RESOLUTION OF EXTREMISATION PROBLEM USING DUALITY PRINCIPLE
O. M. Bamigbola, Idowu Ademola Osinuga · 2009
Given a vector x in a Hilbert space H and a subspace M in H. We wish to find the vec-tor m closest to x in the sense that it maxi-mizes ║x- m ║ is called a minimum norm problem. If M is a closed subspace of Hil-bert space, there is always a unique solution to the minimum problem and the solution satisfies orthogonality condition. Furthermore, we introduced the following two theorems, which centre on the equiva-lence of two extremisation problems: one formular in a normed space X and the other in its dual x’. We remark here that the mini-mum norm problems have been found use-ful extensively in approximation theory, Esti-mation theory, etc. Theorem 1.2 According to (Luenberger 1969) let x be an element in a real normed linear space X. Let d denote its distance from the subspace M. The use of duality principle for characterizing solution of general optimization problem posed in the Hilbert space was considered. The existence and uniqueness of solution are guaranteed by formulat-ing the minimum problem in a dual space. Furthermore, the solution is shown to be aligned.