Abstract
This paper uses a merit function derived from the Fishcher–Burmeister function and formulates box-constrained stochastic variational inequality problems as an optimization problem that minimizes this merit function. A sufficient condition for the existence of a solution to the optimization problem is suggested. Finally, this paper proposes a Monte Carlo sampling method for solving the problem. Under some moderate conditions, comprehensive convergence analysis is included as well.
Highlights
Let l and u be two n-dimensional vectors with components li ∈ R ∪ {−∞} and ui ∈ R ∪ {+∞} satisfying li < ui, and denote, by S, the nonempty and possibly infinite box [l, u] ≔ x ∈ Rn|li ≤ xi ≤ ui, i 1, . . . , n. en, the boxconstrained variational inequality problem (BVIP, for short) is to find a vector x∗ ∈ S such that x − x∗TF x∗ ≥ 0, ∀x ∈ S, (1)where F: Rn ⟶ Rn is a given function. is problem is called the mixed complementarity problem [1]
Motivated by Sun and Womersley [8], Luo and Lin [20] formulated a class of BSVIP as an optimization problem that minimizes the expected residual of the merit function derived on the Fishcher–Burmeister function and proposed a Monte Carlo sampling method for solving the problem
We study a sufficient condition for the existence of a solution to the optimization problem
Summary
Let l and u be two n-dimensional vectors with components li ∈ R ∪ {−∞} and ui ∈ R ∪ {+∞} satisfying li < ui, and denote, by S, the nonempty and possibly infinite box [l, u] ≔ x ∈ Rn|li ≤ xi ≤ ui, i 1, . . . , n. en, the boxconstrained variational inequality problem (BVIP, for short) is to find a vector x∗ ∈ S such that x − x∗TF x∗ ≥ 0, ∀x ∈ S,. Xu [18] applied the well-known sample average approximation (SAA, for short) method to solve the same class of stochastic variational inequality problems (SVIP). Aimed at a practical treatment of the SVIP, box-constrained stochastic variational inequality problem (BSVIP, for short) is meaningful and interesting to study [19]. Motivated by Sun and Womersley [8], Luo and Lin [20] formulated a class of BSVIP as an optimization problem that minimizes the expected residual of the merit function derived on the Fishcher–Burmeister function and proposed a Monte Carlo sampling method for solving the problem.
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