Abstract
This chapter presents the duality theory for optimization problems, by both the minimax and perturbation approach, in a Banach space setting. Under some stability (qualification) hypotheses, it is shown that the dual problem has a nonempty and bounded set of solutions. This leads to the subdifferential calculus, which appears to be nothing but a partial subdifferential rule. Applications are provided to the infimal convolution, as well as recession and perspective functions. The relaxation of some nonconvex problems is analyzed thanks to the Shapley–Folkman theorem.
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Notes
- 1.
Except maybe when the set is a singleton and then the dimension is zero, where this is a matter of definition. However the case when \(A-B\) reduces to a singleton means that both A and B are singletons and then it is easy to separate them.
- 2.
Baire’s lemma tells us that any countable intersection of dense subsets in X is dense, or equivalently, that any countable union of closed sets with empty interiors has an empty interior.
- 3.
Provided by Lionel Thibault, U. Montpellier II.
- 4.
Not to be confused with the duality Lagrangian defined in (1.156).
- 5.
If the minimal number of a nonnegative combination was greater than \(n+p\), adding some linear combination of these elements equal to 0 and with nonzero elements, we could easily find another nonnegative combination of z with fewer nonzero coefficients, which would give a contradiction.
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Bonnans, J.F. (2019). A Convex Optimization Toolbox. In: Convex and Stochastic Optimization. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-14977-2_1
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DOI: https://doi.org/10.1007/978-3-030-14977-2_1
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