scholarly journals Distributionally Robust Model of Energy and Reserve Dispatch Based on Kullback–Leibler Divergence

Electronics ◽  
2019 ◽  
Vol 8 (12) ◽  
pp. 1454 ◽  
Author(s):  
Ce Yang ◽  
Dong Han ◽  
Weiqing Sun ◽  
Kunpeng Tian
Keyword(s):  
Robust Optimization ◽  
Benders Decomposition ◽  
Mixed Integer ◽  
Worst Case ◽  
Total Cost ◽  
Kl Divergence ◽  
Leibler Divergence ◽  
Expected Total Cost ◽  
Distributionally Robust ◽  
Bus System

This paper proposes a distance-based distributionally robust energy and reserve (DB-DRER) dispatch model via Kullback–Leibler (KL) divergence, considering the volatile of renewable energy generation. Firstly, a two-stage optimization model is formulated to minimize the expected total cost of energy and reserve (ER) dispatch. Then, KL divergence is adopted to establish the ambiguity set. Distinguished from conventional robust optimization methodology, the volatile output of renewable power generation is assumed to follow the unknown probability distribution that is restricted in the ambiguity set. DB-DRER aims at minimizing the expected total cost in the worst-case probability distributions of renewables. Combining with the designed empirical distribution function, the proposed DB-DRER model can be reformulated into a mixed integer nonlinear programming (MINLP) problem. Furthermore, using the generalized Benders decomposition, a decomposition method is proposed and sample average approximation (SAA) method is applied to solve this problem. Finally, simulation result of the proposed method is compared with those of stochastic optimization and conventional robust optimization methods on the 6-bus system and IEEE 118-bus system, which demonstrates the effectiveness and advantages of the method proposed.

scholarly journals Performance evaluation for distributionally robust optimization with binary entries

2020 ◽  
Vol 11 (1) ◽  
pp. 1-9
Author(s):  
Shunichi Ohmori ◽  
Kazuho Yoshimoto
Keyword(s):  
Performance Evaluation ◽  
Robust Optimization ◽  
Optimization Technique ◽  
Data Driven ◽  
Optimal Decision ◽  
Distributionally Robust Optimization ◽  
Worst Case ◽  
Leibler Divergence ◽  
Out Of Sample ◽  
Distributionally Robust

We consider the data-driven stochastic programming problem with binary entries where the probability of existence of each entry is not known, instead realization of data is provided. We applied the distributionally robust optimization technique to minimize the worst-case expected cost taken over the ambiguity set based on the Kullback-Leibler divergence. We investigate the out-of-sample performance of the resulting optimal decision and analyze its dependence on the sparsity of the problem.

Optimizing Disruption Tolerance for Rail Transit Networks Under Uncertainty

2021 ◽  
Author(s):  
Lei Xu ◽  
Tsan Sheng (Adam) Ng ◽  
Alberto Costa
Keyword(s):  
Robust Optimization ◽  
Optimization Model ◽  
Optimization Problem ◽  
Mixed Integer ◽  
Rail Transit ◽  
Distributionally Robust Optimization ◽  
Performance Function ◽  
Transit Networks ◽  
Disruption Tolerance ◽  
Distributionally Robust

In this paper, we develop a distributionally robust optimization model for the design of rail transit tactical planning strategies and disruption tolerance enhancement under downtime uncertainty. First, a novel performance function evaluating the rail transit disruption tolerance is proposed. Specifically, the performance function maximizes the worst-case expected downtime that can be tolerated by rail transit networks over a family of probability distributions of random disruption events given a threshold commuter outflow. This tolerance function is then applied to an optimization problem for the planning design of platform downtime protection and bus-bridging services given budget constraints. In particular, our implementation of platform downtime protection strategy relaxes standard assumptions of robust protection made in network fortification and interdiction literature. The resulting optimization problem can be regarded as a special variation of a two-stage distributionally robust optimization model. In order to achieve computational tractability, optimality conditions of the model are identified. This allows us to obtain a linear mixed-integer reformulation that can be solved efficiently by solvers like CPLEX. Finally, we show some insightful results based on the core part of Singapore Mass Rapid Transit Network.

scholarly journals The Distributionally Robust Optimization Reformulation for Stochastic Complementarity Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Liyan Xu ◽  
Bo Yu ◽  
Wei Liu
Keyword(s):  
Robust Optimization ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Linear Complementarity ◽  
Uncertain Parameters ◽  
Distributionally Robust Optimization ◽  
Worst Case ◽  
Nonnegativity Constraints ◽  
Distributionally Robust ◽  
Stochastic Linear Complementarity Problem

We investigate the stochastic linear complementarity problem affinely affected by the uncertain parameters. Assuming that we have only limited information about the uncertain parameters, such as the first two moments or the first two moments as well as the support of the distribution, we formulate the stochastic linear complementarity problem as a distributionally robust optimization reformation which minimizes the worst case of an expected complementarity measure with nonnegativity constraints and a distributionally robust joint chance constraint representing that the probability of the linear mapping being nonnegative is not less than a given probability level. Applying the cone dual theory and S-procedure, we show that the distributionally robust counterpart of the uncertain complementarity problem can be conservatively approximated by the optimization with bilinear matrix inequalities. Preliminary numerical results show that a solution of our method is desirable.

