Variational Theory for Optimization under Stochastic Ambiguity

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

doi:10.1137/16m1060704

Cited by 33 publications

(21 citation statements)

References 42 publications

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“…The next result generalizes a statement in [29] to metric spaces and also tightens it slightly. We define for any C ⊂ X the function ι C : X → IR that has ι C (x) = 0 if x ∈ C and ι C (x) = ∞ otherwise.…”

Section: Proposition (Estimates For Auxiliary Quantity)supporting

confidence: 76%

“…Below, we also let dom f := {x ∈ X : f (x) < ∞}. The next proposition extends a result in [29] from X = IR n to general metric spaces.…”

Section: Distance Estimatesmentioning

confidence: 67%

“…Also, our Lipschitz-stability result for nearoptimal solutions goes beyond that of [7] as it does not require convexity, and we consider near-optimal near-feasible solutions. We refine the estimates of distances between epi-graphs in IR n provided by [24,Chapter 7] and [29], and also make them applicable to general metric spaces. Approximations of lsc functions on IR n by epi-splines is given by [27].…”

Section: Introductionmentioning

confidence: 99%

“…This property is often important in probability density estimation, where the focus is on the modes of the density functions, i.e., the maximizers of the density functions. When fitting cumulative distribution functions, the metric metrizes weak convergence [29]. In both of these cases a reorientation towards upper semicontinuous functions instead of lsc functions is needed.…”

Section: Introductionmentioning

confidence: 99%

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Approximations and solution estimates in optimization

Røyset

2017

Math. Program.

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Abstract. Approximation is central to many optimization problems and the supporting theory provides insight as well as foundation for algorithms. In this paper, we lay out a broad framework for quantifying approximations by viewing finite-and infinite-dimensional constrained minimization problems as instances of extended real-valued lower semicontinuous functions defined on a general metric space. Since the Attouch-Wets distance between such functions quantifies epi-convergence, we are able to obtain estimates of optimal solutions and optimal values through estimates of that distance. In particular, we show that near-optimal and near-feasible solutions are effectively Lipschitz continuous with modulus one in this distance. We construct a general class of approximations of extended real-valued lower semicontinuous functions that can be made arbitrarily accurate and that involve only a finite number of parameters under additional assumptions on the underlying metric space.

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Section: Proposition (Estimates For Auxiliary Quantity)supporting

confidence: 76%

“…Below, we also let dom f := {x ∈ X : f (x) < ∞}. The next proposition extends a result in [29] from X = IR n to general metric spaces.…”

Section: Distance Estimatesmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximations and solution estimates in optimization

Røyset

2017

Math. Program.

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“…We refer to [9, Cor. 2.5] for a result in the convex case and [36,Thm. 5.6] for one under Lipschitz continuity assumptions.…”

Section: Proposition (Inf-projections)mentioning

confidence: 99%

Stability and Error Analysis for Optimization and Generalized Equations

Røyset¹

2020

SIAM J. Optim.

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Stability and error analysis remain challenging for problems that lack regularity properties near solutions, are subject to large perturbations, and might be infinite dimensional. We consider nonconvex optimization and generalized equations defined on metric spaces and develop bounds on solution errors using the truncated Hausdorff distance applied to graphs and epigraphs of the underlying set-valued mappings and functions. In the process, we extend the calculus of such distances to cover compositions and other constructions that arise in nonconvex problems. The results are applied to constrained problems with feasible sets that might have empty interiors, solution of KKT systems, and optimality conditions for difference-of-convex functions and composite functions.

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Statistics with set-valued functions: applications to inverse approximate optimization

Aswani

2018

Math. Program.

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Much of statistics relies upon four key elements: a law of large numbers, a calculus to operationalize stochastic convergence, a central limit theorem, and a framework for constructing local approximations. These elements are wellunderstood for objects in a vector space (e.g., points or functions); however, much statistical theory does not directly translate to sets because they do not form a vector space. Building on probability theory for random sets, this paper uses variational analysis to develop operational tools for statistics with set-valued functions. These tools are first applied to nonparametric estimation (kernel regression of set-valued functions). The second application is to the problem of inverse approximate optimization, in which approximate solutions (corrupted by noise) to an optimization problem are observed and then used to estimate the amount of suboptimality of the solutions and the parameters of the optimization problem that generated the solutions. We show that previous approaches to this problem are statistically inconsistent when the data is corrupted by noise, whereas our approach is consistent under mild conditions.

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Variational Theory for Optimization under Stochastic Ambiguity

Cited by 33 publications

References 42 publications

Approximations and solution estimates in optimization

Approximations and solution estimates in optimization

Stability and Error Analysis for Optimization and Generalized Equations

Statistics with set-valued functions: applications to inverse approximate optimization

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