Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

Corrêa, Rafael; Hantoute, Abderrahim; López, Marco A.

doi:10.1007/s10107-020-01592-0

Cited by 4 publications

(5 citation statements)

References 20 publications

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“…Note that the proof of Lemmas 3.1 and 3.2 is similarly based on those in [12,13]. Moreover, the Lipschitz property of the functions fk (•, μ) and F k given in Proposition 3.2 plays a crucial role in establishing the main results.…”

Section: Thus We Can Find a Diagonal Net (U γ Jmentioning

confidence: 83%

“…The main goal of this section is to establish new robust necessary optimality conditions for the nonconvex and nonsmooth problem (UMP) without the compactness of uncertainty sets and the continuity of uncertain parameters. Our approach is mainly based on the compactification of the sets and the u.s.c regularization of the functions, which have been proposed in [12,13].…”

Section: A Unified Study Of Optimality Conditionsmentioning

confidence: 99%

“…It is worth mentioning that Correa et al [12,13] perfectly removed the compactness of the index set and the continuity of the indicator parameter by using the compactification of the index set and an appropriate enlargement of the original family of data functions, and proposed general formulas for the subdifferential of the supremum of convex functions. Inspired by the work of [12,13], a more precise question is: By virtue of the compactification of uncertainty sets and the appropriate enlargement of original functions, is there any chance for getting rid of the aforementioned assumptions (convexity-concavity of functions, compactness and continuity of uncertain parameters), but keeping alive the possibility of still applying robust optimality conditions developed under them?…”

Section: Introductionmentioning

confidence: 99%

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Unified Robust Necessary Optimality Conditions for Nonconvex Nonsmooth Uncertain Multiobjective Optimization

Wang

Feng

2022

J Optim Theory Appl

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This paper is concerned with nonconvex nonsmooth uncertain multiobjective optimization problems, in which the decision variable of both objective and constraint functions is defined on Banach space while uncertain parameters are defined on arbitrary nonempty (may not be compact) sets. We employ the Stone-Cech compactification of uncertainty sets and the upper semicontinuous regularization of original functions with respect to uncertain parameters, giving rise to unified robust necessary optimality conditions for the local robust weakly efficient solution of the considered problem. Moreover, we derive weak and strong KKT robust necessary conditions via the constraint qualification and the regularity condition, respectively. Several examples are provided to illustrate the validity of our results.

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Section: Thus We Can Find a Diagonal Net (U γ Jmentioning

confidence: 83%

Section: A Unified Study Of Optimality Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Unified Robust Necessary Optimality Conditions for Nonconvex Nonsmooth Uncertain Multiobjective Optimization

Wang

Feng

2022

J Optim Theory Appl

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“…so one could rely on suitable marginal function rules for supremum functions, see e.g. [16,36,37,54], and the sum rule for the singular subdifferential, see e.g. [50,Corollary 2.21], in order to estimate ∂ ∞ χ(x, ȳ, ū) from above.…”

Section: Transformation Based On Lagrange Dualitymentioning

confidence: 99%

Optimality conditions for mixed discrete bilevel optimization problems

2018

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Usually, bilevel optimization problems need to be transformed into singlelevel ones in order to derive optimality conditions and solution algorithms. Among the available approaches, the replacement of the lower-level problem by means of duality relations became popular quite recently. We revisit three realizations of this idea which are based on the lower-level Lagrange, Wolfe, and Mond-Weir dual problem. The resulting single-level surrogate problems are equivalent to the original bilevel optimization problem from the viewpoint of global minimizers under mild assumptions. However, all these reformulations suffer from the appearance of so-called implicit variables, i.e., surrogate variables which do not enter the objective function but appear in the feasible set for modeling purposes. Treating implicit variables as explicit ones has been shown to be problematic when locally optimal solutions, stationary points, and applicable constraint qualifications are compared to the original problem. Indeed, we illustrate that the same difficulties have to be faced when using these duality-based reformulations. Furthermore, we show that the Mangasarian-Fromovitz constraint qualification is likely to be violated at each feasible point of these reformulations, contrasting assertions in some recently published papers.

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“…Finally, the method is illustrated by means of numerous computational examples. • Correa et al [9] establish general formulas for the subdifferential of the pointwise supremum of convex functions covering both the compact continuous and the noncompact non-continuous setting. Moreover the authors provide two applications: to nonconvex Fenchel duality, and to Fritz-John and KKT conditions in convex semi-infinite programming.…”

mentioning

confidence: 99%

Special Issue: Continuous Optimization and Stability Analysis

2021

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Continuous optimization is a vital research area with many active branches such as linear and nonlinear optimization, stochastic optimization, vector optimization, semi-infinite optimization, etc. Stability analysis of optimization problems is of crucial interest in applications, and makes optimization an ideal testing bench for many disciplines such as convex analysis, set-valued analysis, variational analysis, and wellposedness, among others.The present special issue offers some recent developments in this wide area and aims to honor Prof. Marco Antonio López Cerdá on the occasion of his 70th birthday, as well as to show recognition by his colleagues to his contribution to this field. His achievements include some monographs, several surveys and more than 150 scientific articles on optimality, duality, stability and algorithms in semi-infinite programming, Lipschitz-type properties, error bounds, subdifferential calculus, and others such as game theory, robustness, stationarity and regularity.Marco A. López was born in Alcoy (Alicante, Spain) in 1949 and has spent most of his academic career (since 1985) as a full professor of statistics and operations research at the University of Alicante, where nowadays he is Emeritus Professor. Marco is a corresponding fellow of the Spanish Royal Academy of Sciences, an Honorary Research Fellow of the Federation University (Australia), and a Doctor Honoris Causa at the University of Limoges (France). He has also led important research projects, theoretical and applied. From 2008 to 2012 he was the coodinator of i-MATH Consolider B J.

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Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

Cited by 4 publications

References 20 publications

Unified Robust Necessary Optimality Conditions for Nonconvex Nonsmooth Uncertain Multiobjective Optimization

Unified Robust Necessary Optimality Conditions for Nonconvex Nonsmooth Uncertain Multiobjective Optimization

Optimality conditions for mixed discrete bilevel optimization problems

Special Issue: Continuous Optimization and Stability Analysis

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