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Zbl 0756.90081
Fukushima, Masao
Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems.
(English)
[J] Math. Program., Ser. A 53, No.1, 99-110 (1992). ISSN 0025-5610; ISSN 1436-4646/e

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).\par 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.\par A list of sixteen references closes the paper.
[A.G.Azpeitia (Boston)]
MSC 2000:
*90C30 Nonlinear programming
49J40 Variational methods including variational inequalities
90-08 Computational methods (optimization)
90C33 Complementarity problems
90C99 Mathematical programming

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