Differentiability of probability function

Кибзун, А. И.; Uryasev, Stan

doi:10.1080/07362999808809581

Cited by 31 publications

(15 citation statements)

References 15 publications

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“…The differentiability assumptions on μ 0 (H(·, η)) can be guaranteed by assuming continuous differentiability of the function g with respect to both arguments, the existence of the probability density of the random vector V , and by mild regularity conditions (see [9]). Then…”

Section: S(x)mentioning

confidence: 99%

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: S(x)mentioning

confidence: 99%

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

Dentcheva¹,

Henrion²,

́ski³

2007

SIAM J. Optim.

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“…The idea would be to use the general formulas of Kibzun and Uryasev (1998) for the gradients and to expand these formulas for small variances.…”

Section: Resultsmentioning

confidence: 99%

“…This formal derivation can be justified (with the hypothesis that the matrix CðxÞ is invertible and continuously differentiable and CðxÞ is continuously differentiable) by using Remark 2.1 and then Theorem 3.1 of (Kibzun and Uryasev, 1998).…”

Section: The Probabilistic Representation Of the Gradientmentioning

confidence: 99%

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Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations

Garnier¹,

Omrane²,

Rouchdy³

2009

European Journal of Operational Research

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“…It was indicated in [25] that a differentiation formula for an expectation of a continuous function can be obtained by interchanging the gradient and expectation operators. When there exist a monotony relation between y i and an uncertain variable in j, the projection method [23] can be used to compute the chance constraint and its derivatives simultaneously.…”

Section: Mcs-based Probabilistic Analysismentioning

confidence: 99%

Probabilistic analysis for optimal power flow under uncertainty

Zhang

2010

IET Gener. Transm. Distrib.

139

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This study presents a probabilistic analysis to consider the impact of uncertain system parameters on optimal power flow (OPF). A general OPF under uncertainty is formulated as a chance-constrained programming model and its stochastic features are investigated. The impact of the uncertain variables is addressed by the probabilistic analysis with the method of Monte-Carlo simulation (MCS) combined with a deterministic optimisation. In this way, relations between the optimality and reliability in power systems under uncertainty can be achieved. The information derived from the statistical results can be applied in the system risk assessment and operation performance evaluation, which is helpful for making decisions in power system optimal dispatch under uncertainty. Case studies of a 5-bus system and the IEEE 30-bus system are presented to illustrate the probabilistic analysis.

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Differentiability of probability function

Cited by 31 publications

References 15 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

Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations

Probabilistic analysis for optimal power flow under uncertainty

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