Algorithms for solving the Ky Fan inequality EP(f, C) are tackled exploiting the tools developed in the previous chapter. Some assumptions on C and f hold throughout all the chapter in order to provide a unified algorithmic framework. Precisely, C is supposed to be convex and closed, f to be continuous and to satisfy f(x, x) = 0 for any x∈ ℝn while f(x, ⋅) to be τ-convex for any x∈ ℝn for some τ ≥ 0 that does not depend upon the considered point x. Notice that this framework guarantees all the assumptions of the existence Theorems 2.3.4 and 2.3.8 except for the boundedness of C or some kind of monotonicity of f. Indeed, all the algorithms require at least one of them, so that the existence of a solution is always guaranteed.
Algorithms for Equilibria
Bigi G.;Pappalardo M.;Passacantando M.
2019-01-01
Abstract
Algorithms for solving the Ky Fan inequality EP(f, C) are tackled exploiting the tools developed in the previous chapter. Some assumptions on C and f hold throughout all the chapter in order to provide a unified algorithmic framework. Precisely, C is supposed to be convex and closed, f to be continuous and to satisfy f(x, x) = 0 for any x∈ ℝn while f(x, ⋅) to be τ-convex for any x∈ ℝn for some τ ≥ 0 that does not depend upon the considered point x. Notice that this framework guarantees all the assumptions of the existence Theorems 2.3.4 and 2.3.8 except for the boundedness of C or some kind of monotonicity of f. Indeed, all the algorithms require at least one of them, so that the existence of a solution is always guaranteed.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
