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We consider the strong and total Lagrange dualities for infinite quasiconvex optimization problems. By using the epigraphs of the z - quasi-conjugates and the Greenberg-Pierskalla subdifferential of these functions, we introduce some new constraint qualifications. Under the new constraint qualifications, we provide some necessary and sufficient conditions for infinite quasiconvex optimization problems to have the strong and total Lagrange dualities. [ABSTRACT FROM AUTHOR]

Let C be a closed bounded convex subset of a real Banach space X with 0 as its interior and PC the Minkowski functional generated by the set C. For a nonempty set G in X and x ∊ X, g0 ∊ G is called the generalized best approximation to x from G if PC(g0-x) = PC(g - x) for all g ∊ G. In this paper, we will give a distance formula under PC from a point to a closed hyperplane H(x*, α) in X determined by a nonzero continuous linear functional x* in X and a real number α, a representation of the generalized metric projection onto H(x*, α), and investigate the continuity of this generalized metric projection, extending corresponding results for the case of norm. [ABSTRACT FROM AUTHOR]