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72
R.J.: Interiorpoint methods for nonconvex nonlinear programming: orderings and higherorder methods
 Mathematical Programming Ser. B
, 2000
"... Abstract. In this paper, we present the formulation and solution of optimization problems with complementarity constraints using an interiorpoint method for nonconvex nonlinear programming. We identify possible difficulties that could arise, such as unbounded faces of dual variables, linear depend ..."
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Cited by 119 (8 self)
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Abstract. In this paper, we present the formulation and solution of optimization problems with complementarity constraints using an interiorpoint method for nonconvex nonlinear programming. We identify possible difficulties that could arise, such as unbounded faces of dual variables, linear dependence of constraint gradients and initialization issues. We suggest remedies. We include encouraging numerical results on the MacMPEC test suite of problems.
SOLVING MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS AS NONLINEAR PROGRAMS
 OPTIM. METHODS SOFTW. 19, 1540
, 2004
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Interior methods for mathematical programs with complementarity constraints
 SIAM J. Optim
, 2004
"... This paper studies theoretical and practical properties of interiorpenalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is sh ..."
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Cited by 37 (10 self)
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This paper studies theoretical and practical properties of interiorpenalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then extended to an interiorrelaxation approach. Superlinear convergence to strongly stationary points is also established. Two strategies for updating the penalty parameter are proposed, and their efficiency and robustness are studied on an extensive collection of test problems.
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 35 (7 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.
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 30 (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.
Using EPECs to model bilevel games in restructured electricity markets with locational prices
 Oper. Res
, 2007
"... These working papers present preliminary research findings, and you are advised to cite with caution unless you first contact the author regarding possible amendments. Using EPECs to model bilevel games in restructured electricity markets with locational prices ..."
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Cited by 27 (1 self)
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These working papers present preliminary research findings, and you are advised to cite with caution unless you first contact the author regarding possible amendments. Using EPECs to model bilevel games in restructured electricity markets with locational prices
An interior point method for mathematical programs with complementarity constraints (MPCCs
 SIAM Journal on Optimization
"... Abstract. Interior point methods for nonlinear programs (NLP) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard prim ..."
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Cited by 24 (1 self)
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Abstract. Interior point methods for nonlinear programs (NLP) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primaldual algorithm has been adapted with a modified step calculation. The algorithm is shown to be superlinearly convergent in the neighborhood of the solution set under assumptions of MPCCLICQ, strong stationarity and upper level strict complementarity. The modification can be easily accommodated within most nonlinear programming interior point algorithms with identical local behavior. Numerical experience is also presented and holds promise for the proposed method. Key words. Barrier method, MPECs, Complementarity, NLP. AMS subject classifications. 1. Introduction. The MPCC considered in the paper is, min f(x, w, y) x,w,y s.t. h(x, w, y) = 0 x ≥ 0
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 22 (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.
An algorithm for degenerate nonlinear programming with rapid local convergence
 SIAM J. Optim
, 2005
"... Abstract. The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. T ..."
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Cited by 21 (0 self)
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Abstract. The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. The first is any conventional method for nonlinear programming that produces estimates of the Lagrange multipliers at each iteration; the second is a technique for estimating the set of active constraint indices; the third is stabilized LagrangeNewton algorithm with rapid local convergence properties. Results concerning rapid local convergence and global convergence of the proposed framework are proved. The approach improves on existing approaches in that less restrictive assumptions are needed for convergence and/or the computational workload at each iteration is lower.
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 19 (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.