Enumeration Approach for Linear Complementarity Problems Based on a Reformulation-Linearization Technique

Sherali, Hanif D.; Krishnamurthy, Ravi S.; Al-Khayyal, Faiz A.

doi:10.1023/a:1021734613201

Cited by 22 publications

(19 citation statements)

References 25 publications

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“…Lower-bounds are computed in each node by solving some appropriate relaxed convex program in order to alleviate the search in the tree. RLT, SDP and cutting-planes [17,64,82] have been recommended to the LPLCC for such a goal and can also be used for the QPLCC [13]. The MPLCC can also be reduced into a 0 − 1 integer program [35,40,39,61,82] and solved by some appropriate technique.…”

Section: The Mplcc Is Called a Linear (Quadratic) Programming Problemmentioning

confidence: 99%

“…Then the augmented LPLCC is declared infeasible. In particular SDP [16] and RLT [82] techniques may be useful in this extent. Despite promising results in some cases, much research has to be done to assure the general efficiency of these techniques in practice.…”

Section: A Sequential Algorithm For Lplccmentioning

confidence: 99%

“…These improvements have been concerned with the quality of the lower bounds and upper bounds and the branching procedure. Cutting planes [2,6,64,87], RLT [82] and SDP [16,17] have been used for computing better lower bounds than the ones given by the relaxed linear programs. On the other hand, some ideas of combinatorial optimization have been employed to design more efficient branching strategies that lead to better upper bounds for the branch-and-bound method [2,6,23,37].…”

Section: Branch-and-bound Algorithmsmentioning

confidence: 99%

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Optimization With Linear Complementarity Constraints

Júdice

2014

Pesqui. Oper.

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ABSTRACT. A Mathematical Program with Linear Complementarity Constraints (MPLCC)is an optimization problem where a continuously differentiable function is minimized on a set defined by linear constraints and complementarity conditions on pairs of complementary variables. This problem finds many applications in several areas of science, engineering and economics and is also an important tool for the solution of some NP-hard structured and nonconvex optimization problems, such as bilevel, bilinear and nonconvex quadratic programs and the eigenvalue complementarity problem. In this paper some of the most relevant applications of the MPLCC and formulations of nonconvex optimization problems as MPLCCs are first presented.Algorithms for computing a feasible solution, a stationary point and a global minimum for the MPLCC are next discussed. The most important nonlinear programming methods, complementarity algorithms, enumerative techniques and 0 − 1 integer programming approaches for the MPLCC are reviewed. Some comments about the computational performance of these algorithms and a few topics for future research are also included in this survey.

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Section: The Mplcc Is Called a Linear (Quadratic) Programming Problemmentioning

confidence: 99%

Section: A Sequential Algorithm For Lplccmentioning

confidence: 99%

Section: Branch-and-bound Algorithmsmentioning

confidence: 99%

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Optimization With Linear Complementarity Constraints

Júdice

2014

Pesqui. Oper.

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“…Specifically, the corridor length is divided into equal-length subintervals ε v = L /v by a positive integer v > 0. Then the numerical solution of the LCS is obtained by computing the discretized scheme: In the discretization scheme, the derivative can be approximated by the forward difference quotient: Finally, the reformulation-linearization technique is applied to convert the MIBLP into an MILP problem (16). This method consists of two phases: the reformation phase and the linearization phase.…”

Section: Discrete Approximation Of Lcs Modelmentioning

confidence: 99%

Reliability-Based Modeling of Park-and-Ride Service on Linear Travel Corridor

Wang

2013

Transportation Research Record

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The modeling of multimodal choice in a railway–highway system with single park-and-ride service on a linear travel corridor is studied. Commuters choose either auto or railway to travel directly from home to city center or drive to the park-and-ride facility and transfer to railway transit service. Both the traffic congestion on the highway and the crowding on rail transit are considered. The highway capacity is assumed to be stochastic to take into account travel time reliability for use of the auto mode. Commuters are assumed to be distributed uniformly along the corridor. A linear complementarity system to model commuters' mode choice along the corridor and to solve the spatial equilibrium travel pattern is proposed. The formulated linear complementarity system is transformed into a mixed integer linear program to be solved. The modeling approach and solution algorithm are implemented in a small numerical example.

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“…Strikingly, even the weakest level-1 representations unify and dominate a vast array of published formulations for different classes of mixed 0-1 linear and quadratic problems (Adams and Johnson, 1994;Adams andSherali, 1986, 1990). Lower-level forms have motivated very efficient algorithms for 0-1 quadratic (Adams and Sherali, 1986) and mixed 0-1 quadratic programs (Adams and Sherali, 1993), as well as for bilinear programs (Sherali and Alameddine, 1992) and linear complementarity problems (Sherali, Krishnamurthy, and Al-Khayyal, 1998). Various linear reformulations of classic 0-1 optimization problems have been characterized (see, e.g., Adams and Johnson, 1994, Hahn and Grant, 1998, Hahn et al, 2001, Johnson, 1992, and cut strengthening methods (Lougee-Heimer and explained in terms of conditional-logic (Sherali, Adams, and Driscoll, 1998) operations.…”

Section: Introductionmentioning

confidence: 99%

A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems

Adams

Sherali

2005

Ann Oper Res

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We consider linear mixed-integer programs where a subset of the variables are restricted to take on a finite number of general discrete values. For this class of problems, we develop a reformulation-linearization technique (RLT) to generate a hierarchy of linear programming relaxations that spans the spectrum from the continuous relaxation to the convex hull representation. This process involves a reformulation phase in which suitable products using a defined set of Lagrange interpolating polynomials (LIPs) are constructed, accompanied by the application of an identity that generalizes x(1 − x) for the special case of a binary variable x. This is followed by a linearization phase that is based on variable substitutions. The constructs and arguments are distinct from those for the mixed 0-1 RLT, yet they encompass these earlier results. We illustrate the approach through some examples, emphasizing the polyhedral structure afforded by the linearized LIPs. We also consider polynomial mixed-integer programs, exploitation of structure, and conditional-logic enhancements, and provide insight into relationships with a special-structure RLT implementation.

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Enumeration Approach for Linear Complementarity Problems Based on a Reformulation-Linearization Technique

Cited by 22 publications

References 25 publications

Optimization With Linear Complementarity Constraints

Optimization With Linear Complementarity Constraints

Reliability-Based Modeling of Park-and-Ride Service on Linear Travel Corridor

A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems

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