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Interiorpoint algorithms, penalty methods and equilibrium problems
, 2003
"... Abstract. In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)—as nonlinear programs, using an interiorpoint approach. These problems pose theoretical difficultie ..."
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Cited by 19 (4 self)
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Abstract. In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)—as nonlinear programs, using an interiorpoint approach. These problems pose theoretical difficulties for nonlinear solvers, including interiorpoint methods. We examine the use of penalty methods to get around these difficulties and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interiorpoint algorithms encounter in general. 1.
Complementarity Functions and Numerical Experiments on Some Smoothing Newton Methods for Secondordercone Complementarity Problems
, 2003
"... Two results on the secondordercone complementarity problem are presented. We show that the squared smoothing function is strongly semismooth. Under monotonicity and strict feasibility we provide a new proof, based on a penalized natural complementarity function, for the solution set of the secon ..."
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Cited by 18 (9 self)
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Two results on the secondordercone complementarity problem are presented. We show that the squared smoothing function is strongly semismooth. Under monotonicity and strict feasibility we provide a new proof, based on a penalized natural complementarity function, for the solution set of the secondordercone complementarity problem being bounded. Numerical results of squared smoothing Newton algorithms are reported.
NonMonotone TrustRegion Methods for BoundConstrained Semismooth Equations with Applications to Nonlinear Mixed Complementarity Problems
, 1999
"... We develop and analyze a class of trustregion methods for boundconstrained semismooth systems of equations. The algorithm is based on a simply constrained differentiable minimization reformulation. Our global convergence results are developed in a very general setting that allows for nonmonotoni ..."
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Cited by 14 (4 self)
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We develop and analyze a class of trustregion methods for boundconstrained semismooth systems of equations. The algorithm is based on a simply constrained differentiable minimization reformulation. Our global convergence results are developed in a very general setting that allows for nonmonotonicity of the function values at subsequent iterates. We propose a way of computing trial steps by a semismooth Newtonlike method that is augmented by a projection onto the feasible set. Under a DennisMoretype condition we prove that close to a BDregular solution the trustregion algorithm turns into this projected Newton method, which is shown to converge locally qsuperlinearly or quadratically, respectively, depending on the quality of the approximate BDsubdifferentials used. As an important application we discuss in detail how the developed algorithm can be used to solve nonlinear mixed complementarity problems (MCPs). Hereby, the MCP is converted into a boundconstrained semismooth...
Algorithms and Environments for Complementarity
, 2000
"... Complementarity problems arise in a wide variety of disciplines. Prototypical examples include the Wardropian and Walrasian equilibrium models encountered in the engineering and economic disciplines and the first order optimality conditions for nonlinear programs from the optimization community. The ..."
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Cited by 6 (0 self)
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Complementarity problems arise in a wide variety of disciplines. Prototypical examples include the Wardropian and Walrasian equilibrium models encountered in the engineering and economic disciplines and the first order optimality conditions for nonlinear programs from the optimization community. The main focus of this thesis is algorithms and environments for solving complementarity problems. Environments, such as AMPL and GAMS, are used by practitioners to easily write large, complex models. Support for these packages is provided by PATH 4.x and SEMI through the customizable solver interface specified in this thesis. The main design feature is the abstraction of core components from the code with implementations tailored to a particular environment supplied either at compile or run time. This solver interface is then used to develop new links to the MATLAB and NEOS tools. Preprocessing techniques are an integral part of linear and mixed integer programming codes and are primarily used to reduce the size and complexity of a model prior to solving it. For example, wasted computation is avoided when an infeasible model is detected.
Complementarity Problems
 J. Comput. Appl. Math
, 2000
"... This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed eectively with ..."
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Cited by 6 (0 self)
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This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed eectively with complementarity problems. The most important algorithms are presented along with a discussion of when they can be used eectively. We also provide a brief introduction to the study of matrix classes and their relation to linear complementarity problems. Finally, we provide a brief summary of current research trends. Key words: complementarity problems,variational inequalities, matrix classes 1 Introduction The distinguishing feature of a complementarity problem is the set of complementarity conditions. Each of these conditions requires that the product of two or more nonnegative quantities should be zero. (Here, each quantity is either a decision variable, or a function of the decisi...
The Traffic Equilibrium Problem with Nonadditive Costs and Its Monotone Mixed Complementarity Problem Formulation ∗
, 2005
"... Abstract. Various models of traffic equilibrium problems (TEPs) with nonadditive route costs have been proposed in the last decade. However, equilibria of those models are not easy to obtain because the variational inequality problems (VIPs) derived from those models are not monotone in general. In ..."
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Cited by 4 (0 self)
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Abstract. Various models of traffic equilibrium problems (TEPs) with nonadditive route costs have been proposed in the last decade. However, equilibria of those models are not easy to obtain because the variational inequality problems (VIPs) derived from those models are not monotone in general. In this paper, we consider a TEP whose route cost functions are nonadditive disutility functions of time (with money converted to time). We show that the TEP with the disutility functions can be reformulated as a monotone Mixed Complementarity Problem (MCP) under appropriate conditions. We then establish the existence and uniqueness results for an equilibrium of the TEP. Numerical experiments are carried out using various sample networks with different disutility functions for both the singlemode case and the case of two different transportation modes in the network.
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...
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...
A Smooth Newton Method for Nonlinear Programming Problems with Inequality Constraints
"... The paper presents a reformulation of the KarushKuhnTucker (KKT) system associated nonlinear programming problem into an equivalent system of smooth equations. Classical Newton method is applied to solve the system of equations. The superlinear convergence of the primal sequence, generated by prop ..."
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The paper presents a reformulation of the KarushKuhnTucker (KKT) system associated nonlinear programming problem into an equivalent system of smooth equations. Classical Newton method is applied to solve the system of equations. The superlinear convergence of the primal sequence, generated by proposed method, is proved. The preliminary numerical results with a problems test set are presented.