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Homotopy Methods for Solving Variational Inequalities in Unbounded Sets

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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 Cr+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.

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Correspondence to Qing Xu.

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Xu, Q., Yu, B. & Feng, GC. Homotopy Methods for Solving Variational Inequalities in Unbounded Sets. J Glob Optim 31, 121–131 (2005). https://doi.org/10.1007/s10898-004-4272-4

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