Exceptional family and solvability of the second-order cone complementarity problems

Abstract In this paper, we introduce a concept of exceptional family for second-order cone complementarity problems (denoted by SOCCP), which is the particular case of the concept of exceptional family of elements introduced by Isac and Carbone (1999) [25] for the nonlinear complementarity problems. And the exceptional family for second-order cone complementarity problems has some pretty unique properties. Furthermore, we propose a new sufficient existence condition of a solution to SOCCP and give a particular example to show that the new condition is not stronger than known Isac–Carbone’s condition.

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

[2]  G. Isac,et al.  Duality in nonlinear complementarity theory by using inversions and scalar derivatives , 2006 .

[3]  Sandor Z. Németh,et al.  The non-existence of a regular exceptional family of elements. A necessary and sufficient condition. Applications to complementarity theory , 2008, J. Glob. Optim..

[4]  Yun-Bin Zhao,et al.  Exceptional families and finite-dimensional variational inequalities over polyhedral convex sets , 1997 .

[5]  Muhammad Aslam Noor,et al.  Extended general variational inequalities , 2009, Appl. Math. Lett..

[6]  Jen-Chih Yao,et al.  Degree theory for generalized variational inequalities and applications , 2009, Eur. J. Oper. Res..

[7]  G. Isac.  Leray-Schauder type alternatives and the solvability of complementarity problems , 2001 .

[8]  Sandor Z. Németh,et al.  Regular exceptional family of elements with respect to isotone projection cones in Hilbert spaces and complementarity problems , 2008, Optim. Lett..

[9]  S. Plubtieng,et al.  Existence of solutions for generalized variational inequality problems in Banach spaces , 2011 .

[10]  Tony E. Smith,et al.  A solution condition for complementarity problems: with an application to spatial price equilibrium , 1984 .

[11]  G. Isac,et al.  Application of Topological Degree Theory to Complementarity Problems , 1998 .

[12]  George Isac,et al.  Leray–Schauder Type Alternatives, Complementarity Problems and Variational Inequalities , 2006 .

[13]  Vyacheslav Kalashnikov,et al.  Exceptional Families, Topological Degree and Complementarity Problems , 1997, J. Glob. Optim..

[14]  G. Isac,et al.  Complementarity Problems and Variational Inequalities. A Unified Approach of Solvability by an Implicit Leray-Schauder Type Alternative , 2005, J. Glob. Optim..

[15]  Yun-Bin Zhao,et al.  Properties of a Multivalued Mapping Associated with Some Nonmonotone Complementarity Problems , 2000, SIAM J. Control. Optim..

[16]  George Isac,et al.  Exceptional Family of Elements and the Solvability of Variational Inequalities for Unbounded Sets in Infinite Dimensional Hilbert Spaces , 2000 .

[17]  A. Carbone,et al.  Exceptional Families of Elements for Continuous Functions: Some Applications to Complementarity Theory , 1999, J. Glob. Optim..

[18]  George Isac,et al.  Functions Without Exceptional Family of Elements and Complementarity Problems , 1998 .

[19]  G. Isac.  Exceptional Family of Elements, Feasibility, Solvability and Continuous Paths of ε -Solutions for Nonlinear Complementarity Problems , 2000 .

[20]  Sandor Z. Németh.  Exceptional family of elements for general order complementarity problems , 2010, Appl. Math. Comput..

[21]  G. Isac,et al.  Functions without exceptional family of elements and the solvability of variational inequalities on unbounded sets , 2002 .

[22]  Sandor Z. Németh,et al.  Duality in Multivalued Complementarity Theory by Using Inversions and Scalar Derivatives , 2005, J. Glob. Optim..

[23]  Nan-jing Huang,et al.  Non-zero solutions for a class of generalized variational inequalities in reflexive Banach spaces , 2007, Appl. Math. Lett..

[24]  Yunbin Zhao.  Existence of a solution to nonlinear variational inequality under generalized positive homogeneity , 1999, Oper. Res. Lett..

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

[26]  George Isac,et al.  Scalar and Asymptotic Scalar Derivatives: Theory and Applications , 2008 .

[28]  Farid Alizadeh,et al.  Extension of primal-dual interior point algorithms to symmetric cones , 2003, Math. Program..

[29]  Wieslawa T. Obuchowska.  Exceptional Families and Existence Results for Nonlinear Complementarity Problem , 2001, J. Glob. Optim..

[30]  Lei Wang,et al.  Nonzero solutions for a system of variational inequalities in reflexive Banach spaces , 2009, Math. Comput. Model..

[31]  Yunbin Zhao,et al.  Exceptional Family of Elements for a Variational Inequality Problem and its Applications , 1999, J. Glob. Optim..

[32]  G. Isac,et al.  REFE-Acceptable Mappings: Necessary and Sufficient Condition for the Nonexistence of a Regular Exceptional Family of Elements , 2008 .

[33]  George Isac.  Exceptional Families of Elements, Feasibility and Complementarity , 2000 .

[34]  Andrew Brennan,et al.  Necessary and Sufficient Conditions , 2018, Logic in Wonderland.

[35]  L. Faybusovich.  Linear systems in Jordan algebras and primal-dual interior-point algorithms , 1997 .

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

[37]  George Isac,et al.  Duality of Implicit Complementarity Problems by Using Inversions and Scalar Derivatives , 2006 .

[38]  Vyacheslav V. Kalashnikov,et al.  Exceptional Family of Elements, Leray–Schauder Alternative, Pseudomonotone Operators and Complementarity , 2001 .

[39]  George Isac.  Topological Methods in Complementarity Theory , 2009, Encyclopedia of Optimization.

[40]  G. Isac,et al.  Quasi-P*-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems , 2000 .

[41]  Vyacheslav V. Kalashnikov,et al.  Complementarity, Equilibrium, Efficiency and Economics , 2002 .

[42]  Liping Zhang.  Solvability of semidefinite complementarity problems , 2008, Appl. Math. Comput..