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.


65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
49M25 Discrete approximations in optimal control
Full Text: DOI


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