A general theory of surrogate dual and perturbational extended surrogate dual optimization problems

For the optimization problem (P) α = inf h(G), where G ≠ Ø is a subset of a locally convex space $\text{F}\text{and}\text{h: F →}\text{R}\text{?}$, we introduce and study two general concepts of dual problems, encompassing the classical surrogate dual problem. The first one involves only a family of surrogate constraints sets Δ_{G, Φ} ? F (Φ ∈ W), where W ? R^X, X being a locally convex space. The second one uses a perturbation functional $\text{?: F × X →}\text{R}\text{?}$ and a family of sets $\text{}\text{?}\text{gD}{}_{\mathrm{(F,x0),\Phi }}\text{? F × X (Φ ∈ W)}$, where W ? R^X. We give duality theorems, introduce Lagrangians, and show some relations between these problems and the dual problems to (P) defined with the aid of a perturbation and a concept of conjugation of functionals.