Extending the QCR method to general mixed-integer programs

Billionnet, Alain; Elloumi, Sourour; Lambert, Amélie

doi:10.1007/s10107-010-0381-7

Cited by 67 publications

(76 citation statements)

References 23 publications

(40 reference statements)

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“…The geometric investigation in Li et al [36] for binary quadratic programs provides some theoretical support for QCR from another angle. Billionnet et al [9] extended the QCR approach to general mixed-integer programs by using binary decomposition.…”

Section: Combination Of Lift-and-convexification and Qcrmentioning

confidence: 99%

“…We then further combine our lift-and-convexification approach and the quadratic convex reformulation (QCR) [10,11] approach in the literature to form an even tighter reformulation. The QCR approach has been applied to zero-one quadratic programs [11] and integer quadratic programs [9]. While the QCR approach cannot be directly applied to problem (P), it can be successfully applied on top of our new lift-and-convexification reformulation.…”

Section: Introductionmentioning

confidence: 99%

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Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Wu¹,

Sun²,

Li³

2017

SIAM J. Optim.

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We consider in this paper a class of semi-continuous quadratic programming problems which arises in many real-world applications such as production planning, portfolio selection and subset selection in regression. We propose a lift-and-convexification approach to derive an equivalent reformulation of the original problem. This liftand-convexification approach lifts the quadratic term involving x only in the original objective function f (x, y) to a quadratic function of both x and y and convexifies this equivalent objective function. While the continuous relaxation of our new reformulation attains the same tight bound as achieved by the continuous relaxation of the well known perspective reformulation, the new reformulation also retains the linearly constrained quadratic programming structure of the original mix-integer problem. This prominent feature improves the performance of branch-and-bound algorithms by providing the same tightness at the root node as the state-of-the-art perspective reformulation and offering much faster processing time at children nodes. We further combine the liftand-convexification approach and the quadratic convex reformulation approach in the literature to form an even tighter reformulation. Promising results from our computational tests in both portfolio selection and subset selection problems numerically verify the benefits from these theoretical features of our new reformulations.

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Section: Combination Of Lift-and-convexification and Qcrmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Wu¹,

Sun²,

Li³

2017

SIAM J. Optim.

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“…They show that an optimal reformulation can be derived from the dual of an SDP relaxation. Billionnet et al [137] then show that the method can be extended to general MIQPs, provided that the integer-constrained variables are bounded and the part of the objective function associated with the continuous variables is convex.…”

Section: Some Additional Techniquesmentioning

confidence: 99%

Non-convex mixed-integer nonlinear programming: A survey

Burer

Letchford

2012

Surveys in Operations Research and Management Science

379

238

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“…. , n. Related problems of this type arise in many applications, including portfolio selection [5], sparse least-squares [22], optimal control [20], and unit-commitment for power generation [15]. The optimization problem (1) can be very difficult to solve to optimality.…”

Section: Introductionmentioning

confidence: 99%

On Valid Inequalities for Quadratic Programming with Continuous Variables and Binary Indicators

Dong

Linderoth

2013

Lecture Notes in Computer Science

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Abstract. In this paper we study valid inequalities for a set that involves a continuous vector variable x ∈ [0, 1] n , its associated quadratic form xx T , and binary indicators on whether or not x > 0. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MIQPs). Valid inequalities for this set can be obtained by lifting inequalities for a related set without binary variables (QPB), that was studied by Burer and Letchford. After closing a theoretical gap about QPB, we characterize the strength of different classes of lifted QPB inequalities. We show that one class, lifted-posdiag-QPB inequalities, capture no new information from the binary indicators. However, we demonstrate the importance of the other class, called lifted-concave-QPB inequalities, in two ways. First, all lifted-concave-QPB inequalities define the relevant convex hull for the case of convex quadratic programming with indicators. Second, we show that all perspective constraints are a special case of lifted-concave-QPB inequalities, and we further show that adding the perspective constraints to a semidefinite programming (SDP) relaxation of convex quadratic programs with binary indicators results in a problem whose bound is equivalent to the recent optimal diagonal splitting approach of Zheng et al.. Finally, we show the separation problem for lifted-concave-QPB inequalities is tractable if the number of binary variables involved in the inequality is small. Our study points out a direction to generalize perspective cuts to deal with non-separable nonconvex quadratic functions with indicators in global optimization. Several interesting questions arise from our results, which we detail in our concluding section.

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Extending the QCR method to general mixed-integer programs

Cited by 67 publications

References 23 publications

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Non-convex mixed-integer nonlinear programming: A survey

On Valid Inequalities for Quadratic Programming with Continuous Variables and Binary Indicators

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