Nonmonotone Trust-Region Methods for Bound-Constrained Semismooth Equations with Applications to Nonlinear Mixed Complementarity Problems

We develop and analyze a class of trust-region methods for bound-constrained 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 Newton-like method that is augmented by a projection onto the feasible set. Under a Dennis--More-type condition we prove that close to a regular solution the trust-region algorithm turns into this projected Newton method, which is shown to converge locally q-superlinearly or quadratically, respectively, depending on the quality of the approximate subdifferentials used. As an important application we discuss how the developed algorithm can be used to solve nonlinear mixed complementarity problems (MCPs). Hereby, the MCP is converted into a bound-constrained semismooth equation by means of an NCP-function. The efficiency of our algorithm is documented by numerical results for a subset of the MCPLIB problem collection.

[1]  Patrick T. Harker,et al.  A nonsmooth Newton method for variational inequalities, I: Theory , 1994, Math. Program..

[2]  Jorge J. Moré,et al.  Global Methods for Nonlinear Complementarity Problems , 1994, Math. Oper. Res..

[3]  Liqun Qi,et al.  Trust Region Algorithms for Solving Nonsmooth Equations , 1995, SIAM J. Optim..

[4]  P. Toint.  Global Convergence of a a of Trust-Region Methods for Nonconvex Minimization in Hilbert Space , 1988 .

[5]  Jong-Shi Pang,et al.  NE/SQP: A robust algorithm for the nonlinear complementarity problem , 1993, Math. Program..

[6]  L. Qi.  Regular Pseudo-Smooth NCP and BVIP Functions and Globally and Quadratically Convergent Generalized Newton Methods for Complementarity and Variational Inequality Problems , 1999 .

[7]  Jiming Liu.  Strong Stability in Variational Inequalities , 1995 .

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

[9]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[10]  Liqun Qi,et al.  Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations , 1993, Math. Oper. Res..

[11]  B. Kummer.  NEWTON's METHOD FOR NON-DIFFERENTIABLE FUNCTIONS , 1988, Advances in Mathematical Optimization.

[12]  Christian Kanzow,et al.  A QP-free constrained Newton-type method for variational inequality problems , 1999, Math. Program..

[13]  R. Mifflin.  Semismooth and Semiconvex Functions in Constrained Optimization , 1977 .

[14]  Jong-Shi Pang,et al.  Nonsmooth Equations: Motivation and Algorithms , 1993, SIAM J. Optim..

[15]  J. M. Martínez,et al.  A new trust region algorithm for bound constrained minimization , 1994 .

[16]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[17]  W. Ziemer.  Weakly differentiable functions , 1989 .

[18]  S. M. Robinson.  Newton's method for a class of nonsmooth functions , 1994 .

[19]  Michael C. Ferris,et al.  Feasible descent algorithms for mixed complementarity problems , 1999, Math. Program..

[20]  Patrick T. Harker,et al.  A nonsmooth Newton method for variational inequalities, II: Numerical results , 1994, Math. Program..

[21]  Houyuan Jiang,et al.  A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems , 1997 .

[22]  Philippe L. Toint,et al.  Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints , 1997, Math. Program..

[23]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

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

[25]  Gerardo Toraldo,et al.  Convergence properties of trust region methods for linear and convex constraints , 1990, Math. Program..

[26]  Francisco Facchinei,et al.  A semismooth equation approach to the solution of nonlinear complementarity problems , 1996, Math. Program..

[27]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[28]  L. Grippo,et al.  A nonmonotone line search technique for Newton's method , 1986 .

[29]  A. Fischer.  A special newton-type optimization method , 1992 .

[30]  P. Toint,et al.  Correction to the paper on global convergence of a class of trust region algorithms for optimization with simple bounds , 1989 .

[31]  Michael C. Ferris,et al.  A Comparison of Large Scale Mixed Complementarity Problem Solvers , 1997, Comput. Optim. Appl..

[32]  Jong-Shi Pang,et al.  Newton's Method for B-Differentiable Equations , 1990, Math. Oper. Res..

[33]  Andreas Fischer,et al.  Solution of monotone complementarity problems with locally Lipschitzian functions , 1997, Math. Program..

[34]  S. Dirkse,et al.  Mcplib: a collection of nonlinear mixed complementarity problems , 1995 .

[35]  Francisco Facchinei,et al.  A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems , 1997, Math. Program..

[36]  P. Toint,et al.  Global convergence of a class of trust region algorithms for optimization with simple bounds , 1988 .

[37]  Defeng Sun,et al.  On NCP-Functions , 1999, Comput. Optim. Appl..

[38]  M. Heinkenschloss,et al.  Global Convergence of Trust-Region Interior-Point Algorithms for Infinite-Dimensional Nonconvex Mini , 1999 .

[39]  Chih-Jen Lin,et al.  Newton's Method for Large Bound-Constrained Optimization Problems , 1999, SIAM J. Optim..

[40]  J. Pang,et al.  A Trust Region Method for Constrained Nonsmooth Equations , 1994 .

[41]  B. Kummer.  Newton’s Method Based on Generalized Derivatives for Nonsmooth Functions: Convergence Analysis , 1992 .

[42]  Jong-Shi Pang,et al.  A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems , 1991, Math. Program..

[43]  S. Billups.  Algorithms for complementarity problems and generalized equations , 1996 .

[44]  Michael C. Ferris,et al.  Engineering and Economic Applications of Complementarity Problems , 1997, SIAM Rev..

[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]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..

[47]  Masao Fukushima,et al.  A Trust Region Method for Solving Generalized Complementarity Problems , 1998, SIAM J. Optim..