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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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