On Optimization Problems in Quasi-uniform Spaces

This paper is concerned with optimization problems in T₀ quasi-uniform spaces. Many optimization problems such as vector optimization, set-valued optimization, are unified in the quasi-uniform space to have a simple expression. Firstly, the proposition that a quasi-uniform space is T₀ if and only if the intersection of all entourages of the quasi-uniformity is antisymmetric is proved. Secondly, the notion of extremum in quasi-uniform spaces is given by the partial order, and some equivalent conditions of extremum are obtained. Finally, optimization problems in T_c quasi-uniform spaces are put forward, and the conclusion that infimum of a lower semi- continuous mapping from a compact set to a T_c quasi- uniform space can be reached is given.