Results 1  10
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13
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...
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.
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 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 6 (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
Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints
, 2009
"... We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismo ..."
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Cited by 5 (4 self)
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We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismooth and BDregular at the solution under reasonable assumptions. Thus, fast local convergence can be obtained by applying the semismooth Newton method. Moreover, it turns out that the squared residual of the Lagrange system is continuously differentiable (even though the system itself is not), which opens the way for a natural globalization of the local algorithm.
An activeset Newton method for mathematical programs with complementarity constraints
 SIAM J. on Optimization
"... Abstract. For a mathematical program with complementarity constraints (MPCC), we propose an activeset Newton method, which has the property of local quadratic convergence under the MPCC linear independence constraint qualification (MPCCLICQ) and the standard secondorder sufficient condition (SOSC ..."
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Cited by 3 (3 self)
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Abstract. For a mathematical program with complementarity constraints (MPCC), we propose an activeset Newton method, which has the property of local quadratic convergence under the MPCC linear independence constraint qualification (MPCCLICQ) and the standard secondorder sufficient condition (SOSC) for optimality. Under MPCCLICQ, this SOSC is equivalent to the piecewise SOSC on branches of MPCC, which is weaker than the special MPCCSOSC often employed in the literature. The piecewise SOSC is also more natural than MPCCSOSC because, unlike the latter, it has an appropriate secondorder necessary condition as its counterpart. In particular, our assumptions for local quadratic convergence are weaker than those required by standard SQP when applied to MPCC and are equivalent to assumptions required by piecewise SQP for MPCC. Moreover, each iteration of our method consists of solving a linear system of equations instead of a quadratic program. Some globalization issues of the local scheme are also discussed, and illustrative examples and numerical experiments are presented.
Optimization with Equilibrium Constraints: A Piecewise SQP Approach, PSQP
, 1998
"... Introduction. The piecewise sequential quadratic programming (PSQP) method is a numerical method for solving certain mathematical programs with equilibrium constraints (MPEC), based on the classical sequential quadratic programming (SQP) method for nonlinear programs (NLP) [2, 12]. This descriptio ..."
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Cited by 2 (0 self)
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Introduction. The piecewise sequential quadratic programming (PSQP) method is a numerical method for solving certain mathematical programs with equilibrium constraints (MPEC), based on the classical sequential quadratic programming (SQP) method for nonlinear programs (NLP) [2, 12]. This description draws on both [9] and [4], which extend the original proposal for PSQP [13] that was restricted to the case of MPEC with linear complementarity constraints. See [7] for a brief account of an application of PSQP to a problem in civil engineering. It's performance on randomly generated quadratic programs with affine equilibrium constraints is documented in [4] and also in [9, 10]. PSQP can be applied directly to any MPEC whose lowerlevel problem is a mixed complementarity problem, and indirectly to any MPEC where the lowerlevel problem is a variational inequality (VI) that can be written via its KarushKuhnTucker (KKT
A merit function piecewise SQP algorithm for solving mathematical programs with equilibrium constraints
 J. Optim. Theory Appl
, 2006
"... Abstract. In this paper we propose a merit function piecewise SQP algorithm for solving mathematical programs with equilibrium constraints (MPECs) formulated as mathematical programs with complementarity constraints. Under some mild conditions, the new algorithm is globally convergent to a piecewis ..."
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Cited by 2 (1 self)
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Abstract. In this paper we propose a merit function piecewise SQP algorithm for solving mathematical programs with equilibrium constraints (MPECs) formulated as mathematical programs with complementarity constraints. Under some mild conditions, the new algorithm is globally convergent to a piecewise stationary point. Moreover if the partial MPECLICQ is satisfied at the accumulation point then the accumulation point is a Sstationary point.