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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 * S n such that F(x * ),x-x * 0, x n where S is closed and convex and F: n n . Now with G being any n×n positive definite matrix consider the program (2): min{ϕ(y):yS} where ϕ(y)=F(x),(y-x)+1 2(y-x),G(y-x), and let -f(x): n be the optimal objective value of ϕ(y) in (2).

The main result is now: (i) f(x)0, xS, and (ii) x * solves (1) if and only if it solves the program (3): min{f(x):xS} and that happens if and only if f(x * )=0, x * 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:
90C30Nonlinear programming
49J40Variational methods including variational inequalities
90-08Computational methods (optimization)
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C99Mathematical programming
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