Optimality Functions in Stochastic Programming

Røyset, Johannes Ø.

doi:10.21236/ada513135

Cited by 7 publications

(12 citation statements)

References 20 publications

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“…Related to our work we mention [32,37,33,2,24,42,43,50,40,41,47,48,49,15,36,9,7,20,19,10,3]. Except [36,19,20], all other papers assume light-tailed data.…”

Section: Related Workmentioning

confidence: 99%

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Sample average approximation with heavier tails I: non-asymptotic bounds with weak assumptions and stochastic constraints

Oliveira¹,

Thompson²

2017

Preprint

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We derive exponential finite-sample nonasymptotic deviation inequalities for the sample average approximation (SAA) estimator's near-optimal solution set and optimal value. In that respect, we give three contributions. First, our bounds do not require sub-Gaussian assumptions as in previous literature of stochastic optimization (SO). Instead, we just assume random Hölder continuity and a heavy-tailed distribution with finite 2nd moments, a framework more suited for risk-averse portfolio optimization. Second, we derive new deviation inequalities for SO problems with expected-valued stochastic constraints which guarantee joint near feasibility and optimality in terms of the original problem. Unlike previous works, we do not assume a metric regular (MR) solution set, strong growth conditions on the objective nor an indirect problem reformulation (such as penalization or first order conditions, both of which are often necessary but not sufficient conditions). Instead, we just assume a MR feasible set, making our analysis general enough for many classes of problems. A finite-sample near feasibility and optimality deviation is established for MR sets which are nonconvex or which are convex and non-strictly feasible. For strictly feasible convex sets, we obtain finite-sample exact feasibility and near optimality guarantees. The feasible set's MR constant is an additional condition number. For convex sets, we present localized feasibility guarantees in terms of smaller metric entropies. Third, we obtain a general uniform concentration inequality for heavy-tailed Hölder continuous functions using empirical process theory. This is the main tool in our analysis but it is also a result of independent interest.

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“…Related to our work we mention [32,37,33,2,24,42,43,50,40,41,47,48,49,15,36,9,7,20,19,10,3]. Except [36,19,20], all other papers assume light-tailed data.…”

Section: Related Workmentioning

confidence: 99%

“…Related to our work we mention [32,37,33,2,24,42,43,50,40,41,47,48,49,15,36,9,7,20,19,10,3]. Except [36,19,20], all other papers assume light-tailed data. However, their analysis is asymptotic so that assuming only finite second moments is expected.…”

Section: Related Workmentioning

confidence: 99%

Sample average approximation with heavier tails I: non-asymptotic bounds with weak assumptions and stochastic constraints

Oliveira¹,

Thompson²

2017

Preprint

View full text Add to dashboard Cite

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“…A refinement of SAA methods based on a careful analysis of the schedule sequence and the error tolerance of the optimization method was given in [24]. An approach that offers a quantitative measure of SAA solutions is presented in [26], where optimality functions for general stochastic programs (expected value objective and constraint functions) are considered. A method presented in [25] states an auxiliary problem that is solved before the optimization process is started.…”

Section: Introductionmentioning

confidence: 99%

Inexact Restoration approach for minimization with inexact evaluation of the objective function

Krejić¹,

Martínez²

2015

Math. Comp.

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A new method is introduced for minimizing a function that can be computed only inexactly, with different levels of accuracy. The challenge is to evaluate the (potentially very expensive) objective function with low accuracy as far as this does not interfere with the goal of getting high accuracy minimization at the end. For achieving this goal the problem is reformulated in terms of constrained optimization and handled with an Inexact Restoration technique. Convergence is proved and numerical experiments motivated by Electronic Structure Calculations are presented, which indicate that the new method overcomes current approaches for solving large-scale problems.

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“…Our choice of the term "consistent approximations" is motivated by E. Polak's concept in [40,. There epi-convergence of approximating objective functions combined with an outer epi-limit of approximating optimality functions (after a sign change) are defined as consistent; see [49,52,25,39,54] for applications in stochastic optimization, semi-infinite programming, nonsmooth optimization, and optimal control. Since optimality functions are essentially gap-functions of generalized equations representing optimality conditions, Polak's concept is, roughly, equivalent to weak consistency as we define it below.…”

Section: Introductionmentioning

confidence: 99%

Consistent Approximations in Composite Optimization

Røyset¹

2022

Preprint

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Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.

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Optimality Functions in Stochastic Programming

Cited by 7 publications

References 20 publications

Sample average approximation with heavier tails I: non-asymptotic bounds with weak assumptions and stochastic constraints

Sample average approximation with heavier tails I: non-asymptotic bounds with weak assumptions and stochastic constraints

Inexact Restoration approach for minimization with inexact evaluation of the objective function

Consistent Approximations in Composite Optimization

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