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Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. (English) Zbl 0756.90081

The variational inequality problem is (1): Find ${x}^{*}\in S\subset {ℝ}^{n}$ such that $〈F\left({x}^{*}\right),x-{x}^{*}〉\ge 0$, $\forall x\in {ℝ}^{n}$ where $S\ne \varnothing$ is closed and convex and $F:{ℝ}^{n}\to {ℝ}^{n}$. Now with $G$ being any $n×n$ positive definite matrix consider the program (2): $min\left\{\varphi \left(y\right):y\in S\right\}$ where $\varphi \left(y\right)=〈F\left(x\right),\left(y-x\right)〉+\frac{1}{2}〈\left(y-x\right),G\left(y-x\right)〉$, and let $-f\left(x\right):{ℝ}^{n}\to ℝ$ be the optimal objective value of $\varphi \left(y\right)$ in (2).

The main result is now: (i) $f\left(x\right)\ge 0$, $\forall x\in S$, and (ii) ${x}^{*}$ solves (1) if and only if it solves the program (3): $min\left\{f\left(x\right):x\in S\right\}$ and that happens if and only if $f\left({x}^{*}\right)=0$, ${x}^{*}\in S$. Moreover $f$ is continuously differentiable (continuous) if $F$ is continuously differentiable (continuous). In the first case descent methods are presented to solve the program (3) by an iteration process.

A list of sixteen references closes the paper.

##### MSC:
 90C30 Nonlinear programming 49J40 Variational methods including variational inequalities 90-08 Computational methods (optimization) 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 90C99 Mathematical programming
##### References:
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