The Semismooth Algorithm for Large Scale Complementarity Problems

Complementarity solvers are continually being challenged by modelers demanding improved reliability and scalability. Building upon a strong theoretical background, the semismooth algorithm has the potential to meet both of these requirements. We discuss relevant theory associated with the algorithm and then describe a sophisticated implementation in detail. Particular emphasis is given to the use of preconditioned iterative methods to solve the (nonsymmetric) systems of linear equations generated at each iteration and robust methods for dealing with singularity. Results on the MCPLIB test suite indicate that the code is reliable and efficient and scales well to very large problems.

[1]  Xiaojun Chen,et al.  A penalized Fischer-Burmeister NCP-function , 2000, Math. Program..

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

[3]  N. Josephy.  Newton's Method for Generalized Equations. , 1979 .

[4]  Roland W. Freund,et al.  QMRPACK: a package of QMR algorithms , 1996, TOMS.

[5]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

[6]  Michael C. Ferris,et al.  Expressing Complementarity Problems in an Algebraic Modeling Language and Communicating Them to Solvers , 1999, SIAM J. Optim..

[7]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[8]  Michael C. Ferris,et al.  NEOS and Condor: solving optimization problems over the Internet , 2000, TOMS.

[9]  Stephen M. Robinson,et al.  Normal Maps Induced by Linear Transformations , 1992, Math. Oper. Res..

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

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

[12]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

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

[14]  C. Lemaréchal,et al.  The watchdog technique for forcing convergence in algorithms for constrained optimization , 1982 .

[15]  A. D. Vany,et al.  Pipeline Access and Market Integration in the Natural Gas Industry: Evidence from Cointegration Tests , 1993 .

[16]  L. Mathiesen.  Computation of economic equilibria by a sequence of linear complementarity problems , 1985 .

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

[18]  M. Ferris,et al.  Complementarity problems in GAMS and the PATH solver 1 This material is based on research supported , 2000 .

[19]  B. AfeArd.  CALCULATING THE SINGULAR VALUES AND PSEUDOINVERSE OF A MATRIX , 2022 .

[20]  S. Dirkse,et al.  The path solver: a nommonotone stabilization scheme for mixed complementarity problems , 1995 .

[21]  M. Ferris,et al.  Nonmonotone stabilization methods for nonlinear equations , 1994 .

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

[23]  C. Kanzow,et al.  Hamburger Beitr Age Zur Angewandten Mathematik a Penalized Fischer-burmeister Ncp-function: Theoretical Investigation and Numerical Results Hamburger Beitr Age Zur Angewandten Mathematik Reihe a Preprints Reihe B Berichte Reihe C Mathematische Modelle Und Simulation Reihe D Elektrische Netzwerke Und , 2007 .

[24]  P. Gill,et al.  Maintaining LU factors of a general sparse matrix , 1987 .

[25]  Michael C. Ferris,et al.  Complementarity and variational problems : state of the art , 1997 .

[26]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

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

[28]  Robert J. Vanderbei,et al.  Linear Programming: Foundations and Extensions , 1998, Kluwer international series in operations research and management service.

[29]  A. S. Manne,et al.  International Trade in Oil, Gas and Carbon Emission Rights: An Intertemporal General Equilibrium Model* , 1994 .

[30]  David Kendrick,et al.  GAMS, a user's guide , 1988, SGNM.

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

[32]  L. Grippo,et al.  A class of nonmonotone stabilization methods in unconstrained optimization , 1991 .

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

[34]  Daniel Ralph,et al.  Global Convergence of Damped Newton's Method for Nonsmooth Equations via the Path Search , 1994, Math. Oper. Res..

[35]  Francisco Facchinei,et al.  A Theoretical and Numerical Comparison of Some Semismooth Algorithms for Complementarity Problems , 2000, Comput. Optim. Appl..

[36]  Michael C. Ferris,et al.  Interfaces to PATH 3.0: Design, Implementation and Usage , 1999, Comput. Optim. Appl..

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

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

[39]  David G. Tarr,et al.  Quantifying the Uruguay Round , 1997 .

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

[41]  T. Rutherford.  Extension of GAMS for complementarity problems arising in applied economic analysis , 1995 .

[42]  Michael A. Saunders,et al.  MINOS 5. 0 user's guide , 1983 .

[43]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.