scholarly journals Distributionally Robust Counterfactual Risk Minimization

2020 ◽  
Vol 34 (04) ◽  
pp. 3850-3857
Author(s):  
Louis Faury ◽  
Ugo Tanielian ◽  
Elvis Dohmatob ◽  
Elena Smirnova ◽  
Flavian Vasile
Keyword(s):  
Decision Making ◽  
Robust Optimization ◽  
Model Uncertainty ◽  
State Of The Art ◽  
Risk Minimization ◽  
Practical Applications ◽  
Uncertainty Measures ◽  
Leibler Divergence ◽  
Benchmark Datasets ◽  
Distributionally Robust

This manuscript introduces the idea of using Distributionally Robust Optimization (DRO) for the Counterfactual Risk Minimization (CRM) problem. Tapping into a rich existing literature, we show that DRO is a principled tool for counterfactual decision making. We also show that well-established solutions to the CRM problem like sample variance penalization schemes are special instances of a more general DRO problem. In this unifying framework, a variety of distributionally robust counterfactual risk estimators can be constructed using various probability distances and divergences as uncertainty measures. We propose the use of Kullback-Leibler divergence as an alternative way to model uncertainty in CRM and derive a new robust counterfactual objective. In our experiments, we show that this approach outperforms the state-of-the-art on four benchmark datasets, validating the relevance of using other uncertainty measures in practical applications.

scholarly journals Robust Optimization Model for Bi-objective Emergency Medical Service Design Problem with Demand Uncertainty

2019 ◽  
Vol 20 (2) ◽  
pp. 95
Author(s):  
Diah Chaerani ◽  
Siti Rabiatul Adawiyah ◽  
Eman Lesmana
Keyword(s):  
Robust Optimization ◽  
Emergency Medical Service ◽  
Design Problem ◽  
Medical Service ◽  
Optimal Solution ◽  
Service Design ◽  
Mixed Integer ◽  
Worst Case ◽  
Robust Counterpart ◽  
Emergency Medical

Bi-objective Emergency Medical Service Design Problem is a problem to determining the location of the station Emergency Medical Service among all candidate station location, the determination of the number of emergency vehicles allocated to stations being built so as to serve medical demand. This problem is a multi-objective problem that has two objective functions that minimize cost and maximize service. In real case there is often uncertainty in the model such as the number of demand. To deal the uncertainty on the bi-objective emergency medical service problem is using Robust Optimization which gave optimal solution even in the worst case. Model Bi-objective Emergency Medical Service Design Problem is formulated using Mixed Integer Programming. In this research, Robust Optimization is formulated for Bi-objective Emergency Medical Service Design Problem through Robust Counterpart formulation by assuming uncertainty in demand is box uncertainty and ellipsoidal uncertainty set. We show that in the case of bi-objective optimization problem, the robust counterpart remains computationally tractable. The example is performed using Lexicographic Method and Branch and Bound Method to obtain optimal solution. 

The Value of Randomized Solutions in Mixed-Integer Distributionally Robust Optimization Problems

2021 ◽  
Author(s):  
Erick Delage ◽  
Ahmed Saif
Keyword(s):  
Robust Optimization ◽  
Facility Location ◽  
Column Generation ◽  
Location Problem ◽  
Optimization Problems ◽  
Facility Location Problem ◽  
Mixed Integer ◽  
Distributionally Robust Optimization ◽  
Continuous Relaxation ◽  
Distributionally Robust

Randomized decision making refers to the process of making decisions randomly according to the outcome of an independent randomization device, such as a dice roll or a coin flip. The concept is unconventional, and somehow counterintuitive, in the domain of mathematical programming, in which deterministic decisions are usually sought even when the problem parameters are uncertain. However, it has recently been shown that using a randomized, rather than a deterministic, strategy in nonconvex distributionally robust optimization (DRO) problems can lead to improvements in their objective values. It is still unknown, though, what is the magnitude of improvement that can be attained through randomization or how to numerically find the optimal randomized strategy. In this paper, we study the value of randomization in mixed-integer DRO problems and show that it is bounded by the improvement achievable through its continuous relaxation. Furthermore, we identify conditions under which the bound is tight. We then develop algorithmic procedures, based on column generation, for solving both single- and two-stage linear DRO problems with randomization that can be used with both moment-based and Wasserstein ambiguity sets. Finally, we apply the proposed algorithm to solve three classical discrete DRO problems: the assignment problem, the uncapacitated facility location problem, and the capacitated facility location problem and report numerical results that show the quality of our bounds, the computational efficiency of the proposed solution method, and the magnitude of performance improvement achieved by randomized decisions. Summary of Contribution: In this paper, we present both theoretical results and algorithmic tools to identify optimal randomized strategies for discrete distributionally robust optimization (DRO) problems and evaluate the performance improvements that can be achieved when using them rather than classical deterministic strategies. On the theory side, we provide improvement bounds based on continuous relaxation and identify the conditions under which these bound are tight. On the algorithmic side, we propose a finitely convergent, two-layer, column-generation algorithm that iterates between identifying feasible solutions and finding extreme realizations of the uncertain parameter. The proposed algorithm was implemented to solve distributionally robust stochastic versions of three classical optimization problems and extensive numerical results are reported. The paper extends a previous, purely theoretical work of the first author on the idea of randomized strategies in nonconvex DRO problems by providing useful bounds and algorithms to solve this kind of problems.

