Facial reduction in partially finite convex programming (Q1334956)

The problem \(f(x)\to \inf\) subject to \(Ax- b\in K\) is considered where \(f\) is an convex functional on an infinite-dimensional locally convex topological vector space \(X\), \(A\) is a linear operator from \(X\) to \(\mathbb{R}^n\) and \(K\) is a closed convex cone of \(\mathbb{R}^n\). For necessary conditions of optimality constraint qualifications are usually required. If these regularity conditions fail reduction techniques due to the description of the feasible region are useful [cf. A. Ben Israel, A. Ben-Tal, S. Zlobec (1981)]. In this paper a facial reduction in the domain space is used maintaining the involved structure. Using the quasi- relative interior instead of the relative interior of a convex set the achieved results are similar to the known ones in finite dimensions. Some applications to constrained approximation problems in separable \(L_p(S, \mu)\) spaces are given.