Properties of a Class of Nonlinear Transformations Over Euclidean Jordan Algebras with Applications to Complementarity Problems

Abstract

For any function φ from ℜ^r to ℜ^r, Tao and Gowda [Math. Oper. Res., 30 (2005), pp. 985–1004] introduced a corresponding nonlinear transformation R_φ over a Euclidean Jordan algebra (which is called a relaxation transformation) and established some useful relations between φ and R_φ. In this paper, we further investigate some interconnections between properties of φ and properties of R_φ, including the properties of continuity, (local) Lipschitz continuity, directional differentiability, (continuous) differentiability, semismoothness, monotonicity, the P₀-property, and the uniform P-property. As an application, we investigate the symmetric cone complementarity problem with a relaxation transformation. A property of the solution set of this class of problems is given. We also investigate a smoothing algorithm for solving this class of problems and show that the algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty.

Keywords:

• Euclidean Jordan algebra
• Relaxation transformation
• Smoothing algorithm
• Symmetric cone complementarity problem

AMS Subject Classification:

• 90C33
• 17C55

ACKNOWLEDGMENTS

This work was partially supported by the National Natural Science Foundation of China (grant numbers 10871144 and 10671010) and the Natural Science Foundation of Tianjin (grant number 07JCYBJC05200).