Error Bounds for Regularized Complementarity Problems

An approach to solving a complementarity problem entails regularizing/perturbing the problem by adding to the given mapping another mapping multiplied by a small positive parameter. We study properties of the limit point of the solution to the regularized problem. We also derive local error bounds on the distance from the solution to its limit point, expressed in terms of the regularization parameter.

[1]  Z. Luo,et al.  Error Bounds for Quadratic Systems , 1999 .

[2]  Shu-Cherng Fang.  An unconstrained convex programming view of linear programming , 1992, ZOR Methods Model. Oper. Res..

[3]  Michael C. Ferris,et al.  QPCOMP: A quadratic programming based solver for mixed complementarity problems , 1997, Math. Program..

[4]  Franco Giannessi,et al.  Optimization and Related Fields , 1986 .

[5]  G. Isac.  Tihonov′s Regularization and the Complementarity Problem in Hilbert Spaces , 1993 .

[6]  O. Mangasarian.  Iterative Solution of Linear Programs , 1981 .

[7]  Shu-Cherng Fang,et al.  Deriving an unconstrained convex program for linear programming , 1992 .

[8]  Samuel Karlin,et al.  Mathematical Methods and Theory in Games, Programming, and Economics , 1961 .

[9]  G. Alberti,et al.  CALCULUS OF VARIATIONS, HOMOGENIZATION AND CONTINUUM MECHANICS , 1994 .

[10]  O. Mangasarian.  Some Applications of Penalty Functions in Mathematical Programming. , 1986 .

[11]  F. Facchinei.  Structural and Stability Properties of P 0 Nonlinear Complementarity Problems , 1998 .

[12]  M. Pinar,et al.  Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming , 1997 .

[13]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[14]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

[15]  D. Sun.  A Regularization Newton Method for Solving Nonlinear Complementarity Problems , 1999 .

[16]  J. Dussault,et al.  Stable exponential-penalty algorithm with superlinear convergence , 1994 .

[17]  S. Fang,et al.  Entropy Optimization and Mathematical Programming , 1997 .

[18]  Andreas Fischer.  Merit Functions and Stability for Complementarity Problems , 1998 .

[19]  Hédy Attouch,et al.  Viscosity Solutions of Minimization Problems , 1996, SIAM J. Optim..

[20]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[21]  P. Subramanian.  A note on least two norm solutions of monotone complementarity problems , 1988 .

[22]  V. Venkateswaran,et al.  An Algorithm for the Linear Complementarity Problem with a $P_0 $-Matrix , 1993, SIAM J. Matrix Anal. Appl..

[23]  Nimrod Megiddo,et al.  Homotopy Continuation Methods for Nonlinear Complementarity Problems , 1991, Math. Oper. Res..

[24]  Michael C. Ferris,et al.  Finite perturbation of convex programs , 1991 .

[25]  O. Mangasarian,et al.  NONLINEAR PERTURBATION OF LINEAR PROGRAMS , 1979 .

[26]  Roberto Cominetti,et al.  Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming , 1997, Math. Oper. Res..

[27]  Nimrod Megiddo,et al.  A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems , 1991, Lecture Notes in Computer Science.

[28]  G. Mitra.  Variational Inequalities and Complementarity Problems — Theory and Application , 1980 .

[29]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[30]  Roberto Cominetti,et al.  Asymptotic analysis of the exponential penalty trajectory in linear programming , 1994, Math. Program..

[31]  Osman Güler,et al.  Existence of Interior Points and Interior Paths in Nonlinear Monotone Complementarity Problems , 1993, Math. Oper. Res..

[32]  J. Frédéric Bonnans,et al.  Asymptotic Analysis of Congested Communication Networks , 2000, Math. Oper. Res..

[33]  P. Tseng,et al.  On the convergence of a matrix splitting algorithm for the symmetric monotone linear complementarity problem , 1991 .

[34]  A. Auslender.  Optimisation : méthodes numériques , 1976 .

[35]  Paul Tseng.  Convergence and Error Bound for Perturbation of Linear Programs , 1999, Comput. Optim. Appl..

[36]  Zhi-Quan Luo,et al.  Error bounds for analytic systems and their applications , 1994, Math. Program..

[37]  Shinji Mizuno,et al.  A General Framework of Continuation Methods for Complementarity Problems , 1993, Math. Oper. Res..

[38]  C. Kanzow,et al.  A Penalized Fischer-Burmeister Ncp-Function: Theoretical Investigation And Numerical Results , 1997 .

[39]  F. Facchinei,et al.  Beyond Monotonicity in Regularization Methods for Nonlinear Complementarity Problems , 1999 .

[40]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[41]  Mustafa Ç. Pinar,et al.  A New Finite Continuation Algorithm for Linear Programming , 1996, SIAM J. Optim..

[42]  Roger J.-B. Wets,et al.  Quantitative stability of variational systems: III.ε-approximate solutions , 1993, Math. Program..

[43]  Shu-Cherng Fang,et al.  Linear programming with entropic perturbation , 1993, ZOR Methods Model. Oper. Res..

[44]  Shinji Mizuno,et al.  Limiting Behavior of Trajectories Generated by a Continuation Method for Monotone Complementarity Problems , 1990, Math. Oper. Res..

[45]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[46]  O. Mangasarian.  Normal solutions of linear programs , 1984 .

[47]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[48]  Mohamed A. Tawhid,et al.  Existence and Limiting Behavior of Trajectories Associated with P0-equations , 1999, Comput. Optim. Appl..

[49]  L. McLinden.  An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem. , 1980 .

[50]  M. Seetharama Gowda,et al.  On the Limiting Behavior of the Trajectory of Regularized Solutions of a P0-Complementarity Problem , 1998 .

[51]  S. M. Robinson.  Generalized equations and their solutions, Part I: Basic theory , 1979 .

[52]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .