RLT: A unified approach for discrete and continuous nonconvex optimization

Sherali, Hanif D.

doi:10.1007/s10479-006-0107-7

Cited by 14 publications

(3 citation statements)

References 25 publications

(21 reference statements)

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“…Logical variables are usually either incorporated by means of big-M constraints or via a convex hull formulation, see [30,49,50,66]. From a different point of view, disjunctive programming formulations can be interpreted as the result of reformulationlinearization technique (RLT) steps [63]. For both, the convex hull relaxation uses perspective functions.…”

Section: Definition 3 (Licq)mentioning

confidence: 99%

On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

Jung

Kirches

Säger

2013

Facets of Combinatorial Optimization

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Logical implications appear in a number of important mixed-integer nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, and vanishing constraints. In addition to the tightness of the respective relaxations, we also address the issue of constraint qualification and the behavior of computational methods for some formulations. As a benchmark, we formulate a truck cruise control problem with logical implications resulting from gearchoice specific constraints. We provide this benchmark problem in AMPL format along with different realistic scenarios. Computational results for this benchmark are used to investigate feasibility gaps, integer feasibility gaps, quality of local solutions, and well-behavedness of numerical methods for the presented reformulations of the benchmark problem. Vanishing constraints give the most satisfactory results.

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Section: Definition 3 (Licq)mentioning

confidence: 99%

On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

Jung

Kirches

Säger

2013

Facets of Combinatorial Optimization

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“…A bilinear function is a particular case of quadratic functions, for which there are several ways to get convex relaxations. McCormick relaxation (McCormick [14]), Reformulation Linearization Technique (RLT) (Sherali [18], Sherali and Alameddin [19]), Semidefinite relaxation (Anstreicher [2], [3], Bao et al [4]), Lagrangian relaxation (Voorhis [24]) etc. are mostly used relaxation strategies of bilinear functions.…”

Section: Introductionmentioning

confidence: 99%

Facets of a mixed-integer bilinear covering set with bounds on variables

Rahman

Mahajan

2019

J Glob Optim

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We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a certain way. This description does not introduce any new variables, but consists of exponentially many inequalities. An extended formulation with a few extra variables and much smaller number of constraints is presented. We also derive a linear time separation algorithm for finding the facet defining inequalities of this convex hull. We study the effectiveness of the new inequalities and the extended formulation using some examples.

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“…Many researchers have proposed using a combined RLT+SDP relaxation method, which has proven to be more effective as compared to using either the RLT or SDP relaxations as a standalone approach (see Anstreicher [6,8], Sherali [65]). After merging the RLT and SDP relaxations given by (1.5) and (1.11) respectively, the enhanced QCQP formulation is given by:…”

Section: Rlt+sdpmentioning

confidence: 99%

Enhancing convexification techniques for solving nonconvex quadratic programming problems

Qi¹

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At the intersection of combinatorial and nonlinear optimization, quadratic programming (QP) plays an important role in practice and optimization theory. With widespread real-world applications including, but not limited to, engineering design, game theory, and economics, quadratic programming has many valuable contributions and has attracted considerable interest over the past few decades. As a more general formulation, quadratically constrained quadratic programming (QCQP) problems arise naturally from a lot of applications such as facility location, production planning, economic equilibria and Euclidean distance

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RLT: A unified approach for discrete and continuous nonconvex optimization

Cited by 14 publications

References 25 publications

On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

Facets of a mixed-integer bilinear covering set with bounds on variables

Enhancing convexification techniques for solving nonconvex quadratic programming problems

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