Some further duality theorems for optimization problems with reverse convex constraint sets

AbstractContinuing our papers (Optimization 18, 1987, 485–499; and Math. Oper. Stat. Ser. Optim. 11, 1980, 221–234) we give some further duality theorems for the primal problem α = inf h({y ϵ F¦u(y) ϵ Ω}), where F is an arbitrary set, h: F → R̄ = [−∞, +∞] a function, Ω a subset of a locally convex space X such that XΩgW is convex, and u: F → X a mapping, and the dual problem β = inf λ(W), where W ⊂- X∗Ω{0} and λ(w) = f h({y ϵ F¦wu(y) ⩾ supw(XΩΩ)}) or λ(w) = inf h({y ϵ FΩwu(y) = sup w(XΩΩ)}) (w ϵ W). We also give an extension to the case when X is an arbitrary set and W ⊂- RxΩ{0}