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Erschienen in:
2013 | OriginalPaper | Buchkapitel
9. Optimization and multipliers
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
The abstract optimization problem min A
f consists of minimizing a cost function
f(x) over the points x belonging to the admissible set
A. The nature of A, and also of the function f, determine whether our problem is classical or modern, discrete or continuous, finite or infinite dimensional, smooth or convex. Optimization is a rich and varied subject with numerous applications. The core mathematical issues, however, are always the same: We study these issues, paying particular attention to the distinction between the deductive and inductive methods, and to the method of multipliers.
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Existence: Is there, in fact, a solution of the problem?
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Necessary conditions: What special properties must a solution have, properties that will help us to identify it?
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Sufficient conditions: Having identified a point that is suspected of being a solution, what tools can we apply to confirm the suspicion?