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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 $x\sp*\in S\subset\bbfR\sp n$ such that $\langle F(x\sp*),x-x\sp*\rangle\ge 0$, $\forall x\in\bbfR\sp n$ where $S\ne\emptyset$ is closed and convex and $F: \bbfR\sp n\to\bbfR\sp n$. Now with $G$ being any $n\times n$ positive definite matrix consider the program (2): $\min\{\phi(y): y\in S\}$ where $\phi(y)=\langle F(x),(y-x)\rangle+{1\over 2}\langle(y-x),G(y-x)\rangle$, and let $-f(x): \bbfR\sp n\to\bbfR$ be the optimal objective value of $\phi(y)$ in (2). The main result is now: (i) $f(x)\ge 0$, $\forall x\in S$, and (ii) $x\sp*$ solves (1) if and only if it solves the program (3): $\min\{f(x): x\in S\}$ and that happens if and only if $f(x\sp*)=0$, $x\sp*\in S$. 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.

90C30Nonlinear programming
49J40Variational methods including variational inequalities
90-08Computational methods (optimization)
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C99Mathematical programming
Full Text: DOI
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