Conditions for Convex Optimization in \(L^p\)-Spaces

This paper establishes the optimality conditions for convex optimization in \(L^p\)-spaces. We prove that if a lower semi-continuous function \(\vartheta\) : L \(\rightarrow\) \(\mathbb{R}\) is Lipschitz continuous in an \(L^p\)-space L, then it is weakly lower semi-continuous and must attain a unique global minimizer for the convex optimization problem \(\frac{min}{q{\epsilon}G}\) \(\vartheta\)(q) on a sequentially bounded convex constraint set . We also provide further necessary conditions for optimality using the concepts of compactness and coercivity of semi-continuous functions on sequentially bounded domains. Additionally, we prove the existence of minimizers using the concepts of Gateux and Frchet differentiability.