization, tensor complementarity problem and tensor generalized eigenvalue complementarity problem. In this paper, we propose a new algorithm for testing copositivity of high order tensors, and then present applications of the algorithm in physics and hypergraphs. For this purpose, we first give several new conditions for copositivity of tensors based on the representative matrix of a simplex. Then a new algorithm is proposed with the help of a proper convex subcone of the copositive tensor cone, which is defined via the copositivity of Z-tensors. Furthermore, by considering a sum-ofsquares program problem, we define two new subsets of the copositive tensor cone and discuss their convexity. As an application of the proposed algorithm, we prove that the coclique number of a uniform hypergraph is equivalent with an optimization problem over the completely positive tensor cone, which implies that the proposed algorithm can be applied to compute an upper bound of the coclique number of a uniform hypergraph. Then we study another application of the proposed algorithm on particle physics in testing copositivity of some potential fields. At last, various numerical examples are given to show the performance of the algorithm.

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Completely positive reformulations for polynomial optimization

Cited by 50 publications

References 46 publications

An exact completely positive programming formulation for the discrete ordered median problem: an extended version

An exact completely positive programming formulation for the discrete ordered median problem: an extended version

Best Nonnegative Rank-One Approximations of Tensors

Copositive tensor detection and its applications in physics and hypergraphs

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