Duality and calculi without exceptions for convex objects

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

On the universal method to solve extremal problems

Some applications of the theory of extremal problems to mathematics and economics are made more accessible to non-experts.1.The following fundamental results are known to all users of mathematical techniques, such as economist, econometricians, engineers and ecologists: the fundamental theorem of algebra, the Lagrange multiplier rule, the implicit function theorem, separation theorems for convex sets, orthogonal diagonalization of symmetric matrices. However, full explanations, including rigorous proofs, are only given in relatively advanced courses for mathematicians. Here, we offer short ans easy proofs. We show that akk these results can be reduced to the task os solving a suitable extremal problem. Then we solve each of the resulting problems by a universal strategy.2. The following three practical results, each earning their discoverers the Nobel prize for Economics, are known to all economists and aonometricians: Nash bargaining, the formula of Black and Scholes for the price of options and the models of Prescott and Kydland on the value of commitment. However, the great value of such applications of the theory of extremal problems deserves to be more generally appreciated. The great impact of these results on real life examples is explained. This, rather than mathematical depth, is the correct criterion for assessing their value.

A comprehensive view on optimization: reasonable descent

Reasonable descent is a novel, transparent approach to a well-established field: the deep methods and applications of the complete analysis of continuous optimization problems. Standard reasonable descents give a unified approach to all standard necessary conditions, including the Lagrange multiplier rule, the Karush-Kuhn-Tucker conditions and the second order conditions. Nonstandard reasonable descents lead to new necessary conditions. These can be used to give surprising proofs of deep central results outside optimization: the fundamental theorem of algebra, the maximum and the minimum principle of complex function theory, the separation theorems for convex sets, the orthogonal diagonalization of symmetric matrices and the implicit function theorem. These optimization proofs compare favorably with the usual proofs and are all based on the same strategy. This paper is addressed to all practitioners of optimization methods from many fields who are interested in fully understanding the foundations of these methods and of the central results above.optimization;fundamental theorem of algebra;Lagrange multiplier;Karush-Kuhn-Tucker;descent;implicit function theorem;necessary conditions;orthogonal diagonalization

A D-induced duality and its applications

This paper attempts to extend the notion of duality for convex cones, by basing it on a predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the inner-product is replaced by a pre-specified conic ordering, defined by a convex cone D, and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the D-induced duality in the paper. We further introduce the notion of D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced dual cones and are convenient to use for some practical applications. Properties of the extended duality, including the extended bi-polar theorem, are proven. Furthermore, attention is paid to the computation and approximation of the D-induced dual objects. We discuss, as examples, applications of the newly introduced D-induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization.bi-polar theorem;conic optimization;convex cones;duality

Novel insights into the multiplier rule

We present the Lagrange multiplier rule, one of the basic optimization methods, in a new way. Novel features include:• Explanation of the true source of the power of the rule: reversal of tasks, but not the use of multipliers.• A natural proof based on a simple picture, but not the usual technical derivation from the implicit function theorem.• A practical method to avoid the cumbersome second order conditions.• Applications from various areas of mathematics, physics, economics.• Some hnts on the use of the rule.bargaining;dynamical systems;economics;finance;multiplier rule;second order condition

Inner and outer approximation of convex sets using alignment

We show that there exists, for each closed bounded convex set C in the Euclidean plane with nonempty interior, a quadrangle Q having the following two properties. Its sides support C at the vertices of a rectangle r and at least three of the vertices of Q lie on the boundary of a rectangle R that is a dilation of r with ratio 2. We will prove that this implies that quadrangle Q is contained in rectangle R and that, consequently, the inner approximation r of C has an area of at least half the area of the outer approximation Q of C. The proof makes use of alignment or Schüttelung, an operation on convex sets

A structural version of the theorem of Hahn-Banach

We consider one of the basic results of functional analysis, the classical theorem of Hahn-Banach. This theorem gives the existence of a continuous linear functional on a given normed vectorspace extending a given continuous linear functional on a subspace with the same norm. In this paper we generalize this existence theorem to a result on the structure of the set of all these extensions.Hahn-Banach theorem

Duality and calculi without exceptions for convex objects

The aim of this paper is to make a contribution to the investigation of the roots and essence of convex analysis, and to the development of the duality formulas of convex calculus. This is done by means of one single method: firstly conify, then work with the calculus of convex cones, which consists of three rules only, and finally deconify. This generates all definitions of convex objects, duality operators, binary operations and duality formulas, all without the usual need to exclude degenerate situations. The duality operator for convex function agrees with the usual one, the Legendre-Fenchel transform, only for proper functions. It has the advantage over the Legendre-Fenchel transform that the duality formula holds for improper convex functions as well. This solves a well-known problem, that has already been considered in Rockafellar's Convex Analysis (R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970). The value of this result is that it leads to the general validity of the formulas of Convex Analysis that depend on the duality formula for convex functions. The approach leads to the systematic inclusion into convex sets of recession directions, and a similar extension for convex functions. The method to construct binary operations given in (ibidem) is formalized, and this leads to some new duality formulas. An existence result for extended solutions of arbitrary convex optimization problems is given. The idea of a similar extension of the duality theory for optimization problems is given