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517.98

A.M. Вершик (1984)

Zapiski naucnych seminarov Leningradskogo

A duality-based approach to elliptic control problems in non-reflexive Banach spaces

Christian Clason, Karl Kunisch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...

A duality-based approach to elliptic control problems in non-reflexive Banach spaces*

Christian Clason, Karl Kunisch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...

A saddle-point approach to the Monge-Kantorovich optimal transport problem

Christian Léonard (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.

A saddle-point approach to the Monge-Kantorovich optimal transport problem

Christian Léonard (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.

A simple proof in Monge-Kantorovich duality theory

D. A. Edwards (2010)

Studia Mathematica

A simple proof is given of a Monge-Kantorovich duality theorem for a lower bounded lower semicontinuous cost function on the product of two completely regular spaces. The proof uses only the Hahn-Banach theorem and some properties of Radon measures, and allows the case of a bounded continuous cost function on a product of completely regular spaces to be treated directly, without the need to consider intermediate cases. Duality for a semicontinuous cost function is then deduced via the use of an...

Alcuni problemi matematici legati alla gestione ottima di un portafoglio

Maurizio Pratelli (2004)

Bollettino dell'Unione Matematica Italiana

In questa conferenza, vengono esposte le idee essenziali che stanno alla base del classico problema di gestire un portafoglio in modo da rendere massima l'utilità media. I metodi tipici del controllo stocastico sono confrontati con le idee della dualità convessa infinito-dimensionale.

Caracterización algebraica de las aristas infinitas en el conjunto dual factible de un PSI-lineal.

Jesús T. Pastor Ciurana (1987)

Trabajos de Investigación Operativa

Las propiedades geométricas del conjunto factible del dual de un problema semiinfinito lineal son análogas a las correspondientes para el caso finito. En este trabajo mostramos cómo, a partir de la caracterización algebraica de vértices y direcciones extremas, se consigue la correspondiente para aristas infinitas, estableciéndose así las bases para una extensión del método simplex a programas semiinfinitos lineales.

Characterizations of ɛ-duality gap statements for constrained optimization problems

Horaţiu-Vasile Boncea, Sorin-Mihai Grad (2013)

Open Mathematics

In this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.

Comparison between different duals in multiobjective fractional programming

Radu Boţ, Robert Chares, Gert Wanka (2007)

Open Mathematics

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow...

Computation of the distance to semi-algebraic sets

Christophe Ferrier (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is devoted to the computation of distance to set, called S, defined by polynomial equations. First we consider the case of quadratic systems. Then, application of results stated for quadratic systems to the quadratic equivalent of polynomial systems (see [5]), allows us to compute distance to semi-algebraic sets. Problem of computing distance can be viewed as non convex minimization problem: d ( u , S ) = inf x S x - u 2 , where u is in n . To have, at least, lower approximation of distance, we consider the dual...

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