Abstract
We suggest new dual algorithms and iterative methods for solving monotone generalized variational inequalities. Instead of working on the primal space, this method performs a dual step on the dual space by using the dual gap function. Under the suitable conditions, we prove the convergence of the proposed algorithms and estimate their complexity to reach an -solution. Some preliminary computational results are reported.
Highlights
Let C be a convex subset of the real Euclidean space Rn, F be a continuous mapping from C into Rn, and φ be a lower semicontinuous convex function from C into R
F x, x − y∗ φ x − φ y∗ ≥ 0, ∀x ∈ C. This generalized variational inequalities become an attractive field for many researchers and have many important applications in electricity markets, transportations, economics, and nonlinear analysis see 1–9
It is well known that the interior quadratic and dual technique are powerfull tools for analyzing and solving the optimization problems see 10–16
Summary
Let C be a convex subset of the real Euclidean space Rn, F be a continuous mapping from C into Rn, and φ be a lower semicontinuous convex function from C into R. It is well known that the interior quadratic and dual technique are powerfull tools for analyzing and solving the optimization problems see 10–16. These techniques have been used to develop proximal iterative algorithm for variational inequalities see 17– 22. In this paper we extend results in 23 to the generalized variational inequality problem GVI in the dual space. We first develop a convergent algorithm for GVI with F being monotone function satisfying a certain Lipschitz type condition on C.
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