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

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

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

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

Optimalisering in financiering, economie en wiskunde: welke toepassingen zijn overtuigend?

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 werk van Kydland en Presscot (`inzicht') en een bewijs van de hoofdstelling van de algebra (`diepte')

On the complexity of the primal self-concordant barrier method

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On the Galois module structure over CM-fields

In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extension N of a given CM-field K are invariant under the involution on the set of all realizations of Δ over K which is induced by complex conjugation on K and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number field K are completely characterized by ramification and Galois module structure

A simple approach to discrete-time infinite horizon problems

In this note, we consider a type of discrete-time infinite horizon problem that has one ingredient only, a constraint correspondence. The value function of a policy has an intuitive mono- tonicity property; this is the essence of the four standard theorems on the functional equation (‘the Bellman equation’). Some insight is offered into the boundedness condition for the value function that occurs in the formulation of these results: it can be interpreted as accountability of the loss of value caused by a non-optimal policy or, alternatively, it can be interpreted as irrelevance of devia- tions, in the distant future, from the considered policy. Without the boundedness condition, there is a gap, which can be viewed as the persistent potential positive impact of deviations, in the distant future, from the considered policy. The general stationary discrete-time infinite horizon optimization problem considered in Stokey and Lucas (1989) can be mapped to this type of problems and so the results in the present paper can be applied to this general class of problems

On the uniform limit condition for discrete-time infinite horizon problems

In this note, a simplified version of the four main results for discrete-time infinite horizon problems, theorems 4.2-4.5 from Stokey, Lucas and Prescott (1989) [SLP], is presented. A novel assumption on these problems is proposed—the uniform limit condition, which is formulated in terms of the data of the problem. It can be used for example before one has started to look for the optimal value function and for an optimal plan or if one cannot find them analytically: one verifies the uniform limit condition and then one disposes of criteria for optimality of the value function and a plan in terms of the functional equation and the boundedness condition. A comparison to [SLP] is made. The version in [SLP] requires one to verify whether a candidate optimal value function satisfies the boundedness condition; it is easier to check the uniform limit condition instead, as is demonstrated by examples. There is essentially no loss of strength or generality compared to [SLP]. The necessary and sufficient conditions for optimality coincide in the present paper but not in [SLP]. The proofs in the present paper are shorter than in [SLP]. An earlier attempt to simplify, in Acemoglu (2009) --here the limit condition is used rather than the uniform limit condition-- is not correct