Duality and calculi without exceptions for convex objects

The aim of this paper is to make a contribution to the\ninvestigation of the roots and essence of convex analysis, and to\nthe development of the duality formulas of convex calculus. This\nis done by means of one single method: firstly conify, then\nwork with the calculus of convex cones, which consists of three\nrules only, and finally deconify. This generates all\ndefinitions of convex objects, duality operators, binary\noperations and duality formulas, all without the usual need\nto exclude degenerate situations. The duality operator for convex\nfunction agrees with the usual one, the Legendre-Fenchel\ntransform, only for proper functions. It has the advantage over\nthe Legendre-Fenchel transform that the duality formula holds for\nimproper convex functions as well. This solves a well-known\nproblem, that has already been considered in Rockafellar's Convex\nAnalysis (R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970). The value of this result is that it leads\nto the general validity of the formulas of Convex Analysis that\ndepend on the duality formula for convex functions. The approach\nleads to the systematic inclusion into convex sets of recession\ndirections, and a similar extension for convex functions. The\nmethod to construct binary operations given in (ibidem) is\nformalized, and this leads to some new duality formulas. An\nexistence result for extended solutions of arbitrary convex\noptimization problems is given. The idea of a similar extension of\nthe duality theory for optimization problems is given.