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

The variational inequality problem is (1): Find $$x^*\in S\subset\mathbb{R}^ n$$ such that $$\langle F(x^*),x-x^*\rangle\geq 0$$, $$\forall x\in\mathbb{R}^ n$$ where $$S\neq\emptyset$$ is closed and convex and $$F: \mathbb{R}^ n\to\mathbb{R}^ n$$. 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 $$-f(x): \mathbb{R}^ n\to\mathbb{R}$$ be the optimal objective value of $$\phi(y)$$ in (2).
The main result is now: (i) $$f(x)\geq 0$$, $$\forall x\in S$$, and (ii) $$x^*$$ solves (1) if and only if it solves the program (3): $$\min\{f(x): x\in S\}$$ and that happens if and only if $$f(x^*)=0$$, $$x^*\in S$$. 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 inequalities 90-08 Computational methods for problems pertaining to operations research and mathematical programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C99 Mathematical programming
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### References:

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