A semidefinite method for tensor complementarity problems

In this paper, we study how to compute all real solutions of the tensor complimentary problem, if there are finite many ones. We formulate the problem as a sequence of polynomial optimization problems. The solutions can be computed sequentially. Each of them can be obtained by solving Lasserre's hierarchy of semidefinite relaxations. A semidefinite algorithm is proposed and its convergence properties are discussed. Some numerical experiments are also presented.

[1]  Monique Laurent,et al.  Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals , 2008, Found. Comput. Math..

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

[3]  C. SIAMJ. A NEW NONSMOOTH EQUATIONS APPROACH TO NONLINEAR COMPLEMENTARITY PROBLEMS∗ , 1997 .

[4]  Raul E. Curto,et al.  Truncated K-moment problems in several variables , 2005 .

[5]  Liqun Qi,et al.  Tensor Complementarity Problem and Semi-positive Tensors , 2015, J. Optim. Theory Appl..

[6]  Jiawang Nie,et al.  The hierarchy of local minimums in polynomial optimization , 2013, Mathematical Programming.

[7]  Naihua Xiu,et al.  The sparsest solutions to Z-tensor complementarity problems , 2015, Optim. Lett..

[8]  O. Mangasarian Equivalence of the Complementarity Problem to a System of Nonlinear Equations , 1976 .

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

[10]  J. William Helton,et al.  A Semidefinite Approach for Truncated K-Moment Problems , 2012, Foundations of Computational Mathematics.

[11]  Jiawang Nie,et al.  Certifying convergence of Lasserre’s hierarchy via flat truncation , 2011, Math. Program..

[12]  M. Fukushima,et al.  A New Derivative-Free Descent Method for the Nonlinear Complementarity Problem , 2000 .

[13]  Olvi L. Mangasarian,et al.  Nonlinear complementarity as unconstrained and constrained minimization , 1993, Math. Program..

[14]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[15]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[16]  Didier Henrion,et al.  GloptiPoly 3: moments, optimization and semidefinite programming , 2007, Optim. Methods Softw..

[17]  Yimin Wei,et al.  Positive-Definite Tensors to Nonlinear Complementarity Problems , 2015, J. Optim. Theory Appl..

[18]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[19]  P. Rostalski,et al.  Semidefinite characterization and computation of real radical ideals , 2006 .

[20]  Yong Wang,et al.  Global Uniqueness and Solvability for Tensor Complementarity Problems , 2015, J. Optim. Theory Appl..

[21]  Liqun Qi,et al.  Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors , 2014, 1412.0113.

[22]  Defeng Sun,et al.  A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities , 2000, Math. Program..

[23]  Jiawang Nie,et al.  Polynomial Optimization with Real Varieties , 2012, SIAM J. Optim..