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Asymptotic Consistency for Nonconvex Risk-Averse stochastic Optimization with Infinite Dimensional Decision Spaces

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arXiv 2022年

作者： Milz, Johannes Surowiec, Thomas M. H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology AtlantaGA30332 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the *** Codes 90C15, 90C06, 62F12, 35Q93, 49M41, 49J52 © 2022, CC BY-SA.

关键词： stochastic programming

The smoothed Monte Carlo method in robustness optimization

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OPTIMIZATION METHODS & SOFTWARE 2008年第5期23卷 717-729页

作者： Hendrix, Eligius M. T. Olieman, Niels J. Univ Wageningen & Res Ctr Operat Res Logistiek Groep Wageningen Netherlands

The concept of robustness as the probability mass of a design-dependent set has been introduced in the literature. Optimization of robustness can be seen as finding the design that has the highest robustness. The reference method for estimating the robustness is the Monte Carlo (MC) simulation, and the drawback for its direct use in nonlinear optimization is the lack of derivative information and the appearance of discontinuities. An alternative for MC is presented, called the smoothed Monte Carlo estimation. It is proved that the resulting estimate function is made continuous in relevant design points and facilitates the use of standard nonlinear optimization algorithms. The whole procedure is illustrated numerically.

关键词： Monte Carlo robustness non-smooth optimization stochastic programming

A Comparison of two Hydro Scheduling Algorithms, SDDP and LpSim

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A Comparison of two Hydro Scheduling Algorithms, SDDP and Lp...

International Universities Power Engineering Conference

作者： Gunnar Geir Pétursson Kristján Jónasson Egill Benedikt Hreinsson úlfar Linnet University of Iceland Landsvirkjun Iceland

ISBN: (纸本)9781479932559

In this study we present the results of a comparison between the two methods implemented in two hydro scheduling algorithms, SDDP and LpSim. Both methods are carried out for a range of reservoir inflow series and variable load. The yardstick we use for the comparison of the methods is the amount of load each respective program can satisfy while fulfilling restrictions to maximum load shedding and minimum reservoir volumes. Iceland's isolated power system was selected for this case study. Its electricity production is pre-dominantly hydro (74%), the rest being geothermal (26%) with intermittent thermal generation options (~0.02% in production, 2.4% in installed capacity). The system is not connected with other countries. In January 2013, wind power officially became the third renewable source for electricity in Iceland. In the near future the possibility of a submarine power cable connecting Iceland to mainland Europe may further increase the dynamics of Iceland's currently isolated power system. If constructed, future power stations in the lower Thjórsá river basin will add more levels to the cascade of hydro plants there, decreasing the accuracy of hydro scheduling algorithms that rely on aggregate reservoir approximations. It is therefore imperative to investigate the performance of the currently used medium- and long-term hydro scheduling algorithm.

关键词： Comparison Deterministic programming hydro scheduling SDDP stochastic programming water values

Models and Algorithm for stochastic Minimum Weight Dominating Set Problem

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Models and Algorithm for Stochastic Minimum Weight Dominatin...

第六届中国智能计算大会

作者： Yue Zhou Yaodong Ni School of Information Technology and Management University of International Business and Economics Beijing 100029China

The minimum weight dominating set problem is a classical optimization problem in graph theory,which has many real *** real life,there always exist various kinds of uncertainty,thus it is necessary to take uncertainties into account when studying dominating *** this paper,the stochastic minimum weight dominating set problem is *** propose the concepts of α-minimum weight dominating set and the most minimum weight dominating *** to the two decision criteria,different types of decision model are *** produce a hybrid intelligent algorithm integrating stochastic simulation with genetic algorithm to solve the proposed ***,numerical experiments are given to illustrate the effectiveness of the algorithm.

关键词： dominating set stochastic programming stochastic simulation genetic algorithm

Optimizing service restoration in distribution systems with uncertain repair time and demand

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arXiv 2018年

作者： Arif, Anmar Ma, Shanshan Wang, Zhaoyu Wang, Jianhui Ryan, Sarah M. Chen, Chen Department of Electrical and Computer Engineering Iowa State University AmesIA50011 United States Department of Electrical Engineering King Saud University Riyadh11451 Saudi Arabia Department of Industrial and Manufacturing Systems Engineering Iowa State University AmesIA50010 United States Energy Systems Division Argonne National Laboratory LemontIL60439 United States

This paper proposes a novel method to co-optimize distribution system operation and repair crew routing for outage restoration after extreme weather events. A two-stage stochastic mixed integer linear program is developed. The first stage is to dispatch the repair crews to the damaged components. The second stage is distribution system restoration using distributed generators, and reconfiguration. We consider demand uncertainty in terms of a truncated normal forecast error distribution, and model the uncertainty of the repair time using a lognormal distribution. A new decomposition approach, combined with the Progressive Hedging algorithm, is developed for solving largescale outage management problems in an effective and timely manner. The proposed method is validated on modified IEEE 34- and 8500-bus distribution test systems. Copyright © 2018, The Authors. All rights reserved.

