Bilevel stochastic linear programming problems with quantile criterion

Ivanov, Sergey V.

doi:10.1134/s0005117914010081

Cited by 14 publications

(15 citation statements)

References 15 publications

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“…Furthermore, as ε n ↓ 0, (u n , w n , v n ) is feasible for P(ε n ′ ) for any n ′ ≥ n. Thus, we may assume that (23)ṽ ⊤ W w n − Bu n − b ≤ ε n 2 holds for any n ∈ N without loss of generality. (21), (22) and (23) imply that (u n + α m d u , w n + α m d w ,ṽ) is feasible for P(ε n ) for any m, n ≥ N .…”

Section: A Regularization Scheme For Bilevel Linear Problemsmentioning

confidence: 99%

Risk-Averse Models in Bilevel Stochastic Linear Programming

Burtscheidt¹,

Claus²,

Dempe³

2020

SIAM J. Optim.

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We consider bilevel linear problems, where some parameters are stochastic, and the leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader's outcome can be modeled by a random variable, which we evaluate based on some lawinvariant convex risk measure. A qualitative stability result under perturbations of the underlying probability distribution is presented. Moreover, for the expectation, the expected excess, and the upper semideviation, we establish Lipschitz continuity as well as sufficient conditions for differentiability. Finally, for finite discrete distributions, we reformulate the bilevel stochastic problems as standard bilevel problems and propose a regularization scheme for bilevel linear problems.

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Section: A Regularization Scheme For Bilevel Linear Problemsmentioning

confidence: 99%

Risk-Averse Models in Bilevel Stochastic Linear Programming

Burtscheidt¹,

Claus²,

Dempe³

2020

SIAM J. Optim.

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“…Because the function Φ(u, x, y) is bilinear and problem (35) containts additional probabilistic constraints given by function (29), the method described in [10] cannot be applied to solve the problem. Therefore, to solve this problem, we will use the method described in this paper.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…To solve it or to investigate its properties, the problem can be transformed into another optimization problem using the Karush–Kuhn–Tucker conditions, a generalized equation or the optimal value function of the follower's problem, see for example . Despite its practical relevance there are only a small number of bilevel optimization problems with fuzzy , random, or stochastic data .…”

Section: Introductionmentioning

confidence: 99%

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Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

Dempe

Ivanov

Наумов

2017

Appl Stoch Models Bus & Ind

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The paper is devoted to the stochastic optimistic bilevel optimization problem with quantile criterion in the upper level problem. If the probability distribution is finite, the problem can be transformed into a mixed-integer nonlinear optimization problem. We formulate assumptions guaranteeing that an optimal solution exists. A production planning problem is used to illustrate usefulness of the model.

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“…Ivanov [11] proposed a setting for a bi-level stochastic linear programming problem with quantile criterion and presented a deterministic equivalent of the problem for the case of the scalar random parameter. He showed an equivalent problem in the form of the two-stage stochastic discrete distribution of random parameters, the problem reduces to a mixed linear programming problem.…”

Section: Literature Of Past Workmentioning

confidence: 99%

A New Model for Probabilistic Multi-Period Multi-Objective Project Selection Problem

Khalili‐Damghani¹,

Poortarigh²,

Pakgohar³

2018

dtetr

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The project selection problem is considered as one of the most imperative decisions for investor organizations. Due to non-deterministic nature of some criteria in the real world projects in this paper, a new model for project selection problem is proposed in which some parameters are assumed probabilistic. This model is formulated as a non-linear, multi-objective, multi-period, zero-one programming model. Then the epsilon constraint method and an algorithm are applied to check the Pareto front and to find optimal solutions. A case study is conducted to illustrate the applicability and effectiveness of the approach, with the results presented and analysed. Since the proposed model is more compatible with real world problems, the results are more tangible and trustable compared with deterministic cases. Implications of the proposed approach are discussed and suggestions for further work are outlined.Keywords: Project selection, multi objective programming, chance constraint, epsilon constraint method. INTRODUCTIONDecision making about selection or rejection of a project is considerable from both theoretical and practical aspects and it depends on satisfaction of financial and nonfinancial constraints of the problem. In the real world project selection problems there are normally more than one objective function. This makes the solution algorithm more complicated and time consuming. The purpose of this paper is decision making about selection of N projects in T periods of time such that the profit gets maximum and total cost of equipment, human resources and used raw materials become minimum. In real world problems many of parameters are unlikely to be deterministic and ignoring the stochastic model can lead to unreliable results. In this paper a new framework is proposed for modeling a project selection problem relying on the concept of risk and by applying a linear approximation on probabilistic constraints. The mentioned model is called "Mean-Risk" model which the main idea of it is maximizing the expected profit of selecting a project such that risk curve of this selection always be below of the confidence curve. Here without loss of generality and just for simplification, the confidence is assumed a linear function. Since some parameters are probabilistic, the constraints include these parameters are also probabilistic constraints. Chance constrained programming was developed as a means of describing constraints in the form of probability levels of attainment. Consideration of chance constraints allows the decision maker to consider objectives in terms of their attainment probability. This approach changes constraints with stochastic parameters to constraints with a confidence level as the threshold of decision maker, using variance-covariance matrix. If α is a predetermined confidence level desired by the decision maker, the implication is that a constraint will have a probability of satisfaction of α. After transforming the problem from probabilistic mode to deterministic mode, the resulting model i...

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Bilevel stochastic linear programming problems with quantile criterion

Cited by 14 publications

References 15 publications

Risk-Averse Models in Bilevel Stochastic Linear Programming

Risk-Averse Models in Bilevel Stochastic Linear Programming

Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

A New Model for Probabilistic Multi-Period Multi-Objective Project Selection Problem

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