A tutorial on properties of the epigraph reformulation

This paper systematically surveys useful properties of the epigraph reformulation for optimization problems, and complements them by some new results. We focus on the complete compatibility of the original formulation and the epigraph reformulation with respect to solvability and unsolvability, the compatibility with respect to some, but not all, basic constraint qualifications, the formulation of first-order optimality conditions for problems with max-type objective function, and the interpretation of feasibility and optimality cuts along epigraphs in the framework of cutting plane methods. Finally we introduce a generalized epigraph reformulation which is particularly useful for treating nonsmooth summands of objective and constraint functions independently in the reformulation. • Treats complete compatibility of solvability and unsolvability cases in the epigraph reformulation of optimization models. • Treats compatibility with respect to some, but not all, basic constraint qualifications. • Derivation of first order optimality conditions for problems with max-type objective. • Interpretation of feasibility and optimality cuts along epigraphs. • Introduction of a generalized epigraph reformulation.