Differentiable non-convex functions and general variational inequalities. (English) Zbl 1147.65047

Noor, Muhammad Aslam

Appl. Math. Comput. 199, No. 2, 623-630 (2008).

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.

Reviewer: Hans Benker (Merseburg)

MSC:

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

Keywords:

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

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