Variational Inequalities and Related Projections

  • Sjur D. Flåm


The variational inequality problem is to find a point x * in X such that
$$\begin{array}{*{20}{c}} {\langle f(x*),x - x*\rangle \geqslant 0} & {for all x \in X.} \\ \end{array}$$
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
$$f({x^{k}}) + \frac{1}{{{\tau _{k}}}}G(x - {x^{k}})$$
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,
$${x^{{k + 1}}} = {P_{X}}({x^{k}} - {\tau _{k}}{G^{{ - 1}}}f({x^{k}}))$$
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).




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© Springer Science+Business Media New York 1992

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  • Sjur D. Flåm

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