| dc.description.abstract | The study of semi-continuity and optimization has garnered significant attention from mathematicians for a prolonged period. While characterization of semi-continuity has been conducted in topological spaces and Hilbert spaces, it has not been studied in Lp -spaces. Similarly, while optimality conditions for convex optimization have been established in Hilbert spaces, Hausdorff spaces and normed spaces, they have not been determined in Lp -spaces. The aim of this study was to characterize semi- continuous functions and convex optimization in Lp -spaces. The specific objectives included: characterizing lower semi-continuous (lsc) functions in Lp -spaces; characterizing upper semi-continuous (usc) functions in Lp - spaces; and establishing conditions for convex optimization in Lp -spaces. The research methodology involved the use of Fatou’s Lemma and Dini’s theorem to characterize lsc functions, and Beer’s theorem which was used in characterizing usc functions. Technical approaches included the use of K K T conditions for optimality to establish conditions for convex optimization in Lp -spaces. The study has shown that if the epigraph of an Lp -space function is closed then, the function is lsc in the space. Additionally, it has been demonstrated that a function ϑ contained in a convex subset of an Lp -spaces L is usc if it is convex. The study has further shown that a Lipschitz-continuous function in a sequentially bounded subspace of a convex Lp -space L is lsc and has a local minimizer. It has also been proven that if ϑ(q) is Frechet differentiable in a convex Lp -space L, then q is a stationary point of ϑ(q) and forms its local minimizer. These results have potential applications in mathematical analysis, particularly in norm approximation which is useful in image and signal processing. | en |
