Optimization by level set methods v:duality theorems for perturbed optimization problems2

Abstract

We consider the optimization problem (P), α =inf h(G),, where G≠ O is a subset of a locally convex space Fand h:F→ R, “embedded” into a family of optimization problems where Xis a locally convex space and φ:F× X→ R.For surrogate dual, respectively. Lagrangian dual problems (Q), β = sup λ(X *), to (P),, defined with the aid of this embedding, we give necessary and sufficient conditions for α = β,involving functionals φ e X *and level sets of f, respectively of [ftilde](x,t), = f(x), + t(xe X,te R),, or involving surrogate e-subdifferentials (which we introduce here),, respectively e-subdifferentials of f(e ≧ 0),. We give applications to, optimization problems perturbed by multifunctions and to optimization problems for systems, obtaining conditions for surrogate duality in terms of functionals φ e X *and the level sets of h.