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...

We consider the classical duality operators for convex objects such as the polar of a convex set containing the origin, the dual norm, the Fenchel-transform of a convex function and the conjugate of a convex cone. We give a new, sharper, unified treatment of the theory of these operators,...

Our aim is to give a simple view on the basics and applications of convex analysis. The essential feature of this account is the systematic use of the possibility to associate to each convex object---such as a convex set, a convex function or a convex extremal problem--- a cone, without loss of...

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...

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...

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...

In deze paper wordt de stelling onderbouwd dat er drie redenen zijn waarom een toepassing van optimaliseringsmethoden overtuigend is: `nut', `inzicht' en `diepte'. Ieder van de drie wordt geillustreerd met eenkarakteristiek voorbeeld: de prijsformule voor opties van Black en Scholes (`nut'), het...

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...

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...

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...