Differentiable non-convex functions and general variational inequalities.

*(English)*Zbl 1147.65047The 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

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

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.

Reviewer: Hans Benker (Merseburg)

##### MSC:

65K10 | Optimization techniques (numerical methods) |

49J40 | Variational methods including variational inequalities |

49M25 | Discrete approximations in calculus of variations |