Properties of a Class of Nonlinear Transformations Over Euclidean Jordan Algebras with Applications to Complementarity Problems |
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Authors: | Nan Lu Jiye Han |
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Institution: | 1. Department of Mathematics, School of Science , Tianjin University , Tianjin, People's Republic of China;2. Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences , Beijing, People's Republic of China |
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Abstract: | For any function φ from ?r to ?r, 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. |
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Keywords: | Euclidean Jordan algebra Relaxation transformation Smoothing algorithm Symmetric cone complementarity problem |
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