Abstract
Let X be a completely regular topological space. Denote, as usual, by C(X) the family of all bounded continuous real-valued functions in X. The space C(X) equipped with the sup-norm ||f||∞ = sup{| f(x)|: x ∈ X}, f ∈ C(X), becomes a Banach space. Each f ∈ C(X) determines a minimization problem: find x0 ∈ X with f(x 0) = inf {f(x) : x ∈ X} =: inf (X, f). We designate this problem by (X, f). Among the different properties of the minimization problem (X, f) the following ones are of special interest in the theory of optimization: (a) (X, f) has a solution (existence of the solution); (b) the solution set for (X, f) is a singleton (uniqueness of the solution); (c) if f(x*) is close to inf (X, f), then x* is a good approximation of the solution of (X, f) (stability of the solution—see bellow the precise definition).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.