Lopsided convergence: an extension and its quantification

Røyset, Johannes Ø.; Wets, Roger J.-B.

doi:10.1007/s10107-018-1275-3

Cited by 14 publications

(19 citation statements)

References 23 publications

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“…It is apparent from this proposition, which is a collection of results from [33], that lopsided convergence becomes a central property in the study of approximations of (SAP). It is well-known that the infimum over a compact set of an extended real-valued lower semicontinuous (lsc) function is attained.…”

Section: (Iv) If the Convergence Is Tight And Infmentioning

confidence: 93%

“…We set out to apply our recent extension of the variational theory of lopsided convergence [33] for the first time in the context of optimization under stochastic ambiguity, where it appears especially well suited. In this section, we first recall definitions and essential facts about lopsided convergence pertaining to the present context; see [33] for details and proofs.…”

Section: Lopsided Convergencementioning

confidence: 99%

“…In this section, we first recall definitions and essential facts about lopsided convergence pertaining to the present context; see [33] for details and proofs. Second, we develop results for optimization under stochastic ambiguity, including sufficient conditions for lopsided convergence.…”

Section: Lopsided Convergencementioning

confidence: 99%

“…The notion originated with [2] and later was modified and extended in [15,16,34,33]. We show that lopsided convergence can be used to establish the existence of solutions of optimization problems with stochastic ambiguity as well as to prove the convergence of solutions of approximate problems to those of an original problem under mild assumptions.…”

Section: Introductionmentioning

confidence: 98%

“…Third, we estimate, for the first time, the lop-distance between bifunctions, a quantification of lopsided convergence developed in [33]. Utilizing these estimates, the paper bounds the difference between optimal solutions and optimal values of two optimization problems with stochastic ambiguity.…”

Section: Introductionmentioning

confidence: 99%

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Variational Theory for Optimization under Stochastic Ambiguity

Røyset

Wets

2017

SIAM J. Optim.

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Abstract.Stochastic ambiguity provides a rich class of uncertainty models that includes those in stochastic, robust, risk-based, and semi-infinite optimization, and that accounts for both uncertainty about parameter values as well as incompleteness of the description of uncertainty. We provide a novel, unifying perspective on optimization under stochastic ambiguity that rests on two pillars. First, the paper models ambiguity by decision-dependent collections of cumulative distribution functions viewed as subsets of a metric space of upper semicontinuous functions. We derive a series of results for this setting including estimates of the metric, the hypo-distance, and a new proof of the equivalence with weak convergence. Second, we utilize the theory of lopsided convergence to establish existence, convergence, and approximation of solutions of optimization problems with stochastic ambiguity. For the first time, we estimate the lop-distance between bifunctions and show that this leads to bounds on the solution quality for problems with stochastic ambiguity. Among other consequences, these results facilitate the study of the "price of robustness" and related quantities.

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Section: (Iv) If the Convergence Is Tight And Infmentioning

confidence: 93%

Section: Lopsided Convergencementioning

confidence: 99%

Section: Lopsided Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Variational Theory for Optimization under Stochastic Ambiguity

Røyset

Wets

2017

SIAM J. Optim.

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A Generalized $$\Gamma $$-Convergence Concept for a Class of Equilibrium Problems

Hintermüller,

Stengl

2024

J Nonlinear Sci

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A novel generalization of $$\Gamma $$ Γ -convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our $$\Gamma $$ Γ -convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature.

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Consistency of statistical estimators of solutions to stochastic optimization problems

Diem

2022

J Glob Optim

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Lopsided convergence: an extension and its quantification

Cited by 14 publications

References 23 publications

Variational Theory for Optimization under Stochastic Ambiguity

Variational Theory for Optimization under Stochastic Ambiguity

A Generalized $$\Gamma $$-Convergence Concept for a Class of Equilibrium Problems

Consistency of statistical estimators of solutions to stochastic optimization problems

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