关键词： stochastic programming

Residuals-based distributionally robust optimization with covariate information

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arXiv 2020年

作者： Kannan, Rohit Bayraksan, Güzin Luedtke, James R. Los Alamos National Laboratory Los AlamosNM United States Department of Integrated Systems Engineering The Ohio State University ColumbusOH United States Department of Industrial & Systems Engineering Wisconsin Institute for Discovery University of Wisconsin-Madison MadisonWI United States

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is *** Codes 90C15, 90C47 Copyright © 2020, The Authors. All rights reserved.

关键词： stochastic programming

On the Viability of stochastic Economic Dispatch for Real-Time Energy Market Clearing

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arXiv 2023年

作者： Zhao, Haoruo Tanneau, Mathieu Van Hentenryck, Pascal H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology 755 Ferst Dr NW AtlantaGA30318 United States

Over the past decade, the rapid adoption of intermittent renewable energy sources (RES), especially wind and solar generation, has posed challenges in managing real-time uncertainty and variability. In the U.S., Independent System Operators (ISOs) solve a security-constrained economic dispatch (SCED) every five minutes to clear real-time electricity markets, co-optimizing energy dispatch and reserve to minimize costs while meeting physical and reliability constraints. All SCED formulations in the U.S. are deterministic and mostly consider a single time period, limiting their effectiveness in managing real-time operational uncertainty from RES intermittency. This limitation is highlighted by the recent introduction of multiple short-term ramping products in U.S. markets, aiming to bridge the gap between deterministic and stochastic SCED formulations. While stochastic formulations address uncertainty in a unified and endogenous manner, their adoption has been hindered by high computational costs and, to a lesser extent, the availability of probabilistic forecasts. This paper revisits these concerns and demonstrates that stochastic economic dispatch is now a viable technology for real-time market clearing. It introduces the stochastic look-ahead dispatch (SLAD) formulation for real-time market clearing and presents an accelerated Benders’ decomposition to solve it efficiently. Extensive experiments on a real, industry-sized transmission grid demonstrate the computational scalability of the proposed approach, with SLAD instances being solved in under 5 minutes. Furthermore, results show that SLAD provides more than 50% additional savings compared to flexiramp products and is more robust to the forecasting methodology. Therefore, SLAD is a promising approach for uncertainty management in real-time electricity markets. Copyright © 2023, The Authors. All rights reserved.

关键词： stochastic programming

Stability and Sample-based Approximations of Composite stochastic Optimization Problems

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arXiv 2022年

作者： Dentcheva, Darinka Lin, Yang Penev, Spiridon Department of Mathematical Sciences Stevens Institute of Technology HobokenNJ03070 United States School of Mathematics and Statistics UNSW Data Science Hub UNSW Sydney SydneyNSW2052 Australia

Optimization under uncertainty and risk is indispensable in many practical situations. Our paper addresses stability of optimization problems using composite risk functionals which are subjected to measure perturbations. Our main focus is the asymptotic behavior of data-driven formulations with empirical or smoothing estimators such as kernels or wavelets applied to some or to all functions of the compositions. We analyze the properties of the new estimators and we establish strong law of large numbers, consistency, and bias reduction potential under fairly general assumptions. Our results are germane to risk-averse optimization and to data science in general. © 2022, CC BY.

关键词： stochastic programming

Offline Learning of Decision Functions in Multiplayer Games with Expectation Constraints

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arXiv 2024年

作者： Huang, Yuanhanqing Hu, Jianghai Elmore Family School of Electrical and Computer Engineering Purdue University West LafayetteIN47907 United States

We explore a class of stochastic multiplayer games where each player in the game aims to optimize its objective under uncertainty and adheres to some expectation constraints. The study employs an offline learning paradigm, leveraging a pre-existing dataset containing auxiliary features. While prior research in deterministic and stochastic multiplayer games primarily explored vector-valued decisions, this work departs by considering function-valued decisions that incorporate auxiliary features as input. We leverage the law of large deviations and degree theory to establish the almost sure convergence of the offline learning solution to the true solution as the number of data samples increases. Finally, we demonstrate the validity of our method via a multi-account portfolio optimization problem. Copyright © 2024, The Authors. All rights reserved.

关键词： stochastic programming