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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...
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...
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.
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 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...
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.
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