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Differentiable non-convex functions and general variational inequalities. (English) Zbl 1147.65047

The author introduces a new class of non-convex functions: The function $F:K\subset H\to H$ is said to be $g$-convex, if there exists a function $g$ such that

$F\left(u+t\left(g\left(v\right)-u\right)\right)\le \left(1-t\right)F\left(u\right)+tF\left(g\left(v\right)\right)\phantom{\rule{0.166667em}{0ex}}\forall u,v\in H:u,g\left(v\right)\in K,\phantom{\rule{1.em}{0ex}}t\in \left[0,1\right]$

where $K$ is a $g$-convex set. It is proved that the minimum of differentiable $g$-convex functions can be characterized by a class of variational inequalities, which is called the general variational inequality. Using the projection technique, the equivalence between the general variational inequalities and the fixed-point problems as well as with the Wiener-Hopf equations is established. This equivalence is used to suggest and analyze some iterative algorithms for solving the general variational inequalities.

MSC:
 65K10 Optimization techniques (numerical methods) 49J40 Variational methods including variational inequalities 49M25 Discrete approximations in calculus of variations