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A BoxConstrained Optimization Algorithm With Negative Curvature Directions and Spectral Projected Gradients
, 2001
"... A practical algorithm for boxconstrained optimization is introduced. The algorithm combines an activeset strategy with spectral projected gradient iterations. In the interior of each face a strategy that deals eciently with negative curvature is employed. Global convergence results are given. ..."
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Cited by 27 (5 self)
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A practical algorithm for boxconstrained optimization is introduced. The algorithm combines an activeset strategy with spectral projected gradient iterations. In the interior of each face a strategy that deals eciently with negative curvature is employed. Global convergence results are given. Numerical results are presented. Keywords: box constrained minimization, active set methods, spectral projected gradients, dogleg path methods. AMS Subject Classication: 49M07, 49M10, 65K, 90C06, 90C20. 1
On the Solution of Mathematical Programming Problems With Equilibrium Constraints
, 2001
"... Mathematical programming problems with equilibrium constraints (MPEC) are nonlinear programming problems where the constraints have a form that is analogous to firstorder optimality conditions of constrained optimization. We prove that, under reasonable sufficient conditions, stationary points of t ..."
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Cited by 12 (3 self)
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Mathematical programming problems with equilibrium constraints (MPEC) are nonlinear programming problems where the constraints have a form that is analogous to firstorder optimality conditions of constrained optimization. We prove that, under reasonable sufficient conditions, stationary points of the sum of squares of the constraints are feasible points of the MPEC. In usual formulations of MPEC all the feasible points are nonregular in the sense that they do not satisfy the MangasarianFromovitz constraint qualification of nonlinear programming. Therefore, all the feasible points satisfy the classical FritzJohn necessary optimality conditions. In principle, this can cause serious difficulties for nonlinear programming algorithms applied to MPEC. However, we show that most feasible points do not satisfy a recently introduced stronger optimality condition for nonlinear programming. This is the reason why, in general, nonlinear programming algorithms are successful when applied to MPEC. Keywords. Mathematical programming with equilibrium constraints, optimality conditions, minimization algorithms, reformulation. AMS: 90C33, 90C30
Reformulation Of Variational Inequalities On A Simplex And Compactification Of Complementarity Problems
 SIAM Journal on Optimization
, 2000
"... . Many variational inequality problems (VIP) can be reduced, by a compactification procedure, to a VIP on the canonical simplex. Reformulations of this problem are studied, including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized FischerBurmeis ..."
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Cited by 3 (1 self)
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. Many variational inequality problems (VIP) can be reduced, by a compactification procedure, to a VIP on the canonical simplex. Reformulations of this problem are studied, including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized FischerBurmeister function. It is proved that bounded level set results hold for these reformulations, under quite general assumptions on the operator. Therefore, it can be guaranteed that minimization algorithms generate bounded sequences and, under monotonicity conditions, these algorithms necessarily find solutions of the original problem. Some numerical experiments are presented. Key words. Variational inequalities, complementarity, minimization algorithms, reformulation. AMS subject classifications. 90C33, 90C30 1. Introduction. We are interested in reformulations of variational inequality problems (VIP) where the domain is a simplex. The main motivation is that variational inequalities on generaliz...
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"... limitedmemory multipoint symmetric secant method for bound constrained optimization ..."
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limitedmemory multipoint symmetric secant method for bound constrained optimization
A Globally Convergent Sequential Linear Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints 1
"... Abstract: This paper presents a sequential linear programming algorithm for computing a stationary point of a mathematical program with linear equilibrium constraints. The algorithm is based on a formulation of equilibrium constraints as a system of semismooth equations by means of a perturbed Fishe ..."
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Abstract: This paper presents a sequential linear programming algorithm for computing a stationary point of a mathematical program with linear equilibrium constraints. The algorithm is based on a formulation of equilibrium constraints as a system of semismooth equations by means of a perturbed FisherBurmeister functional. Using only data of the problem, we introduce a method to update the parameter that characterizes the aforesaid perturbed functional. Some computational results are reported.
On the Regularization of Mixed Complementarity Problems
, 1999
"... A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MCP). Monotonicity of the VIP implies that the MCP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity probl ..."
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A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MCP). Monotonicity of the VIP implies that the MCP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP. Keywords. Variational inequalities, complementarity, perturbations, inexact solutions, minimization algorithms, reformulation. AMS: 90C33, 90C30 1 Introduction The variational inequality problem was introduced as a tool in the study of partial differential equations [21]. Modern applications of the VIP include Department of Computer Scienc...
unknown title
, 2001
"... A boxconstrained optimization algorithm with negative curvature directions and spectral projected gradients ..."
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A boxconstrained optimization algorithm with negative curvature directions and spectral projected gradients