Results 1  10
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20
Some properties of regularization and penalization schemes for MPECs
 Optimization Methods and Software
, 2004
"... Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a secondorde ..."
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Cited by 19 (2 self)
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Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a secondorder sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. Existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
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
Elasticmode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties
 Math. Program
, 2005
"... Abstract. The elasticmode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first and secondorder necessary optimality conditions for the ..."
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Cited by 11 (1 self)
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Abstract. The elasticmode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first and secondorder necessary optimality conditions for the original problem are also first and secondorder points of the elasticmode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first or secondorder point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, Mstationarity, or Cstationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elasticmode formulation is solved inexactly. Key words. Nonlinear programming, equilibrium constraints, complementarity constraints, elasticmode formulation, strong stationarity, Cstationarity, Mstationarity. AMS subject classifications 49M30, 49M37, 65K05, 90C30, 90C33 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 outcomesâ€”infeasibility, unboundedness, or solvabilityâ€”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.
An interiorpoint method for MPECs based on strictly feasible relaxations
 Preprint ANL/MCSP11500404, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2004
"... Abstract. An interiorpoint method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty ..."
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Cited by 6 (0 self)
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Abstract. An interiorpoint method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fastlocal convergence properties of the algorithm. Key words. nonlinear programming, mathematical programs with equilibrium constraints, constrained minimization, interiorpoint methods, primaldual methods, barrier methods
On the global minimization of the valueatrisk
 Optimization Methods and Software
"... In this paper, we consider the nonconvex minimization problem of the valueatrisk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower ..."
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Cited by 6 (4 self)
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In this paper, we consider the nonconvex minimization problem of the valueatrisk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined bounding procedures can be used to compute the latter value to global optimality. A numerical example is provided to illustrate the methodology. Dedication. With great pleasure we dedicate this paper to a respected pioneer of our field, Professor Olvi L. Mangasarian, on the occasion of his 70th birthday. The two topics of this paper, LPECs and smoothing methods, are examples of the vast contributions that Olvi has made in optimization, which have benefited us in many ways and which will continue to benefit us in the future. Happy 70th birthday, Olvi! 1
A robust SQP method for mathematical programs with linear complementarity constraints
 Computational Optimization and Applications
, 2003
"... Abstract. The relationship between the mathematical program with linear complementarity constraints (MPCC) and its inequality relaxation is studied. A new sequential quadratic programming (SQP) method is presented for solving the MPCC based on this relationship. A certain SQP technique is introduced ..."
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Cited by 6 (0 self)
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Abstract. The relationship between the mathematical program with linear complementarity constraints (MPCC) and its inequality relaxation is studied. A new sequential quadratic programming (SQP) method is presented for solving the MPCC based on this relationship. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived without assuming the linear independence constraint qualification for MPEC and nondegeneracy of the complementarity constraints. Preliminary numerical results are reported. Key words: mathematical programs with equilibrium constraints, sequential quadratic programming, complementarity, constraint qualification, degeneracy
Model selection via bilevel programming
 Proceedings of the IEEE International Joint Conference on Neural Networks
, 2006
"... Support vector machines and related classification models require the solution of convex optimization problems that have one or more regularization hyperparameters. Typically, the hyperparameters are selected to minimize cross validated estimates of the outofsample classification error of the mo ..."
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Cited by 5 (5 self)
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Support vector machines and related classification models require the solution of convex optimization problems that have one or more regularization hyperparameters. Typically, the hyperparameters are selected to minimize cross validated estimates of the outofsample classification error of the model. This crossvalidation optimization problem can be formulated as a bilevel program in which the outerlevel objective minimizes the average number of misclassified points across the crossvalidation folds, subject to innerlevel constraints such that the classification functions for each fold are (exactly or nearly) optimal for the selected hyperparameters. Feature selection is included in the bilevel program in the form of bound constraints in the weights. The resulting bilevel problem is converted to a mathematical program with linear equilibrium constraints, which is solved using stateoftheart optimization methods. This approach is significantly more versatile than commonly used grid search procedures, enabling, in particular, the use of models with many hyperparameters. Numerical results demonstrate the practicality of this approach for model selection in machine learning.
A Hybrid Algorithm with Active Set Identification for Mathematical Programs with Complementarity Constraints
, 2002
"... We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. To this end, we first introduce an active set identif ..."
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Cited by 4 (1 self)
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We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. To this end, we first introduce an active set identification technique. Then, by applying this technique to a smoothing continuation method presented by Fukushima and Pang (1999), we propose a hybrid method for solving MPCC. Under reasonable assumptions, the hybrid algorithm is shown to possess a finite termination property. Numerical experience shows that the proposed approach is quite e#ective.
Hybrid approach with active set identification for mathematical programs with complementarity constraints
 J. of Optimization Theory and Applications
"... Abstract. We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. We apply an active set identification techn ..."
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Cited by 2 (0 self)
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Abstract. We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. We apply an active set identification technique to a smoothing continuation method (Fukushima and Pang, 1999) and propose a hybrid algorithm for solving MPCC. We also develop two kinds of modifications, one of which makes use of an index addition strategy and the other adopts an index subtraction strategy. We show that, under reasonable assumptions, all the proposed algorithms possess a finite termination property. Further discussions and computational results are given as well. Key Words. mathematical program with complementarity constraints, MPCCLICQ, weak secondorder necessary condition, (B, M, C) stationarity, asymptotically weak nondegeneracy, identification function. 1