A new method for solving Pareto eigenvalue complementarity problems

In this paper, we introduce a new method, called the Lattice Projection Method (LPM), for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems. The LPM is compared to SNMmin and SNMFB, two methods widely discussed in the literature for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool (Dolan, Moré in Math. Program. 91:201–213, 2002). The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market (Boisvert et al. in The quality of numerical software: assessment and enhancement, pp. 125–137, 1997), are computing time, number of iterations, number of failures and maximum number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.

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

[2]  Paul Horst,et al.  Relations amongm sets of measures , 1961 .

[3]  Joaquim J. Júdice,et al.  The directional instability problem in systems with frictional contacts , 2004 .

[4]  Ronald F. Boisvert,et al.  The Quality of Numerical Software: Assessment and Enhancement , 1996, Quality of Numerical Software.

[5]  Chong-sun Kim.  Canonical Analysis of Several Sets of Variables , 1973 .

[6]  Moody T. Chu,et al.  On a Multivariate Eigenvalue Problem, Part I: Algebraic Theory and a Power Method , 1993, SIAM J. Sci. Comput..

[7]  Joaquim Júdice,et al.  The symmetric eigenvalue complementarity problem , 2003, Math. Comput..

[8]  M. Raous,et al.  Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction , 1999 .

[9]  Alberto Seeger,et al.  On eigenvalues induced by a cone constraint , 2003 .

[10]  Yutaka Tanaka,et al.  Some generalized methods of optimal scaling and their asymptotic theories: The case of multiple responses-multiple factors , 1978 .

[11]  Karl Meerbergen,et al.  The Quadratic Eigenvalue Problem , 2001, SIAM Rev..

[12]  M. Fukushima,et al.  New NCP-Functions and Their Properties , 1997 .

[13]  A. Seeger.  Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions , 1999 .

[14]  Samir Adly,et al.  A nonsmooth algorithm for cone-constrained eigenvalue problems , 2011, Comput. Optim. Appl..

[15]  Alberto Seeger,et al.  ON CARDINALITY OF PARETO SPECTRA , 2011 .

[16]  Kimmo Berg,et al.  European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS) , 2004 .

[17]  Hanif D. Sherali,et al.  On the asymmetric eigenvalue complementarity problem , 2009, Optim. Methods Softw..

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

[19]  H. Hotelling.  The most predictable criterion. , 1935 .

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

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

[22]  H. Hotelling.  Relations Between Two Sets of Variates , 1936 .

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

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

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

[26]  J. A. C. Martins,et al.  Stability of finite-dimensional nonlinear elastic systems with unilateral contact and friction , 2000 .

[27]  Hanif D. Sherali,et al.  The eigenvalue complementarity problem , 2007, Comput. Optim. Appl..

[28]  Alberto Seeger,et al.  Numerical resolution of cone-constrained eigenvalue problems , 2009 .

[29]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[30]  Richard F. Barrett,et al.  Matrix Market: a web resource for test matrix collections , 1996, Quality of Numerical Software.

[31]  Helmut Kleinmichel,et al.  A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems , 1998, Comput. Optim. Appl..

[32]  Joaquim Júdice,et al.  On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm , 2008, Numerical Algorithms.

[33]  Alberto Seeger,et al.  Cone-constrained eigenvalue problems: theory and algorithms , 2010, Comput. Optim. Appl..

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

[35]  M. Chu,et al.  On a Multivariate Eigenvalue Problem � I Algebraic Theory and a Power Method , 2004 .

[36]  Michael C. Ferris,et al.  A pathsearch damped Newton method for computing general equilibria , 1996, Ann. Oper. Res..

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

[38]  Mohamed Hanafi,et al.  Global optimality of the successive Maxbet algorithm , 2003 .

[39]  Michael Ulbrich,et al.  Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces , 2011, MOS-SIAM Series on Optimization.

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

[41]  Alberto Seeger,et al.  Spectral analysis of coupled linear complementarity problems , 2010 .

[42]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

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