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17
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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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.
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 (1 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
Leaderfollower equilibria for electric power and NOx allowances markets
 Computational Management Science
, 2004
"... This paper investigates the ability of the largest producer in an electricity market to manipulate both the electricity and emission allowances markets to its advantage. A Stackelberg game to analyze this situation is constructed in which the largest firm plays the role of the leader, while the medi ..."
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Cited by 12 (4 self)
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This paper investigates the ability of the largest producer in an electricity market to manipulate both the electricity and emission allowances markets to its advantage. A Stackelberg game to analyze this situation is constructed in which the largest firm plays the role of the leader, while the mediumsized firms are treated as Cournot followers with pricetaking fringes that behave competitively in both markets. Since there is no explicit representation of the bestreply function for each follower, this Stackelberg game is formulated as a largescale mathematical program with equilibrium constraints. The bestreply functions are implicitly represented by a set of nonlinear complementarity conditions. Analysis of the computed solution for the Pennsylvania New Jersey Maryland electricity market shows that the leader can gain substantial profits by withholding allowances and driving up NOx allowance costs for rival producers. The allowances price is higher than the corresponding price in the NashCournot case, although the electricity prices are essentially the same. 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 8 (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
A note on multiobjective optimization and complementarity constraints
 PREPRINT ANL/MCSP12900905, MATHEMATICS AND COMPUTER SCIENCE DIVISION, ARGONNE NATIONAL LABORATORY
, 2005
"... We propose a new approach to convex nonlinear multiobjective optimization that captures the geometry of the Pareto set by generating a discrete set of Pareto points optimally. We show that the problem of finding an optimal representation of the Pareto surface can be formulated as a mathematical prog ..."
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We propose a new approach to convex nonlinear multiobjective optimization that captures the geometry of the Pareto set by generating a discrete set of Pareto points optimally. We show that the problem of finding an optimal representation of the Pareto surface can be formulated as a mathematical program with complementarity constraints. The complementarity constraints arise from modeling the set of Pareto points, and the objective maximizes some quality measure of this discrete set. We present encouraging numerical experience on a range of test problem collected from the literature.
On Perspective Functions and Vanishing Constraints in MixedInteger Nonlinear Optimal Control
"... Abstract Logical implications appear in a number of important mixedinteger nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an o ..."
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Abstract Logical implications appear in a number of important mixedinteger nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, and vanishing constraints. In addition to the tightness of the respective relaxations, we also address the issue of constraint qualification and the behavior of computational methods for some formulations. As a benchmark, we formulate a truck cruise control problem with logical implications resulting from gearchoice specific constraints. We provide this benchmark problem in AMPL format along with different realistic scenarios. Computational results for this benchmark are used to investigate feasibility gaps, integer feasibility gaps, quality of local solutions, and wellbehavedness of numerical methods for the presented reformulations of the benchmark problem. Vanishing constraints give the most satisfactory results. 1
A NEW RELAXATION SCHEME FORMATHEMATICAL PROGRAMSWITH EQUILIBRIUM CONSTRAINTS
"... Abstract. We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently nondegenerate complementarity components ..."
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Abstract. We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently nondegenerate complementarity components and relaxes only the remaining complementarity conditions. A positive parameter determines to what extent the complementarity conditions are relaxed. The relaxation scheme is such that a strongly stationary solution of the MPEC is also a solution of the relaxed problem if the relaxation parameter is chosen sufficiently small. We discuss the properties of the resulting parameterized nonlinear programs and compare stationary points and solutions. We further prove that a limit point of a sequence of stationary points of a sequence of relaxed problems is Cstationary if it satisfies a socalled MPECconstant rank constraint qualification and it is Mstationary if it satisfies the MPEClinear independence constraint qualification and the stationary points satisfy a second order sufficient condition. From this relaxation scheme, a numerical approach is derived that is applied to a comprehensive testset. The numerical results show that the approach combines good efficiency with high robustness. 1. Introduction. We
AND THE COMMITTEE ON GRADUATE STUDIES
, 2015
"... Redistributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution3.0 United States License. ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degr ..."
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Redistributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution3.0 United States License. ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Walter Murray, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate