Equilibrium problems involving the Lorentz cone

We study a general equilibrium model formulated as a smooth system of equations coupled with complementarity conditions relative to the $$n$$-dimensional Lorentz cone. For the purpose of analysis, as well as for the design of algorithms, we exploit the fact that the Lorentz cone is representable as a cone of squares in a suitable Euclidean Jordan algebra.

[1]  M. Malik,et al.  On Q and R0 properties of a quadratic representation in linear complementarity problems over the second-order cone , 2005 .

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

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

[4]  Peter Lancaster,et al.  Lambda-matrices and vibrating systems , 2002 .

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

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

[7]  P. Parrilo,et al.  Symmetry groups, semidefinite programs, and sums of squares , 2002, math/0211450.

[8]  Olvi L. Mangasarian,et al.  Absolute value programming , 2007, Comput. Optim. Appl..

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

[10]  Oleg A. Prokopyev,et al.  On equivalent reformulations for absolute value equations , 2009, Comput. Optim. Appl..

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

[12]  O. Mangasarian,et al.  Absolute value equations , 2006 .

[13]  Roberto Andreani,et al.  Box-constrained minimization reformulations of complementarity problems in second-order cones , 2008, J. Glob. Optim..

[14]  Pablo A. Parrilo,et al.  Computing sum of squares decompositions with rational coefficients , 2008 .

[15]  Jean B. Lasserre,et al.  Moments and sums of squares for polynomial optimization and related problems , 2009, J. Glob. Optim..

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

[17]  H. Walker,et al.  Least-change secant update methods for undetermined systems , 1990 .

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

[19]  Qiong Zhang,et al.  A generalized Newton method for absolute value equations associated with second order cones , 2011, J. Comput. Appl. Math..

[20]  Olvi L. Mangasarian,et al.  A generalized Newton method for absolute value equations , 2009, Optim. Lett..

[21]  A. Markus,et al.  Introduction to the Spectral Theory of Polynomial Operator Pencils , 2012 .

[22]  Naihua Xiu,et al.  The solution set structure of monotone linear complementarity problems over second-order cone , 2008, Oper. Res. Lett..

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

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

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

[26]  Jiri Rohn,et al.  A theorem of the alternatives for the equation |Ax| − |B||x| = b , 2004, Optimization Letters.