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29
An implementable activeset algorithm for computing a Bstationary point of a mathematical program with linear complementarity constraints
 SIAM J. Optim
"... Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster poi ..."
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Cited by 22 (4 self)
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Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster points are Bstationary points of the problem. However, the proof has a gap and only shows that each cluster point is an Mstationary point. We discuss this gap and show that Bstationarity can be achieved if the algorithm is modified and an additional error bound condition holds. Key words. MPEC, Bstationary point, ɛactive set, error bound AMS subject classifications. 65K05, 90C30, 90C33
Numerical experience with solving MPECs as NLPs
 Department of Mathematics and Computer Science, University of Dundee, Dundee
, 2002
"... This paper describes numerical experience with solving MPECs as NLPs on a large collection of test problems. The key idea is to use offtheshelf NLP solvers to tackle large instances of MPECs. It is shown that SQP methods are very well suited to solving MPECs and at present outperform Interior Poin ..."
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Cited by 19 (1 self)
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This paper describes numerical experience with solving MPECs as NLPs on a large collection of test problems. The key idea is to use offtheshelf NLP solvers to tackle large instances of MPECs. It is shown that SQP methods are very well suited to solving MPECs and at present outperform Interior Point solvers both in terms of speed and reliability. All NLP solvers also compare very favourably to special MPEC solvers on tests published in the literature.
Stochastic mathematical programs with equilibrium constraints, modeling and . . .
 SCHOOL OF INDUSTRIAL AND SYSTEM ENGINEERING, GEORGIA INSTITUTE OF TECHNOLOGY
, 2005
"... In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We sho ..."
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Cited by 19 (5 self)
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In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized KarushKuhnTucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally we present some preliminary numerical test results.
THE THEORY OF 2REGULARITY FOR MAPPINGS WITH LIPSCHITZIAN DERIVATIVES AND ITS APPLICATIONS TO OPTIMALITY CONDITIONS
, 2002
"... We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2regularity (a certain kind of secondorder regularity) for a once differentiable mapping whose derivative is Lipschitz continuous ..."
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Cited by 17 (15 self)
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We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2regularity (a certain kind of secondorder regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to “righthand side ” perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2regular) equality constraints and optimality conditions for related optimization problems. The class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equationbased reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.
Generalized stationary points and an interiorpoint method for mathematical programs with equilibrium constraints
 Industrial Engineering & Management Sciences, Northwestern University
, 2005
"... Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier proble ..."
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Cited by 15 (0 self)
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Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or the linear independence constraint qualification for MPEC (MPECLICQ). Under certain general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interiorpoint algorithm failed to find a stationary point. Key words: Global convergence, interiorpoint methods, mathematical programming with equilibrium constraints, stationary point
Complementarity And Related Problems: A Survey
, 1998
"... This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to ..."
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Cited by 14 (0 self)
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This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.
Smoothing Implicit Programming Approaches For Stochastic Mathematical Programs With Linear Complementarity Constraints
, 2003
"... In this paper, we consider the stochastic mathematical program with equilibrium constraints (SMPEC) , which can be thought as a generalization of the mathematical program with equilibrium constraints. Many decision problems can be formulated as SMPECs in practice. We discuss both lowerlevel waitan ..."
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Cited by 7 (3 self)
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In this paper, we consider the stochastic mathematical program with equilibrium constraints (SMPEC) , which can be thought as a generalization of the mathematical program with equilibrium constraints. Many decision problems can be formulated as SMPECs in practice. We discuss both lowerlevel waitandsee and hereand now decision problems. For the lowerlevel waitandsee model, we propose a smoothing implicit programming method and establish a comprehensive convergence theory. For the hereandnow decision problem, we apply a penalty technique and suggest a similar method. We show that the two methods possess similar convergence properties.
A survey of some nonsmooth equations and smoothing Newton methods
 Progress in Optimization, volume 30 of Applied Optimization
, 1999
"... In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1 ..."
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Cited by 7 (2 self)
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In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1
On the Global Solution of Linear Programs with Linear Complementarity Constraints
, 2007
"... This paper presents a parameterfree integerprogramming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three ..."
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Cited by 6 (3 self)
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This paper presents a parameterfree integerprogramming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvability—of an LPEC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the form of satisfiability constraints are obtained. The feasibility problem of these inequalities and the carefully guided linear programming relaxations of the LPEC are the workhorse of the algorithm, which also employs a specialized procedure for the sparsification of the satifiability cuts. We establish the finite termination of the algorithm and report computational results using the algorithm for solving randomly generated LPECs of reasonable sizes. The results establish that the algorithm can handle infeasible, unbounded, and solvable LPECs effectively.
Bilevel model selection for support vector machines
, 2007
"... Abstract. The successful application of Support Vector Machines (SVMs), kernel methods and other statistical machine learning methods requires selection of model parameters based on estimates of the generalization error. This paper presents a novel approach to systematic model selection through bile ..."
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Cited by 6 (5 self)
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Abstract. The successful application of Support Vector Machines (SVMs), kernel methods and other statistical machine learning methods requires selection of model parameters based on estimates of the generalization error. This paper presents a novel approach to systematic model selection through bilevel optimization. We show how modelling tasks for widely used machine learning methods can be formulated as bilevel optimization problems and describe how the approach can address a broad range of tasks—among which are parameter, feature and kernel selection In addition, we also discuss the challenges in implementing these approaches and enumerate opportunities for future work in this emerging research area. 1.