Computational Optimization and Applications Manuscript No. on an Enumerative Algorithm for Solving Eigenvalue Complementarity Problems

In this paper, we discuss the solution of linear and quadratic eigenvalue complementarity problems (EiCPs) using an enumerative algorithm of the type introduced by Júdice et al. (Optim. Methods Softw. 24:549–586, 2009). Procedures for computing the interval that contains all the eigenvalues of the linear EiCP are first presented. A nonlinear programming (NLP) model for the quadratic EiCP is formulated next, and a necessary and sufficient condition for a stationary point of the NLP to be a solution of the quadratic EiCP is established. An extension of the enumerative algorithm for the quadratic EiCP is also developed, which solves this problem by computing a global minimum for the NLP formulation. Some computational experience is presented to highlight the efficiency and efficacy of the proposed enumerative algorithm for solving linear and quadratic EiCPs.

[1]  M. Seetharama Gowda,et al.  On the finiteness of the cone spectrum of certain linear transformations on Euclidean Jordan algebras , 2009 .

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

[3]  Joaquim Júdice,et al.  Efficient DC programming approaches for the asymmetric eigenvalue complementarity problem , 2013, Optim. Methods Softw..

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

[5]  Francisco Marcellán,et al.  A new numerical quadrature formula on the unit circle , 2007, Numerical Algorithms.

[6]  Joaquim Júdice,et al.  A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem , 2012, Comput. Optim. Appl..

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

[8]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[9]  Nikolaos V. Sahinidis,et al.  Exact Algorithms for Global Optimization of Mixed-Integer Nonlinear Programs , 2002 .

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

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

[12]  Alberto Seeger,et al.  Quadratic Eigenvalue Problems under Conic Constraints , 2011, SIAM J. Matrix Anal. Appl..

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

[14]  Bela Martos,et al.  Nonlinear programming theory and methods , 1977 .

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

[16]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[17]  Alberto Seeger,et al.  Local minima of quadratic forms on convex cones , 2009, J. Glob. Optim..

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

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

[20]  Golub Gene H. Et.Al.  Matrix Computations, 3rd Edition , 2007 .

[21]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

[22]  K. Lommatzsch.  Martos, B., Nonlinear Programming Theory and Methods, 280 S., Budapest. Akadémiai Kiadó. 1975. Ft 180,- , 1976 .

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

[24]  Hanif D. Sherali,et al.  A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique , 1992, J. Glob. Optim..

[25]  Hector A. Rosales-Macedo.  Nonlinear Programming: Theory and Algorithms (2nd Edition) , 1993 .

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

[27]  Silvério S. Rosa,et al.  Variational Inequality Formulation of the Asymmetric Eigenvalue Complementarity Problem and its Solution By Means of Gap Functions , 2011 .

[28]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

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

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

[31]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

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