Valadier-like Formulas for the Supremum Function II: The Compactly Indexed Case
- Authors: Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 26, no. 1 (2019), p. 299-324
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- Description: Continuing with the work on the subdifferential of the pointwise supremum of convex functions, started in part I of this paper [R. Correa, A. Hantoute, M. A. Lopez, Valadier-like formulas for the supremum function I, J. Convex Analysis 25 (2018) 1253-1278], we focus now on the compactly indexed case. We assume that the index set is compact and that the data functions are upper semicontinuous with respect to the index variable (actually, this assumption will only affect the set of epsilon-active indices at the reference point). As in the previous work, we do not require any continuity assumption with respect to the decision variable. The current compact setting gives rise to more explicit formulas, which only involve subdifferentials at the reference point of active data functions. Other formulas are derived under weak continuity assumptions. These formulas reduce to the characterization given by M. Valadier [Sous-differentiels d'une borne superieure et d'une somme continue de fonctions convexes, C. R. Acad. Sci. Paris Ser. A-B Math. 268 (1969) 39-42, Theorem 2], when the supremum function is continuous.
Holder error bounds and holder calmness with applications to convex semi-infinite optimization
- Authors: Kruger, Alexander , Lopez, Marco , Yang, Xiaoqi , Zhu, Jiangxing
- Date: 2019
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 27, no. 4 (Dec 2019), p. 995-1023
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- Description: Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Holder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Holder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Holder calmness modulus of the argmin mapping in the framework of linear programming.
Relaxed Lagrangian duality in convex infinite optimization: Reverse strong duality and optimality
- Authors: Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Volle, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Journal of Applied and Numerical Optimization Vol. , no. (2021), p.
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- Description: We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems.