On conic QPCCs, conic QCQPs and completely positive programs

Bai, Lijie; Mitchell, John E.; Pang, Jong-Shi

doi:10.1007/s10107-015-0951-9

Cited by 27 publications

(28 citation statements)

References 56 publications

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“…We will now prove that strict copositivity of at least one Q i guarantees attainability of (2.2), even without the assumption that the objective is bounded below on the feasible set. This result complements prior investigations [36] and a recent study [3]. Let us first establish the following auxiliary result.…”

Section: A Two-fold Characterization Of Semi-lagrangian Dualsupporting

confidence: 81%

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Bomze¹

2015

SIAM J. Optim.

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We study nonconvex quadratic minimization problems under (possibly nonconvex) quadratic and linear constraints, characterizing both Lagrangian and semi-Lagrangian dual bounds in terms of conic optimization. While the Lagrangian dual is equivalent to the SDP relaxation (which has been known for quite a while, although the presented form, incorporating explicitly linear constraints, seems to be novel), we show that the semi-Lagrangian dual is equivalent to a natural copositive relaxation (and this has apparently not been observed before). This way, we arrive at conic bounds tighter than the usual Lagrangian dual (and thus than the SDP) bounds. Any of the known tractable inner approximations of the copositive cone can be used for this tightening, but in particular, the above-mentioned characterization with explicit linear constraints is a natural, much cheaper, relaxation than the usual zero-order approximation by doubly nonnegative (DNN) matrices and still improves upon the Lagrangian dual bounds. These approximations are based on LMIs on matrices of basically the original order plus additional linear constraints (in contrast to more familiar sum-of-squares or moment approximation hierarchies) and thus may have merits in particular for large instances where it is important to employ only a few inequality constraints (e.g., n instead of n(n−1) 2 for the DNN relaxation). Further, we specify sufficient conditions for tightness of the semi-Lagrangian relaxation and show that copositivity of the slack matrix guarantees global optimality for KKT points of this problem, thus significantly improving upon a well-known secondorder global optimality condition.

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Section: A Two-fold Characterization Of Semi-lagrangian Dualsupporting

confidence: 81%

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Bomze¹

2015

SIAM J. Optim.

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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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“…It is shown in Bai et al [2] that (4) can be reformulated as a convex conic completely positive optimization problem. However, the cone in the completely positive formulation does not have a polynomial-time separation oracle.…”

Section: Semidefinite Cone Complementarity Formulation For Rank Minimmentioning

confidence: 99%

A penalty method for rank minimization problems in symmetric matrices

Shen

Mitchell

2018

Comput Optim Appl

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The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point.We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.

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On conic QPCCs, conic QCQPs and completely positive programs

Cited by 27 publications

References 56 publications

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

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

A penalty method for rank minimization problems in symmetric matrices

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