Abstract
Given two convex lower semicontinuous extended real valued functions F and h defined on locally convex spaces, we provide a dual transcription of the relation
Some results in this direction are obtained in the first part of the paper (Lemma 2, Theorem 1). These results then are applied to the case when the left-hand-side in (⋆) is the sum of two convex functions with a convex composite one (Theorem 2). In the spirit of previous works (Hiriart-Urruty and Phelps, J Funct Anal 118:154–166, 1993; Penot, J Convex Anal 3:207–219, 1996, 2005; Thibault, 1995, SIAM J Control Optim 35:1434–1444, 1997, etc.) we give in Theorem 3 a formula for the subdifferential of such a function without any qualification condition. As a consequence of that, we extend to the nonreflexive setting a recent result (Jeyakumar et al., J Glob Optim 36:127–137 2006, Theorem 3.2) about subgradient optimality conditions without constraint qualifications. Finally, we apply Theorem 2 to obtain Farkas-type lemmas and new results on DC, convex, semi-definite, and linear optimization problems.
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Dinh, N., Goberna, M.A., López, M.A. et al. Convex Inequalities Without Constraint Qualification nor Closedness Condition, and Their Applications in Optimization. Set-Valued Anal 18, 423–445 (2010). https://doi.org/10.1007/s11228-010-0166-4
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DOI: https://doi.org/10.1007/s11228-010-0166-4
Keywords
- Convex inequalities
- Subdifferential mapping
- Farkas-type lemmas
- DC programming
- Semi-definite programming
- Convex and linear optimization problems