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

Author： Lu Nan

Publisher： Taylor & Francis Ltd

ISSN： 0163-0563

Source： Numerical Functional Analysis and Optimization, Vol.30, Iss.7-8, 2009-07, pp. : 799-821

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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.