- Title
- From the Farkas lemma to the Hahn-Banach theorem
- Creator
- Dinh, Nguyen; Goberna, Miguel; López, Marco; Mo, T. H.
- Date
- 2014
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/63117
- Identifier
- vital:6434
- Identifier
-
https://doi.org/10.1137/120901805
- Identifier
- ISSN:1052-6234
- Abstract
- This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ° g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.
- Publisher
- Society for Industrial and Applied Mathematics Publications
- Relation
- SIAM Journal on Optimization Vol. 24, no. 2 (2014), p. 678-701
- Rights
- © 2014 Society for Industrial and Applied Mathematics
- Rights
- Open Access
- Rights
- This metadata is freely available under a CCO license
- Subject
- Farkas lemma; Hahn-Banach theorem; Hahn-Banach-Lagrange theorem; Mazur-Orlicz theorem; 0102 Applied Mathematics; 0103 Numerical and Computational Mathematics
- Full Text
- Reviewed
- Hits: 1651
- Visitors: 1953
- Downloads: 356
| Thumbnail | File | Description | Size | Format | |||
|---|---|---|---|---|---|---|---|
| View Details Download | SOURCE1 | Published Version | 322 KB | Adobe Acrobat PDF | View Details Download |