SOME FURTHER DUALITY THEOREMS FOR OPTIMIZATION PROBLEMS WITH REVERSE CONVEX CONSTRAINT SETS

Continuing 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{lunate} F|u(y) ε{lunate} Ω}), where F is an arbitrary set, h: F → R ̄ = [-∞, +∞] 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{lunate} F|wu(y) ≥ supw(XΩΩ)}) or λ(w) = inf h({y ε{lunate} FΩwu(y) = sup w(XΩΩ)}) (w ε{lunate} W). We also give an extension to the case when X is an arbitrary set and W ⊂- RxΩ{0}. © 1992.