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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:KHH is said to be g-convex, if there exists a function g such that

F(u+t(g(v)-u))(1-t)F(u)+tF(g(v))u,vH:u,g(v)K,t[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:
65K10Optimization techniques (numerical methods)
49J40Variational methods including variational inequalities
49M25Discrete approximations in calculus of variations