Representing quadratically constrained quadratic programs as generalized copositive programs

Burer, Samuel; Dong, Hongbo

doi:10.1016/j.orl.2012.02.001

Cited by 54 publications

(66 citation statements)

References 16 publications

(14 reference statements)

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“…Anstreischer and Burer [2] further investigated various computable representations for the convex hull of quadratic forms. Burer and Dong [12] reformulated QCQPs as co-positive programs. In [24], Lasserre developed a hierarchical SDP relaxations for polynomial optimization that converges to the global optimal solution of the original problem.…”

Section: Proposition 11 the Wclo Problem Is Np-hardmentioning

confidence: 99%

A nonlinear semidefinite optimization relaxation for the worst-case linear optimization under uncertainties

Peng

Zhu

2014

Math. Program.

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In this paper, we consider the so-called worst-case linear optimization (WCLO) with uncertainties in the right-hand-side of the constraints. Such a problem arises from numerous applications such as systemic risk estimate in finance and stochastic optimization. We first show the problem is NP-hard and present a coarse semidefinite relaxation (SDR) for WCLO. An iterative procedure is introduced to sequentially refine the relaxation model based on the solution of the current relaxation model by simply changing some parameters in the coarse SDR. We show that the sequence of the proposed SDRs will converge to some nonlinear semidefinite optimization problem (SDO). A bi-section search algorithm is proposed to solve the resulting nonlinear SDO. Our preliminary experimental results illustrate that the nonlinear SDR can provide very tight bounds for the original WCLO and is able to locate the exact global solution in most cases within a few iterations on average.

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Section: Proposition 11 the Wclo Problem Is Np-hardmentioning

confidence: 99%

A nonlinear semidefinite optimization relaxation for the worst-case linear optimization under uncertainties

Peng

Zhu

2014

Math. Program.

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“…The maximum stable set problem in [11], a graph tri-partitioning problem in [14], and the quadratic assignment problem [14], were considered and reduced to CPPs. More recently, general QOPs with quadratic constraints were represented as generalized CPPs in [6]. However, it is not well understood yet whether a general class of QOPs can be formulated as CPPs.…”

Section: Introductionmentioning

confidence: 99%

A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming

Arima¹,

Kim²,

Kojima³

2013

SIAM J. Optim.

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Abstract.We propose a class of quadratic optimization problems whose exact optimal objective values can be computed by their completely positive cone programming relaxations. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one inhomogeneous equality where a quadratic form is set to be a positive constant. For the equality constraints, "a hierarchy of copositvity" condition is assumed. This model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex, and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations. In particular, it generalizes the recent results on quadratic optimization problems by Burer and the set-semidefinite representation by Eichfelder and Povh. Key words.Copositve programming, quadratic optimization problem with quadratic constraints, a hierarchy of copositivity. AMS Classification. 90C20, 90C25, 90C26

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“…Later, Burer and Dong [9] extended this equivalence to general nonconvex quadratically constrained quadratic program whose feasible region is nonempty and bounded. From the proof of Proposition 5.3, the cone of completely positive matrices can be imbedded into the cone of quartic SOQ forms.…”

Section: Proposition 52mentioning

confidence: 99%

On Cones of Nonnegative Quartic Forms

Jiang

Zhang

2015

Found Comput Math

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Historically, much of the theory and practice in nonlinear optimization has revolved around the quadratic models. Though quadratic functions are nonlinear polynomials, they are well structured and easy to deal with. Limitations of the quadratics, however, become increasingly binding as higher degree nonlinearity is imperative in modern applications of optimization. In the recent years, one observes a surge of research activities in polynomial optimization, and modeling with quartic or higher order polynomial functions has been more commonly accepted. On the theoretical side, there are also major recent progresses on polynomial functions and optimization. For instance, Ahmadi et al. [2] proved that checking the convexity of a quartic polynomial function is strongly NP-hard in general, which settles a long-standing open question. In this paper we proceed to studying six fundamentally important convex cones of quartic functions in the space of symmetric quartic tensors, including the cone of nonnegative quartic polynomials, the sum of squared polynomials, the convex quartic polynomials, and the sum of fourth powered polynomials. It turns out that these convex cones coagulate into a chain in decreasing order. The complexity status of these cones is sorted out as well. Finally, potential applications of the new results to solve highly nonlinear and/or combinatorial optimization problems are discussed.

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Representing quadratically constrained quadratic programs as generalized copositive programs

Cited by 54 publications

References 16 publications

A nonlinear semidefinite optimization relaxation for the worst-case linear optimization under uncertainties

A nonlinear semidefinite optimization relaxation for the worst-case linear optimization under uncertainties

A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming

On Cones of Nonnegative Quartic Forms

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