Weak sharp solutions for variational inequalities in Banach spaces. (English) Zbl 1200.49007
Summary: We introduce the notion of a weak sharp set of solutions to a Variational Inequality Problem (VIP) in a reflexive, strictly convex and smooth Banach space, and present its several equivalent conditions. We also prove, under some continuity and monotonicity assumptions, that if any sequence generated by an algorithm for solving a VIP converges to a weak sharp solution, then we can obtain solutions for a VIP by solving a finite number of convex optimization subproblems with linear objective. Moreover, in order to characterize finite convergence of an iterative algorithm, we introduce the notion of a weak subsharp set of solutions of a VIP, which is more general than that of weak sharp solutions in Hilbert spaces. We establish a sufficient and necessary condition for the finite convergence of an algorithm for solving a VIP which satisfies that the sequence generated by which converges to a weak subsharp solution of a VIP, and show that the proximal point algorithm satisfies this condition. As a consequence, we prove that the proximal point algorithm possesses the finite convergence property whenever the sequence generated by which converges to a weak subsharp solution of a VIP.
MSC:
| 49J40 | Variational inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 49M30 | Other numerical methods in calculus of variations (MSC2010) |
Keywords:
variational inequalities; weakly sharp solutions; error bound; proximal point algorithms; finite convergenceReferences:
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