Abstract
We present a new method for solving the box-constrained stochastic linear variational inequality problem with three special types of uncertainty sets. Most previous methods, such as the expected value and expected residual minimization, need the probability distribution information of the stochastic variables. In contrast, we give the robust reformulation and reformulate the problem as a quadratically constrained quadratic program or convex program with a conic quadratic inequality quadratic program, which is tractable in optimization theory.
Highlights
Variational inequality theory is an important branch in operations research
To obtain the solutions of VI(l, u, F), many methods are presented based on the KKT system, which is given as follows: xi – li ≥, Fi(x) + yi ≥, (xi – li) Fi(x) + yi =, ( )
The demands are generally difficult to be determined in supply chain network, because they vary with the change of income level [, ]
Summary
Variational inequality theory is an important branch in operations research. Recall that the variational inequality problem, denoted by VI(X, F), is to find a vector x∗ ∈ X such that x – x∗ T F x∗ ≥ , ∀x ∈ X, ( )where X ⊆ Rn is a non-empty closed convex set, F : X → Rn is a given function. In order to meet the needs in practice, many researchers begin to consider the following stochastic variational inequality problem, denoted by SVI(X, F), which requires an x∗ such that x – x∗ T F x∗, ω ≥ , ∀x ∈ X, ω ∈ , a.s., ( ) By employing KKT system ( ), we give the robust reformulation of SLVI(l, u, F) as follows: min max tT , yT
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