Stability analysis for stochastic programs

Römisch, Werner; Schultz, Rüdiger

doi:10.1007/bf02204819

Cited by 87 publications

(32 citation statements)

References 26 publications

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“…On the other hand, the specific structure of dominance constraints is significantly different from the structure of finitely many probabilistic constraints. Our stability analysis follows similar patterns to those in [8,22,23], where the focus was on probabilistic constraints. However, a straightforward application of those results (a recent overview of which can be found in [21]) is not possible due to the specific structure of problem (1.3).…”

Section: Introduction the Notion Of Stochastic Ordering (Or Stochastmentioning

confidence: 72%

Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints

Dentcheva¹,

Henrion²,

́ski³

2007

SIAM J. Optim.

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Abstract. We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type. Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the dominance constraints.

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Section: Introduction the Notion Of Stochastic Ordering (Or Stochastmentioning

confidence: 72%

Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints

Dentcheva¹,

Henrion²,

́ski³

2007

SIAM J. Optim.

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“…Our purpose is to analyse convergence of the optimal value and the optimal solutions of problem (2.11) when P N converges to P. In the case when P reduces to a singleton of the true probability measure, the convergence analysis is well documented in the literature of stochastic programming; see [22,23,37,38] and references therein. Our focus here is the case when P is a set, e.g., constructed through some moment conditions and P N is an approximation regime with some parameters being estimated through empirical data or samples.…”

Section: Preliminariesmentioning

confidence: 99%

“…We do so by showing that P (H(·)) is both upper and lower semicontinuous. Since H(x) is closed for any x ∈ X, P → {x ∈ X : P (H(x)) ≥ p} has a closed graph for fixed p ∈ IR [38,Proposition 3.1]. This means that the set {x ∈ X : P (H(x)) ≥ p} is closed, hence the upper semicontinuity of P (H(·)) holds [36,Page 13].…”

Section: Continuity Of Probability Function P (H(·))mentioning

confidence: 99%

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

Guo¹,

Xu²,

Zhang³

2017

SIAM J. Optim.

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Abstract. Convergence analysis for optimization problems with chance constraints concerns impact of variation of probability measure in the chance constraints on the optimal value and the optimal solutions and research on this topic has been well documented in the literature of stochastic programming. In this paper, we extend such analysis to optimization problems with distributionally robust chance constraints where the true probability distribution is unknown, but it is possible to construct an ambiguity set of probability distributions and the chance constraint is based on the most conservative selection of probability distribution from the ambiguity set. The convergence analysis focuses on impact of the variation of the ambiguity set on the optimal value and the optimal solutions. We start by deriving general convergence results under abstract conditions such as continuity of the robust probability function and uniform convergence of the robust probability functions and followed with detailed analysis of these conditions. Two sufficient conditions have been derived with one applicable to both continuous and discrete probability distribution and the other to continuous distribution. Case studies are carried out for ambiguity sets being constructed through moments and samples.

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“…These results were detailed mainly for separable linear probabilistic programs and α-concave probability distributions, see e.g. [16], [25], [26].…”

Section: Theorem 1 (Theorem 439 In [31]) Let the Functions Gmentioning

confidence: 99%

Robustness Analysis of Stochastic Programs with Joint Probabilistic Constraints

Dupačová

2013

IFIP Advances in Information and Communication Technology

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Abstract. Due to their frequently observed lack of convexity and/or smoothness, stochastic programs with joint probabilistic constraints have been considered as a hard type of constrained optimization problems, which are rather demanding both from the computational and robustness point of view. Dependence of the set of solutions on the probability distribution rules out the straightforward construction of the convexitybased global contamination bounds for the optimal value; at least local results for probabilistic programs of a special structure will be derived. Several alternative approaches to output analysis will be mentioned.

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Stability analysis for stochastic programs

Cited by 87 publications

References 26 publications

Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints

Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

Robustness Analysis of Stochastic Programs with Joint Probabilistic Constraints

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