A Distributionally Robust Optimization Model for Unit Commitment Based on Kullback–Leibler Divergence

2018 ◽  
Vol 33 (5) ◽  
pp. 5147-5160 ◽  
Author(s):  
Yuwei Chen ◽  
Qinglai Guo ◽  
Hongbin Sun ◽  
Zhengshuo Li ◽  
Wenchuan Wu ◽  
...  
Keyword(s):  
Robust Optimization ◽  
Optimization Model ◽  
Unit Commitment ◽  
Distributionally Robust Optimization ◽  
Leibler Divergence ◽  
Distributionally Robust

scholarly journals Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

2021 ◽  
Author(s):  
Jose Blanchet ◽  
Karthyek Murthy ◽  
Fan Zhang
Keyword(s):  
Robust Optimization ◽  
Optimal Transport ◽  
Decision Rules ◽  
Stochastic Gradient Descent ◽  
Transport Cost ◽  
Distributionally Robust Optimization ◽  
Worst Case ◽  
Strongly Convex ◽  
Distributionally Robust ◽  
Convexity Assumptions

We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent.

scholarly journals Conic reformulations for Kullback-Leibler divergence constrained distributionally robust optimization and applications

2021 ◽  
Vol 11 (2) ◽  
pp. 139-151
Author(s):  
Burak Kocuk
Keyword(s):  
Robust Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Empirical Distribution ◽  
Computational Study ◽  
Facility Location Problem ◽  
Optimization Approach ◽  
Distributionally Robust Optimization ◽  
Leibler Divergence ◽  
Distributionally Robust

In this paper, we consider a Kullback-Leibler divergence constrained distributionally robust optimization model. This model considers an ambiguity set that consists of all distributions whose Kullback-Leibler divergence to an empirical distribution is bounded. Utilizing the fact that this divergence measure has an exponential cone representation, we obtain the robust counterpart of the Kullback-Leibler divergence constrained distributionally robust optimization problem as a dual exponential cone constrained program under mild assumptions on the underlying optimization problem. The resulting conic reformulation of the original optimization problem can be directly solved by a commercial conic programming solver. We specialize our generic formulation to two classical optimization problems, namely, the Newsvendor Problem and the Uncapacitated Facility Location Problem. Our computational study in an out-of-sample analysis shows that the solutions obtained via the distributionally robust optimization approach yield significantly better performance in terms of the dispersion of the cost realizations while the central tendency deteriorates only slightly compared to the solutions obtained by stochastic programming.

scholarly journals An Extreme Scenario Method for Robust Transmission Expansion Planning with Wind Power Uncertainty

Energies ◽  
2018 ◽  
Vol 11 (8) ◽  
pp. 2116 ◽  
Author(s):  
Zipeng Liang ◽  
Haoyong Chen ◽  
Xiaojuan Wang ◽  
Idris Ibn Idris ◽  
Bifei Tan ◽  
...  
Keyword(s):  
Wind Power ◽  
Transmission Lines ◽  
Benders Decomposition ◽  
Electrical Power ◽  
Mixed Integer ◽  
Two Stage ◽  
Operating Variables ◽  
Expansion Planning ◽  
Power Resources ◽  
Bus System

The rapid incorporation of wind power resources in electrical power networks has significantly increased the volatility of transmission systems due to the inherent uncertainty associated with wind power. This paper addresses this issue by proposing a transmission network expansion planning (TEP) model that integrates wind power resources, and that seeks to minimize the sum of investment costs and operation costs while accounting for the costs associated with the pollution emissions of generator infrastructure. Auxiliary relaxation variables are introduced to transform the established model into a mixed integer linear programming problem. Furthermore, the novel concept of extreme wind power scenarios is defined, theoretically justified, and then employed to establish a two-stage robust TEP method. The decision-making variables of prospective transmission lines are determined in the first stage, so as to ensure that the operating variables in the second stage can adapt to wind power fluctuations. A Benders’ decomposition algorithm is developed to solve the proposed two-stage model. Finally, extensive numerical studies are conducted with Garver’s 6-bus system, a modified IEEE RTS79 system and IEEE 118-bus system, and the computational results demonstrate the effectiveness and practicability of the proposed method.

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