Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

# Advanced Search

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 0841.49005
Zolezzi, T.
Well-posedness criteria in optimization with application to the calculus of variations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 25, No.5, 437-453 (1995). ISSN 0362-546X

In this paper, for the global optimization problem $(X, J)$, to minimize the proper extended real-valued function $J: X\to (- \infty, \infty)$ over the given subset $X$ of a normed linear space equipped with the strong convergence, well-posedness criteria are derived. The given problem is embedded into a smoothly parametrized family $(X, I(., p))$ of minimization problems, where $p$ is a parameter belonging to a given Banach space $L$, and $p^*$ is the parameter value to which the given unperturbed problem corresponds, i.e., $I(x, p^*)= J(x)$ $\forall x\in X$. Defining the value function $V(p)= \inf\{I(x, p)\mid x\in X\}$ the author gives the following definition of well-posedness.\par $(X, J)$ is well-posed with respect to the embedding iff $V(p)> -\infty$, $\forall p\in L$, and there exists a unique $x^*= \arg\min(X, J)$ and for every sequence $p_n\to p^*$ and every sequence $x_n\in X$ such that $I(x_n, p_n)- V(p_n)\to 0$ as $n\to \infty$ we have $x_n\to x^*$ in $X$.\par This definition is stronger than the Tikhonov well-posedness. In the following, the defined well-posedness is related under suitable conditions to the differentiability properties of $V$ at $p^*$. These abstract results are applied to one-dimensional problems of the calculus of variations.
[H.Benker (Merseburg)]
MSC 2000:
*49J27 Optimal control problems in abstract spaces (existence)
49K99 Necessary and sufficient conditions for optimality
90C99 Mathematical programming

Keywords: global optimization problem; well-posedness criteria

Login Username: Password:

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences
Published by Springer-Verlag | Webmaster