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 a conic approach to convex analysis.

Abstract. The aim of this paper is to make an attempt to justify the main results from Convex Analysis by one elegant tool, the conification method, which consists of three steps: conify, work with convex cones, deconify. It is based on the fact that the standard operations and facts (`the calculi') are much simpler for special convex sets, convex cones. By considering suitable classes of convex cones, we get the standard operations and facts for all situations in the complete generality that is required. The main advantages of this conification method are that the standard operations---linear image, inverse linear image, closure, the duality operator, the binary operations and the inf-operator---are defined on all objects of each class of convex objects---convex sets, convex functions, convex cones and sublinear functions---and that moreover the standard facts---such as the duality theorem---hold for all closed convex objects. This requires that the analysis is carried out in the context of convex objects over cosmic space, the space that is obtained from ordinary space by adding a horizon, representing the directions of ordinary space.

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

Crystallization of Isotactic Poly(methylmethacrylate) in Monolayers and Thin Films

Langmuir-Blodgett monolayers of isotactic PMMA exhibit a pressure-induced transition upon compression, that can be described in terms of a two-dimensional crystallization process, analogous to a normal melt crystallization. These water surface crystallized monolayers can be used to prepare highly crystalline thin films of isotactic PMMA with tailor-made orientational characteristics.

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

The Lagrange multiplier rule revisited

In this paper we give a short novel proof of the well-known Lagrange multiplier rule, discuss the sources of the power of this rule and consider several applications of this rule. The new proof does not use the implicit function theorem and combines the advantages of two of the most well-known proofs: it provides the useful geometric insight of the elimination approach based on differentiable curves and technically it is not more complicated than the simple penalty approach. Then we emphasize that the power of the rule is the reversal of order of the natural tasks, elimination and differentiation. This turns the hardest task,elimination, from a nonlinear problem into a linear one. This phenomenon is illustrated by several convincing examples of applications of the rule to various areas. Finally we give three hints on the use of the rule.Lagrange multiplier rule;compactness;optimization

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. Basic properties of the extended duality, including the extended bi-polar theorem, are proven. Examples are give to show the applications of the new results.Duality;Convex cones;Bi-polar theorem;Conic optimization

Matrix convex functions with applications to weighted centers for semidefinite programming

In this paper, we develop various calculus rules for general smooth matrix-valued functions and for the class of matrix convex (or concave) functions first introduced by Loewner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function -log X to study a new notion of weighted convex centers for semidefinite programming (SDP) and show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the calculus rules for matrix convex functions can be used in the implementation of barrier methods for optimization problems involving nonlinear matrix functions.matrix convexity;matrix monotonicity;semidefinite programming