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44
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 22 (8 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.
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.
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.
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 18 (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.
Newtontype methods for optimization problems without constraint qualifications
 SIAM Journal on Optimization
, 2004
"... Abstract. We consider equalityconstrained optimization problems, where a given solution may not satisfy any constraint qualification but satisfies the standard secondorder sufficient condition for optimality. Based on local identification of the rank of the constraints degeneracy via the singular ..."
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Cited by 16 (12 self)
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Abstract. We consider equalityconstrained optimization problems, where a given solution may not satisfy any constraint qualification but satisfies the standard secondorder sufficient condition for optimality. Based on local identification of the rank of the constraints degeneracy via the singularvalue decomposition, we derive a modified primaldual optimality system whose solution is locally unique, nondegenerate, and thus can be found by standard Newtontype techniques. Using identification of active constraints, we further extend our approach to mixed equality and inequalityconstrained problems, and to mathematical programs with complementarity constraints (MPCC). In particular, for MPCC we obtain a local algorithm with quadratic convergence under the secondorder sufficient condition only, without any constraint qualifications, not even the special MPCC constraint qualifications.
On attraction of Newtontype iterates to multipliers violating secondorder sufficiency conditions
, 2009
"... Assuming that the primal part of the sequence generated by a Newtontype (e.g., SQP) method applied to an equalityconstrained problem converges to a solution where the constraints are degenerate, we investigate whether the dual part of the sequence is attracted by those Lagrange multipliers which s ..."
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Cited by 16 (15 self)
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Assuming that the primal part of the sequence generated by a Newtontype (e.g., SQP) method applied to an equalityconstrained problem converges to a solution where the constraints are degenerate, we investigate whether the dual part of the sequence is attracted by those Lagrange multipliers which satisfy secondorder sufficient condition (SOSC) for optimality, or by those multipliers which violate it. This question is relevant at least for two reasons: one is speed of convergence of standard methods; the other is applicability of some recently proposed approaches for handling degenerate constraints. We show that for the class of damped Newton methods, convergence of the dual sequence to multipliers satisfying SOSC is unlikely to occur. We support our findings by numerical experiments. We also suggest a simple auxiliary procedure for computing multiplier estimates, which does not have this
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
A comparison of electricity market design in networks. Cambridge Working
 Department of Applied Economics, University of Cambridge
, 2003
"... In the real world two classes of market designs are implemented to trade electricity in transmission constrained networks. Analytical results show that in two node networks integrated market designs reduce the ability of electricity generators to exercise market power relative to separated market de ..."
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Cited by 11 (1 self)
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In the real world two classes of market designs are implemented to trade electricity in transmission constrained networks. Analytical results show that in two node networks integrated market designs reduce the ability of electricity generators to exercise market power relative to separated market designs. In multi node networks countervailing effects make an analytic analysis difficult. We present a formulation of both market designs as an equilibrium problem with equilibrium constraints. We find that in a realistic network, prices are lower with the integrated market design. 1
Examples of dual behaviour of Newtontype methods on optimization problems with degenerate constraints
 Computational Optimization and Applications
"... discuss possible scenarios of behaviour of the dual part of sequences generated by primaldual Newtontype methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a so ..."
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Cited by 11 (9 self)
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discuss possible scenarios of behaviour of the dual part of sequences generated by primaldual Newtontype methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a socalled critical multiplier (which, in particular, violates some secondorder sufficient conditions for optimality), the latter appearing to be a typical scenario when critical multipliers exist; (c) convergence to a noncritical multiplier. The case of mathematical programs with complementarity constraints is also discussed. We illustrate those scenarios with examples, and discuss consequences for the speed of convergence. We also put together a collection of examples of optimization problems with constraints violating some standard constraint qualifications, intended for preliminary testing of existing algorithms on degenerate problems, or for developing special new algorithms designed to deal with constraints degeneracy. Keywords Degenerate constraints · Secondorder sufficiency · Newton method · SQP