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Zbl 1147.65047
Differentiable non-convex functions and general variational inequalities.
(English)
[J] Appl. Math. Comput. 199, No. 2, 623-630 (2008). ISSN 0096-3003

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 $$F(u+ t(g(v)- u))\le(1- t)F(u)+ tF(g(v))\,\forall u,v\in H: u, g(v)\in K,\quad t\in[0,1]$$ 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.
[Hans Benker (Merseburg)]
MSC 2000:
*65K10 Optimization techniques (numerical methods)
49J40 Variational methods including variational inequalities
49M25 Finite difference methods

Keywords: variational inequalities; non-convex functions; fixed-point problem; Wiener-Hopf equations; projection operator; convergence

Cited in: Zbl 1212.52001

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