# Variational Inequalities and Related Projections

Chapter

## Abstract

The variational inequality problem is to find a point Here where τ. Thus, where which can equivalently be written as and the latter variational inequality amounts to equation (10.3).

*x*^{ * }in*X*such that$$\begin{array}{*{20}{c}} {\langle f(x*),x - x*\rangle \geqslant 0} & {for all x \in X.} \\ \end{array}$$

(10.1)

*X*is a nonempty closed convex subset of ℝ^{ n }, and*f*maps ℝ^{ n }into itself, this space being equipped with the ordinary inner product, <^{.},^{.}>. Such problems emerge as necessary optimality conditions in mathematical programming or in noncooperative game theory [2,3,5,9,10]. Most solution methods proceed by linearizing*f at*the current estimate x^{k}produced during iteration*k.*That is to say, in the next iterative step*k + 1*, the function*f(x)*is replaced by an approximate$$f({x^{k}}) + \frac{1}{{{\tau _{k}}}}G(x - {x^{k}})$$

(10.2)

_{k}is a positive scaling parameter and*G is*a positive-definite symmetric matrix. This local representation of*f*has the advantage of leading us to recognize the solution*x*^{ k+1 }, to be furnished at stage*k + 1*, as the unique point in*X*that is closest to$${x^{k}} - {\tau _{k}}{G^{{ - 1}}}f({x^{k}})$$

$${x^{{k + 1}}} = {P_{X}}({x^{k}} - {\tau _{k}}{G^{{ - 1}}}f({x^{k}}))$$

(10.3)

*P*_{ X }denotes the projection operator onto*X.*Indeed, using expression (10.2) in place off and writing x^{k}+^{1}=*x**, inequality (10.1) takes on the form$$\begin{array}{*{20}{c}} {\langle f({x^{k}}) + \frac{1}{{{\tau _{k}}}}G({x^{{k + 1}}} - {x^{k}}),x - {x^{{k + 1}}}\rangle \geqslant 0} & {for all x \in X,} \\ \end{array}$$

$$\begin{array}{*{20}{c}} {\langle {x^{{k + 1}}} - ({x^{k}} - {\tau _{k}}{G^{{ - 1}}}f({x^{k}})),G(x - {x^{{k + 1}}})\rangle \geqslant 0} & {for all x \in X,} \\ \end{array}$$

### Keywords

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© Springer Science+Business Media New York 1992