Journal of Nonlinear and Variational Analysis

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DOI: 10.23952/jnva.11.2027.1.04

Volume 11, Issue 1, 1 February 2027, Pages 45-75

 

Abstract. This paper presents a unified stochastic approximation framework for solving multivalued variational inequalities in real separable Hilbert spaces. We propose an iterative projection scheme that operates over a sequence of expanding closed convex subsets, a formulation motivated by variable-mesh discretization strategies in infinite-dimensional optimization. The proposed algorithm utilizes noisy observations of the underlying operator, accommodating both intrinsic stochasticity (as in the minimization of expected-value functionals) and extrinsic computational errors. We establish the strong convergence of the generated sequence, in both the mean-square and almost sure senses, to the limit of the solutions of the discretized problems, which is a solution of the original variational inequality under standard approximation conditions. Furthermore, we derive explicit rates of convergence for both modes of convergence, and we treat the ill-posed case of vanishing regularization. The versatility of the framework is demonstrated through applications to non-smooth stochastic optimization, support vector machines, optimal control and inverse problems for partial differential equations with random data, and image deblurring.

 

How to Cite this Article:
S. Bai, A.A. Khan, C. Tammer, A. Zălinescu, A unified stochastic approximation framework for mean-square and almost sure convergence with rates in multivalued variational inequalities, J. Nonlinear Var. Anal. 11 (2027), 45-75.