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Variational Inequalities and Related Projections

  • Sjur D. Flåm

Abstract

The variational inequality problem is to find a point 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)
Here 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 xk 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)
where τk 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
$${x^{k}} - {\tau _{k}}{G^{{ - 1}}}f({x^{k}})$$
. Thus,
$${x^{{k + 1}}} = {P_{X}}({x^{k}} - {\tau _{k}}{G^{{ - 1}}}f({x^{k}}))$$
(10.3)
where P X denotes the projection operator onto X. Indeed, using expression (10.2) in place off and writing xk+1= x*, inequality (10.1) takes on the form
$$\begin{array}{*{20}{c}} {\langle f({x^{k}}) + \frac{1}{{{\tau _{k}}}}G({x^{{k + 1}}} - {x^{k}}),x - {x^{{k + 1}}}\rangle \geqslant 0} & {for all x \in X,} \\ \end{array}$$
which can equivalently be written as
$$\begin{array}{*{20}{c}} {\langle {x^{{k + 1}}} - ({x^{k}} - {\tau _{k}}{G^{{ - 1}}}f({x^{k}})),G(x - {x^{{k + 1}}})\rangle \geqslant 0} & {for all x \in X,} \\ \end{array}$$
and the latter variational inequality amounts to equation (10.3).

Keywords

Variational Inequality Variational Inequality Problem Nonempty Closed Convex Subset Interior Point Algorithm Convex Feasibility Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1992

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  • Sjur D. Flåm

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