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109
Local convergence of SQP methods for Mathematical Programs with Equilibrium Constraints
, 2002
"... Recently, it has been shown that Nonlinear Programming solvers can successfully solve a range of Mathematical Programs with Equilibrium Constraints (MPECs). ..."
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Cited by 55 (17 self)
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Recently, it has been shown that Nonlinear Programming solvers can successfully solve a range of Mathematical Programs with Equilibrium Constraints (MPECs).
Algorithms For Complementarity Problems And Generalized Equations
, 1995
"... Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult pr ..."
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Cited by 41 (5 self)
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Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult problems are being proposed that exceed the capabilities of even the best algorithms currently available. There is, therefore, an immediate need to improve the capabilities of complementarity solvers. This thesis addresses this need in two significant ways. First, the thesis proposes and develops a proximal perturbation strategy that enhances the robustness of Newtonbased complementarity solvers. This strategy enables algorithms to reliably find solutions even for problems whose natural merit functions have strict local minima that are not solutions. Based upon this strategy, three new algorithms are proposed for solving nonlinear mixed complementarity problems that represent a significant improvement in robustness over previous algorithms. These algorithms have local Qquadratic convergence behavior, yet depend only on a pseudomonotonicity assumption to achieve global convergence from arbitrary starting points. Using the MCPLIB and GAMSLIB test libraries, we perform extensive computational tests that demonstrate the effectiveness of these algorithms on realistic problems. Second, the thesis extends some previously existing algorithms to solve more general problem classes. Specifically, the NE/SQP method of Pang & Gabriel (1993), the semismooth equations approach of De Luca, Facchinei & Kanz...
A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities
, 2000
"... ..."
A Path to the ArrowDebreu Competitive Market Equilibrium
 MATH. PROGRAMMING
, 2004
"... We present polynomialtime interiorpoint algorithms for solving the Fisher and ArrowDebreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛequilibrium solution. If the p ..."
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Cited by 36 (7 self)
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We present polynomialtime interiorpoint algorithms for solving the Fisher and ArrowDebreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛequilibrium solution. If the problem data are rational numbers and their bitlength is L, then the bound to generate an exact solution is O(n 4 L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound O(n 8 log(1/ɛ)) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the ArrowDebreu equilibrium, similar to the central path developed for linear programming interiorpoint methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixedpoint theorem. The defining equations are bilinear and possess some primaldual structure for the application of the Newtonbased pathfollowing method.
A Semismooth Newton Method For Variational Inequalities: Theoretical Results And Preliminary Numerical Experience
, 1997
"... Variational inequalities over sets defined by systems of equalities and inequalities are considered. A continuously differentiable merit function is proposed whose unconstrained minima coincide with the KKTpoints of the variational inequality. A detailed study of its properties is carried out showi ..."
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Cited by 34 (11 self)
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Variational inequalities over sets defined by systems of equalities and inequalities are considered. A continuously differentiable merit function is proposed whose unconstrained minima coincide with the KKTpoints of the variational inequality. A detailed study of its properties is carried out showing that under mild assumptions this reformulation possesses many desirable features. A simple algorithm is proposed for which it is possible to prove global convergence and a fast local convergence rate. Preliminary numerical results showing viability of the approach are reported.
NEOS and CONDOR: Solving Optimization Problems over the Internet
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1998
"... We discuss the use of Condor, a distributed resource management system, as a provider of computational resources for NEOS, an environment for solving optimization problems over the Internet. We also describe how problems are submitted and processed by NEOS, and then scheduled and solved by Condor ..."
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Cited by 34 (1 self)
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We discuss the use of Condor, a distributed resource management system, as a provider of computational resources for NEOS, an environment for solving optimization problems over the Internet. We also describe how problems are submitted and processed by NEOS, and then scheduled and solved by Condor on available (idle) workstations.
Oligopolistic Competition in Power Networks: A Conjectured Supply Function Approach
 IEEE Transactions on Power Systems
, 2002
"... Conjectured supply function (CSF) models of competition among power generators on a linearized DC network are presented. As a detailed survey of the power market modeling literature shows, CSF models differ from previous approaches in that they represent each GenCo's conjectures regarding how rival ..."
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Cited by 32 (6 self)
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Conjectured supply function (CSF) models of competition among power generators on a linearized DC network are presented. As a detailed survey of the power market modeling literature shows, CSF models differ from previous approaches in that they represent each GenCo's conjectures regarding how rival firms will adjust sales in response to price changes. The CSF approach is a more realistic and flexible framework for modeling imperfect competition than other models for three reasons. First, the models include as a special case the Cournot conjecture that rivals will not change production if prices change; thus, the CSF framework is more general. Second, Cournot models cannot be used when price elasticity of demand is zero, but the proposed models can. Third, unlike supply function equilibrium models, CSF equilibria can be calculated for large transmission networks. Existence and uniqueness properties for prices and profits are reported. An application shows how transmission limits and strategic interactions affect equilibrium prices under forced divestment of generation.
QPCOMP: A Quadratic Programming Based Solver for Mixed Complementarity Problems
 Mathematical Programming
, 1997
"... QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel[14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, th ..."
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Cited by 31 (15 self)
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QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel[14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudomonotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm. 1 Introduction This paper describes a new algorithm for solving the mixed nonlinear complementarity problem (MCP), which provides a significant improvement in robustness over previous superlinearly or quadratically convergent algorithms, while preserving these fast local convergence properties. The MCP is defined in...
A nonsmooth inexact Newton method for the solution of largescale nonlinear complementarity problems
, 1997
"... A new algorithm for the solution of largescale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact LevenbergMarquardttype algorithm for its solution. Under mild assumptions, and requiring o ..."
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Cited by 30 (5 self)
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A new algorithm for the solution of largescale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact LevenbergMarquardttype algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10000 variables are presented. 1 Introduction We consider the complementarity problem NCP(F ), which is to find a vector in IR n satisfying the conditions x 0; F (x) 0; x T F (x) = 0; where F : IR n ! IR n is a continuously differentiable function. Nonlinear complementarity problems have important applications, see, e.g., [11,19], which often call for the solution of largescale problems. During the last few years many methods have been developed for the solution of the non...
Semismooth Newton methods for operator equations in function spaces
, 2000
"... We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our resul ..."
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Cited by 28 (3 self)
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We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and fforder semismoothness from finitedimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi's finitedimensional Csubdifferential to our infinitedimensional framework. We apply these semismoothness results to develop a Newtonlike method for nonsmooth operator equations and prove its local qsuperlinear convergence to regular solutions. If the underlying operator is fforder semismoothness, convergence of qorder 1 + ff is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating e...