Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints

Hoheisel, Tim; Kanzow, Christian; Schwartz, Alexandra

doi:10.1007/s10107-011-0488-5

Cited by 159 publications

(112 citation statements)

References 26 publications

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“…It is well known that these complementarity constraints cause several difficulties, e.g., the constraint qualification (CQ) of Mangasarian-Fromovitz is violated in all feasible points. This class of standard MPCCs is well understood, both theoretically and numerically, we refer to Luo et al [1996], Scheel and Scholtes [2000], Hoheisel et al [2013] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Programs with Complementarity Constraints in Banach Spaces

Wachsmuth

2014

J Optim Theory Appl

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We consider optimization problems in Banach spaces involving a complementarity constraint defined by a convex cone K. By transferring the local decomposition approach, we define strong stationarity conditions and provide a constraint qualification under which these conditions are necessary for optimality. To apply this technique, we provide a new uniqueness result for Lagrange multipliers in Banach spaces. Under the additional assumption that the cone K is polyhedral, we show that our strong stationarity conditions possess a reasonable strength. Finally, we generalize to the case that K is not a cone and apply the theory to two examples.

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Section: Introductionmentioning

confidence: 99%

Mathematical Programs with Complementarity Constraints in Banach Spaces

Wachsmuth

2014

J Optim Theory Appl

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

confidence: 99%

“…This definition has implied a number of stationary concepts associated to the MPLCC [30,38,60,70,72,76,83,89,90]. Among them, strongly stationary, M-stationary and B-stationary points [30,38,76] are the most important and are briefly surveyed in this paper. Many algorithms have been developed in the past several years for computing stationary points of MPLCC [4,7,15,22,26,30,29,32,31,34,42,49,50,53,56,60,70,76,78,79].…”

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

confidence: 99%

“…However, a certificate for B-stationarity may be quite demanding for a degenerate feasible solution when its associated set I x ∩ I w has a relatively large number of elements [30]. The concepts of Strongly Stationary Point (SSP) and M-Stationary Point (MSP) have been introduced in the literature [30,38,76,83] and are more accessible for embedding with an algorithm. Their definitions are as follows: (68)- (73) with…”

Section: Finding a Stationary Pointmentioning

confidence: 99%

“…Note that any global minimum of MPLCC (1)- (5) is an MSP of f on K [38], but it may be not a SSP [83]. Furthermore any SSP is an MSP [38] and a BSP [30].…”

Section: Finding a Stationary Pointmentioning

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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