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Optimierung konvexer Funktionen mit Stabilitätsbetrachtungen

100%

Kosmol P.

Instytut Matematyczny Polskiej Akademii Nauk

INHALTSVERZEICHNIS Einleitung....................................................................................................................... 5 § 1. Existenz der Minimallösungen................................................................... 7 § 2. Modulare Optimierung......................................................................................... 9 § 3. Der Satz von Banach-Steinhaus für konvexe Funktionen.............................. 17 § 4. Stabilität bei nichtlinearer Optimierung............................................................ 23 Literaturverzeichnis ............................................................................................. 37

Pointwise minimization of supplemented variational problems

61%

Kosmol P., Müller-Wichards D.

Colloquium Mathematicum

2004

tom 101

nr 1

25-49

We describe an approach to variational problems, where the solutions appear as pointwise (finite-dimensional) minima for fixed t of the supplemented Lagrangian. The minimization is performed simultaneously with respect to the state variable x and ẋ, as opposed to Pontryagin's maximum principle, where optimization is done only with respect to the ẋ-variable. We use the idea of the equivalent problems of Carathéodory employing suitable (and simple) supplements to the original minimization problem. Whereas Carathéodory considers equivalent problems by use of solutions of the Hamilton-Jacobi partial differential equations, we shall demonstrate that quadratic supplements can be constructed such that the supplemented Lagrangian is convex in the vicinity of the solution. In this way, the fundamental theorems of the calculus of variations are obtained. In particular, we avoid any employment of field theory.

Ograniczanie wyników

1 Colloquium Mathematicum

2 Kosmol P.

1 Müller-Wichards D.

1 2004

1 1976

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Optimierung konvexer Funktionen mit Stabilitätsbetrachtungen

Pointwise minimization of supplemented variational problems