Abstract
For any function φ from ℜr to ℜr, Tao and Gowda [Math. Oper. Res., 30 (2005), pp. 985–1004] introduced a corresponding nonlinear transformation Rφ over a Euclidean Jordan algebra (which is called a relaxation transformation) and established some useful relations between φ and Rφ. In this paper, we further investigate some interconnections between properties of φ and properties of Rφ, 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.
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).