In this paper, we consider a data-driven distributionally robust two-stagestochasticlinear optimization problem over 1-Wasserstein ball centered at a discrete empirical distribution. Differently from the traditional...
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In this paper, we consider a data-driven distributionally robust two-stagestochasticlinear optimization problem over 1-Wasserstein ball centered at a discrete empirical distribution. Differently from the traditional two-stagestochasticprogramming which involves the expected recourse function as the preference criterion and hence is risk-neutral, we take the conditional value-at-risk (CVaR) as the risk measure in order to model its effects on decision making problems. We mainly explore tractable reformulations for the proposed robust two-stagestochasticprogramming with mean-CVaR criterion by analyzing the first case where uncertainties are only in the objective function and then the second case where uncertainties are only in the constraints. We demonstrate that the first model can be exactly reformulated as a deterministic convex programming. Furthermore, it is shown that under several different support sets, the resulting convex optimization problems can be converted into computationally tractable conic programmings. Besides, the second model is generally NP-hard since checking constraint feasibility can be reduced to a norm maximization problem over a polytope. However, even with the case of uncertainty in constraints, tractable conic reformulations can be established when the extreme points of the polytope are known. Finally, we present numerical results to discuss how to control the risk for the best decisions and illustrate the computational effectiveness and superiority of the proposed models.
two-stagestochasticlinear program with recourse is represented as a convex programming problem. The problem is often large-scale because involves an expectation. Moreover objective function is not necessary differen...
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two-stagestochasticlinear program with recourse is represented as a convex programming problem. The problem is often large-scale because involves an expectation. Moreover objective function is not necessary differentiable. In order to reduce difficulties in solving, the inner problem is approached by quadratic problem which are differentiable and are least two-norm solution problem of inner recourse linear problem. This paper proposes newton method to obtain least two-norm solution of recourse problems by using penalty problem of recourse problem.
stochasticprogramming is an optimization technique used in the presence of uncertainty and it typically leads to very large problem sizes. In this paper, a modified version of the L-shaped method was used to solve tw...
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stochasticprogramming is an optimization technique used in the presence of uncertainty and it typically leads to very large problem sizes. In this paper, a modified version of the L-shaped method was used to solve two-stagestochasticlinear programs with recourse, based on the projection method and the augmented Lagrangian method. Using this modified version of the L-shaped method allows us to reduce the number of iterations and the time of solving a two-stagestochasticlinear program with fixed recourse, in comparison with traditional methods. (C) 2015 Elsevier Inc. All rights reserved.
Mean-risk stochasticlinearprogramming provides a framework for controlling cost variability in problems involving sequential decision making under uncertainty. It goes beyond the classical expected value framework b...
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Mean-risk stochasticlinearprogramming provides a framework for controlling cost variability in problems involving sequential decision making under uncertainty. It goes beyond the classical expected value framework by including risk measures in the objective function and aims at controlling cost variability in the solution. This allows for modeling risk averseness in variety of applications such as long-term financial planning, scheduling of power systems, supply chain management and portfolio optimization. In this dissertation, we derive stochastic decomposition algorithms for solving mean-risk two-stagestochasticlinear programs (MR-SLP) and mean-risk multistagestochasticlinear programs (MR-MSLP) with deviation and quantile risk measures. stochastic decomposition(SD) is a type of internal sampling method and at every iteration of algorithm only one linear problem is solved for approximating the recourse function. A salient feature of the SD algorithm is that the number of samples is not fixed a priori, which allows to obtain good candidate solutions early in the procedure. We also report on a computational study to evaluate the empirical performance of the SD algorithms for MR-SLP and MR-MSLP with expected excess (EE), quantile deviation (QDEV) and conditional value-at-risk (CVaR) as risk measures. The goal of the study was to analyze for a given instance how SD algorithm performs across different levels of risk, investigate the effect of different risk measures and understand when it is appropriate to use the risk-averse approach. For MR-SLP, the SD algorithm is implemented and applied to standard test instances and it shows that the risk measure QDEV has more impact on expected cost and the cost associated with extreme scenarios compared to the impact of CVaR and EE. We also observed that for higher target values, the risk measure EE becomes effective only for a relatively small number of scenarios and has little to no-effect on the optimal solution for smal
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