A New Method for Solving Second-Order Cone Eigenvalue Complementarity Problems

In this paper, we study numerical methods for solving eigenvalue complementarity problems involving the product of second-order cones (or Lorentz cones). We reformulate such problem to find the roots of a semismooth function. An extension of the Lattice Projection Method (LPM) to solve the second-order cone eigenvalue complementarity problem is proposed. The LPM is compared to the semismooth Newton methods, associated to the Fischer–Burmeister and the natural residual functions. The performance profiles highlight the efficiency of the LPM. A globalization of these methods, based on the smoothing and regularization approaches, are discussed.

[1]  J. Frédéric Bonnans,et al.  Perturbation analysis of second-order cone programming problems , 2005, Math. Program..

[2]  Hiroshi Yamashita,et al.  A primal–dual interior point method for nonlinear optimization over second-order cones , 2009, Optim. Methods Softw..

[3]  Paul Tseng,et al.  Analysis of nonsmooth vector-valued functions associated with second-order cones , 2004, Math. Program..

[4]  Masao Fukushima,et al.  An SQP-type algorithm for nonlinear second-order cone programs , 2007, Optim. Lett..

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

[6]  Jein-Shan Chen,et al.  Alternative proofs for some results of vector-valued functions associated with second-order cone , 2005 .

[7]  Sara Bozzini.  Fourth European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004) , 2005 .

[8]  Defeng Sun,et al.  Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems , 2003, Math. Oper. Res..

[9]  F. Facchinei,et al.  A semismooth Newton method for variational in - equalities: The case of box constraints , 1997 .

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

[11]  Masao Fukushima,et al.  Smoothing Functions for Second-Order-Cone Complementarity Problems , 2002, SIAM J. Optim..

[12]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

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

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

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

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

[17]  Samir Adly,et al.  A new method for solving Pareto eigenvalue complementarity problems , 2013, Comput. Optim. Appl..

[18]  Defeng Sun,et al.  Complementarity Functions and Numerical Experiments on Some Smoothing Newton Methods for Second-Order-Cone Complementarity Problems , 2003, Comput. Optim. Appl..

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

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

[21]  H. Upmeier.  ANALYSIS ON SYMMETRIC CONES (Oxford Mathematical Monographs) , 1996 .

[22]  R. Sznajder,et al.  Some P-properties for linear transformations on Euclidean Jordan algebras , 2004 .

[23]  Defeng Sun,et al.  Strong Semismoothness of the Fischer-Burmeister SDC and SOC Complementarity Functions , 2005, Math. Program..

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

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

[26]  Masao Fukushima,et al.  On the Local Convergence of Semismooth Newton Methods for Linear and Nonlinear Second-Order Cone Programs Without Strict Complementarity , 2009, SIAM J. Optim..

[27]  Naihua Xiu,et al.  The Fischer-Burmeister Complementarity Function on Euclidean Jordan Algebras∗ , 2008 .

[28]  J. Faraut,et al.  Analysis on Symmetric Cones , 1995 .

[29]  Jein-Shan Chen,et al.  A Damped Gauss-Newton Method for the Second-Order Cone Complementarity Problem , 2009 .

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

[31]  Masao Fukushima,et al.  A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems , 2005, SIAM J. Optim..

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

[33]  M. Koecher,et al.  The Minnesota Notes on Jordan Algebras and Their Applications , 1999 .

[34]  Paul Tseng,et al.  An unconstrained smooth minimization reformulation of the second-order cone complementarity problem , 2005, Math. Program..

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

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

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

[38]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..