Duality and calculi without exceptions for convex objects

Abstract

The aim of this paper is to make a contribution to theinvestigation of the roots and essence of convex analysis, and tothe development of the duality formulas of convex calculus. Thisis done by means of one single method: firstly conify, thenwork with the calculus of convex cones, which consists of threerules only, and finally deconify. This generates alldefinitions of convex objects, duality operators, binaryoperations and duality formulas, all without the usual needto exclude degenerate situations. The duality operator for convexfunction agrees with the usual one, the Legendre-Fencheltransform, only for proper functions. It has the advantage overthe Legendre-Fenchel transform that the duality formula holds forimproper convex functions as well. This solves a well-knownproblem, that has already been considered in Rockafellar's ConvexAnalysis (R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970). The value of this result is that it leadsto the general validity of the formulas of Convex Analysis thatdepend on the duality formula for convex functions. The approachleads to the systematic inclusion into convex sets of recessiondirections, and a similar extension for convex functions. Themethod to construct binary operations given in (ibidem) isformalized, and this leads to some new duality formulas. Anexistence result for extended solutions of arbitrary convexoptimization problems is given. The idea of a similar extension ofthe duality theory for optimization problems is given.duality;convex functions;convex sets;convex optimization