An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities
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Research Article An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities Juhe Sun, Shaowu Zhang, and Liwei Zhang Received 16 June 2007; Accepted 19 September 2007 Recommended by Nan-Jing Huang
A new monotonicity, M-monotonicity, is introduced, and the resolvant operator of an M-monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI (Sn+ ,F + G) is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable. Copyright © 2007 Juhe Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In recent years, the variational inequality has been addressed in a large variety of problems arising in elasticity, structural analysis, economics, transportation equilibrium, optimization, oceanography, and engineering sciences [1, 2]. Inspired by its wide applications, many researchers have studied the classical variational inequality and generalized it in various directions. Also, many computational methods for solving variational inequalities have been proposed (see [3–8] and the references therein). Among these methods, resolvant operator technique is an important one, which was studied in the 1990s by many researchers (such as [4, 6, 9]), and further studies developed recently [3, 10, 11]. As monotonicity plays an important role in the theory of variational inequality and its generalizations, in this paper, we introduce a new class of monotone operator: Mmonotone operator. The resolvant operator associated with an M-monotone operator is
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Fixed Point Theory and Applications
proved to be Lipschitz-continuous. Applying the resolvant operator technique, we transform the positively semidefinite variational inequality (V I) problem V I(Sn+ ,F + G) into a fixed point problem of a nonexpansive mapping and suggest a proximal point algorithm to solve the fixed point problem. Under the condition that F in the V I problem is strongly monotone and Lipschitz-continuous, we prove that the algorithm has a global convergence. To ensure the proposed proximal point algorithm is implementable, we introduce a path Newton algorithm whose step size is calculated by Armijo rule. In the next section, we recall some results and concepts that will be used in this paper. In Section 3, we introduce the definition of an M-monotone operator, and discuss properties of this kind of operators, espec
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