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16
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.
Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints
 SIAM Journal on Optimization
, 1997
"... Mathematical programs with nonlinear complementarity constraints are reformulated using betterposed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the ex ..."
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Cited by 35 (0 self)
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Mathematical programs with nonlinear complementarity constraints are reformulated using betterposed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations. We present two globally convergent algorithms based on sequential quadratic programming, SQP, as applied in exact penalty methods for nonlinear programs. Global convergence of the implicit smooth SQP method depends on existence of a lowerlevel nondegenerate (strictly complementary) limit point of the iteration sequence. Global convergence of the explicit smooth SQP method depends on a weaker property, i.e. existence of a limit point at which a generalized constraint qualification holds. We also discuss some practical matter...
Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities
 Mathematics of Computation
, 1998
"... Abstract. The smoothing Newton method for solving a system of nonsmooth equations F (x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth functio ..."
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Cited by 34 (16 self)
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Abstract. The smoothing Newton method for solving a system of nonsmooth equations F (x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f(·,εk), and the derivative of f(·,εk) at x k is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if F is semismooth at the solution and f satisfies a Jacobian consistency property. We show that most common smooth functions, such as the GabrielMoré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is P –uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically). 1.
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...
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
Complementarity And Related Problems: A Survey
, 1998
"... This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to ..."
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Cited by 14 (0 self)
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This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.
Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints
, 2005
"... ..."
A modified relaxation scheme for mathematical programs with complementarity constraints
 Annals of Operations Research
, 2002
"... In [1], a modified relaxation method was proposed for mathematical programs with complementarity constraints and some new sufficient conditions for M or Bstationarity were shown. However, due to an ignored sign in the Lagrangian function of the relaxed problem, the proofs of Theorems 3.4 and 3.5 i ..."
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Cited by 4 (4 self)
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In [1], a modified relaxation method was proposed for mathematical programs with complementarity constraints and some new sufficient conditions for M or Bstationarity were shown. However, due to an ignored sign in the Lagrangian function of the relaxed problem, the proofs of Theorems 3.4 and 3.5 in [1] are incorrect. In what follows, we give the corrected proofs. Throughout, we use the same notations as in [1]. Theorem 3.4. Let {ɛk} ⊆ (0, +∞) be convergent to 0 and z k ∈ Fɛk be a stationary point of problem (3) with ɛ = ɛk and multiplier vectors λ k, µ k, δ k, and γ k. Suppose that, for each k, ∇ 2 zLɛk (zk, λ k, µ k, δ k, γ k) is bounded below with constant αk on the corresponding tangent space Tɛk (zk). Let ¯z be an accumulation point of the sequence {z k}. If the sequence {αk} is bounded and the MPECLICQ holds at ¯z, then ¯z is an Mstationary point of problem (1). Proof. Assume that limk→ ∞ z k = ¯z without loss of generality. First of all, we note from Theorem 3.3 that ¯z is a Cstationary point of problem (1). To prove the theorem, we assume to the contrary that ¯z is not Mstationary to problem (1). Then, it follows from the definitions of Cstationarity and Mstationarity that there must exist an i0 ∈ IG(¯z) ∩ IH(¯z) such that
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.