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15
A Smoothing Method For Mathematical Programs With Equilibrium Constraints
, 1996
"... The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalitie ..."
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Cited by 52 (5 self)
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The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalities are considered in this paper. They are transformed into an equivalent onelevel nonsmooth optimization problem. Then, a sequence of smooth, regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced. It is shown that the solutions (stationary points) of the approximate problems converge to a solution (stationary point) of the original MPEC problem. Numerical results showing viability of the approach are reported.
QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints
"... . We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) co ..."
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Cited by 20 (5 self)
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. We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, illconditioning, convexity of the objective, monotonicity and symmetry of the secondlevel problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.maths.mu.OZ.AU/~danny/qpecgendoc.h...
Reformulations in Mathematical Programming: A Computational Approach
"... Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathema ..."
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Cited by 17 (13 self)
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Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of blackbox functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization. 1
REFORMULATIONS IN MATHEMATICAL PROGRAMMING: DEFINITIONS AND SYSTEMATICS
, 2008
"... A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations c ..."
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Cited by 17 (13 self)
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A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are very common in mathematical programming but interestingly they have never been studied under a common framework. This paper attempts to move some steps in this direction. We define a framework for storing and manipulating mathematical programming formulations, give several fundamental definitions categorizing reformulations in essentially four types (optreformulations, narrowings, relaxations and approximations). We establish some theoretical results and give reformulation examples for each type.
Complementarity Constraint Qualifications and Simplified BStationarity Conditions for Mathematical Programs with Equilibrium Constraints
, 1998
"... With the aid of some novel complementarity constraint qualifications, we derive some simplied primaldual characterizations of a Bstationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a prog ..."
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Cited by 15 (6 self)
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With the aid of some novel complementarity constraint qualifications, we derive some simplied primaldual characterizations of a Bstationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a program near a feasible point. The simplied results, which rely heavily on a careful dissection and improved understanding of the tangent cone of the feasible region of the program, bypass the combinatorial characterization that is intrinsic to Bstationarity.
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
Piecewise Sequential Quadratic Programming For Mathematical Programs With . . .
"... We describe some first and secondorder optimality conditions for mathematical programs with equilibrium constraints (MPEC). Mathematical programs with parametric nonlinear complementarity constraints are the focus. Of interest is the result that under a linear independence assumption that is stand ..."
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Cited by 12 (5 self)
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We describe some first and secondorder optimality conditions for mathematical programs with equilibrium constraints (MPEC). Mathematical programs with parametric nonlinear complementarity constraints are the focus. Of interest is the result that under a linear independence assumption that is standard in nonlinear programming, the otherwise combinatorial problem of checking whether a point is stationary for an MPEC is reduced to checking stationarity of single nonlinear program. We also present a piecewise sequential quadratic programming (PSQP) algorithm for solving MPEC. Local quadratic convergence is shown under the linear independence assumption and a secondorder sufficient condition. Some computational results are given. KEY WORDS MPEC, bilevel program, nonlinear complementarity problem, nonlinear program, first and secondorder optimality conditions, linear independence constraint qualification, sequential quadratic programming, quadratic convergence. 2 Chapter 1 1 INTRODUC...
A Bundle Algorithm Applied to Bilevel Programming Problems With NonUnique Lower Level Solutions
, 1996
"... In the paper, the question is investigated if a bundle algorithm can be used to compute approximate solutions for bilevel programming problems where the lower level optimal solution is in general not uniquely determined. To give a positive answer to this question, an appropriate regularization appro ..."
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Cited by 4 (1 self)
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In the paper, the question is investigated if a bundle algorithm can be used to compute approximate solutions for bilevel programming problems where the lower level optimal solution is in general not uniquely determined. To give a positive answer to this question, an appropriate regularization approach is used in the lower level. In the general case, the resulting algorithm computes an approximate solution. If the problem proves to have strongly stable lower level solutions for all parameter values in a certain neighborhood of the stationary solutions of the bilevel problem, convergence to stationary solutions can be shown. Keywords: bilevel programming, parametric optimization, bundle algorithm, nondifferentiable optimization 1
A TrustRegion Method for Nonlinear Bilevel Programming: Algorithm and Computational Experience
, 2005
"... We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upperlevel objective and all constraintdefining functions, as well as a quadratic approximation of the lowerlevel objective ..."
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Cited by 4 (0 self)
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We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upperlevel objective and all constraintdefining functions, as well as a quadratic approximation of the lowerlevel objective. We describe the main features of the algorithm and the resulting software. Numerical experiments tend to confirm the promising behavior of the method.