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Homotopy Methods for Solving Variational Inequalities in Unbounded Sets

Authors:

Email author" target="_blank">Qing?Xu Email author Bo?Yu Guo-Chen?Feng

Institution:

(1) School of Management, Fudan University, Shanghai, 200433, P.R. China;(2) Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning, 116024, P.R. China;(3) Institute of Mathematics, Jilin University, Changchum, Jilin, 130012, P.R. China

Abstract:

In this paper, for solving the finite-dimensional variational inequality problem

$(x-x*)^{T} F(x*)\geq 0, \quad \forall x\in X,$

where F is a $$C^r (r gt; 1)$$

mapping from X to Rⁿ, X = ${ x \in R^{n} : g(x) leq; 0}$ is nonempty (not necessarily bounded) and ${\it g}({\it x}): R^{n} \rightarrow R^{m}$ is a convex C^r+1 mapping, a homotopy method is presented. Under various conditions, existence and convergence of a smooth homotopy path from almost any interior initial point in X to a solution of the variational inequality problem is proven. It leads to an implementable and globally convergent algorithm and gives a new and constructive proof of existence of solution.

Keywords:

Homotopy method Interior point method Variational inequality

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