Some Goldstein's type methods for co-coercive variant variational inequalities
This paper is devoted to the study of the so-called variant variational inequality VVI \((F, \Omega)\), (which is closely related to VI \((\Omega, F)\): find a vector \(u \in \Omega\) such that: \((v-u)^T F(u) \geq 0, \forall v \in \Omega\), where \(\Omega\) is a nonempty closed convex set in \(\mathbb R^n\) and \(F\) is a mapping from \(\mathbb R^n\) into itself) of the form. Find a vector \(u \in \mathbb R^n\) such that (1) \(F(u) \in \Omega, (v-F(u))^T u \geq 0, \forall v \in \Omega\). It is assumed that the solution set \(S^*\) of (1) is nonempty. Since, in general, it is not possible to identify analytic solutions of VI \((\Omega, F)\) and VVI \((\Omega, F)\) without specific structures of \(F\) and \(\Omega\), the researchers are particularly interested in designing numerical algorithms to approach to VI \((\Omega, F)\) and VVI \((\Omega, F)\), subject to certain criteria. The monotonicity of \(F\) plays a crucial role in designing numerical algorithms for solving VI \((\Omega, F)\) and VVI \((\Omega, F)\). With the restrictions that \(F\) is Lipschitz continuous and strongly monotone, the authors use the classical Goldstein method to get iteratively updates \(u^K\) (starting from any \(u^o \in \Omega\)) according to the formula (2) \(u^{K+1} = P_\Omega [u^K - \beta_K F(u^K)]\), where \(\beta_K > 0\) is the scaling parameter and \(P_\Omega (v)\) denotes the projection of \(v\) onto \(\Omega\) under the Euclidean norm, i.e. \(P_\Omega (v) = arg \min \{\|u-v\|: u \in \Omega \}\). The Goldstein method converges globally with linear speed under the condition (3) \(0 < \beta_L \leq \inf^\infty_{K=1} \{\beta_K\} \leq \sup^\infty_{K=1} \{\beta_K\} \leq \beta_U < \frac{2s}{L^2}\). To solve VVI \((\Omega, F)\) with Lipschitz continuous and strongly monotone \(F\), an extended Goldstein method is presented. The corresponding scheme is: (4) (He-Goldstein method) \(u^{k+1} = u^K-\frac{1}{\beta_K}\{F(u^K)- P_\Omega [F(u^K)-\beta_Ku^k]\}\). Similar to (3) the scaling parameters \(\{\beta_K\}\) in (4) should be restricted in certain ranges, relying on the Lipschitz constant and the strong monotonicity constant \(F\). In addition to the monotonicity, the co-coercivity of \(F\) also arises widely in the VI and VVI models in various applications. The goal of this paper is to give the affirmative answer to the natural question if it is possible to extend the Goldstein type method (4) to solve the co-coercive VVI \(( \Omega,F)\). Main result: The authors show that when the involved co-coercive modulus of VVI is known, the Goldstein method can be improved by identifying a refined step-size at each iteration. Hence, an improved Goldstein type method is proposed for co-coercive VVIs with known co-coercive modulus. Finally, some numerical results to verify the efficiency of the proposed Goldstein type methods for VVIs, are also presented.
Lipschitz and strongly monotone operators
scientific article; zbMATH DE number 5840217
scientific article