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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
$F(u+ t(g(v)- u))\leq(1- t)F(u)+ tF(g(v))\,\forall u,v\in H: u, g(v)\in K,\quad t\in[0,1]$
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 Numerical optimization and variational techniques 49J40 Variational inequalities 49M25 Discrete approximations in optimal control
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