Informs Annual Meeting 2017

TB81

INFORMS Houston – 2017

TB81

We present a nonlinear stochastic programming formulation for the contingency- constrained optimal power flow problem. Using a non-linear AC power flow model, we construct a nonlinear, multi-scenario optimization formulation where each scenario considers nominal operation followed by a failure an individual transmission element. Given the number of potential failures in the network, these problems are large, yet must be solved rapidly. We demonstrate that this multi-scenario problem can be solved quickly using a parallel decomposition approach based on progressive hedging and nonlinear interior-point methods. Parallel and serial timing results are shown using Matpower test cases. 2 - Evaluation of Decision Systems in the Presence of Uncertainty David L. Woodruff, University of California-Davis, Graduate School of Management, One Shields Avenue, Davis, CA, 95616, United States, dlwoodruff@ucdavis.edu When inputs are uncertain, the performance of a decision system requires consideration both of the way stochastic processes are modelled as well as the optimization models and methods. In this talk we will describe ongoing work to provide software to support this sort of validation. 3 - Internal Decomposition Approaches for Nonlinear Programming Problems in Power Systems Carl Laird, R&D Software and Engineering, Sandia National Laboratories, Albuquerque, NM, United States, cdlaird@sandia.gov Stocahstic programming is an important tool for advanced design and operation of power transmission systems. To ensure optimal solutions are feasible for the real system, there is a need to include rigorous nonlinear models of AC transmission within the multi-scenario problem. In this presentation, we discuss key applications of nonlinear stochastic programming in power systems and compare different decomposition strategies for these large-scale nonlinear optimization problems. 4 - A Framework for Modeling and Optimizing Dynamic Systems under Uncertainty Bethany Nicholson, Sandia National Laboratories, 11600 Academy Rd NE, Apt 3924, Albuquerque, NM, 87111, United States, blnicho@sandia.gov, John Siirola There are many classes of optimization problems that are not easily represented in most Algebraic modeling languages (AMLs). These problems are typically reformulated before implementation, requiring significant effort from the modeler and obscuring the original problem structure or context. In this talk we show how the Pyomo AML can be used to represent complex models using high-level constructs. We describe the model transformations available in Pyomo for transforming high-level constructs to solvable forms, we demonstrate the combination of Pyomo extensions for stochastic programming and dynamic optimization, and we show the scalability of this framework on large-scale models. Chair: Jagdish Ramakrishnan, Walmart Labs, 1000 National Ave, Unit 214, San Bruno, CA, 94066, United States, jagdish.ram@gmail.com 1 - Stochastic Variance Reduction Method with Low Memory and Operation Cost for Smooth Finite Sum Optimization Jourdain Lamperski, Massachusetts Institute of Technology, 540 Memorial Drive, Cambridge, MA, 02139, United States, jourdain@mit.edu We propose a stochastic variance reduction method for the minimization of a smooth finite sum of convex functions. Current stochastic variance reduction methods enjoy a faster convergence rate than “vanilla” methods like stochastic gradient descent, but incur either a higher operation or memory cost. We show in the context of logistic regression that our method preserves the operation and memory complexity of stochastic gradient descent while enjoying the faster sublinear rate of convergence. Various aspects of the convergence analysis generalize to the setting of smooth finite sum optimization. We present several numerical results that support our theoretical analysis. 2 - Topology Optimization using Conditional Value at Risk Geoffrey Oxberry, Member of the Technical Staff, Lawrence Traditional approaches to formulating stochastic topology optimization problems for additive manufacturing tend to use linear combinations of the mean and standard deviation of a design figure of merit (e.g., compliance). However, this choice of objective/constraint underweights tail events for non-normal random variables. We propose replacing these mean-plus-standard-deviation expressions with the conditional value-at-risk (CVaR), and argue that it better compensates for worst-case tail events. Livermore National Laboratory, 2934 Worthing Common, Livermore, CA, 94550, United States, oxberry1@llnl.gov TB83 382C Optimization, Robust Contributed Session

382A Distributionally Robust Optimization: Theory and Applications Sponsored: Optimization, Optimization Under Uncertainty Sponsored Session Chair: Ruiwei Jiang, University of Michigan, University of Michigan, Ann Arbor, MI, 48109, United States, ruiwei@umich.edu 1 - Distributionally Robust Optimization with Principal Component Analysis Jianqiang Cheng, University of Arizona, jqcheng@email.arizona.edu, Richard Li-Yang Chen, Habib Najm, Ali Pinar, Cosmin Safta, Jean-Paul Watson In this talk, we propose a new approximation method to solve distributionally robust optimization problems with moment-based ambiguity sets. Our approximation method relies on principal component analysis (PCA) for optimal lower dimensional representation of variability in random samples. We show that the PCA approximation yields a relaxation of the original problem and derive theoretical bounds on the gap between the original problem and its PCA approximation. Furthermore, an extensive numerical study shows the strength of the proposed approximation method in terms of solution quality and runtime. 2 - Optimized Bonferroni Approximations of Distributionally Robust Joint Chance Constraints Weijun Xie, Georgia Institute of Technology, 4301 Lakeshore Xing NE, Atlanta, GA, 30324, United States, wxie33@gatech.edu, Shabbir Ahmed, Ruiwei Jiang In this paper, we study a conservative approximation - referred to as a Bonferroni approximation - of a joint chance constraint, i.e. a chance constraint involving a system of multiple uncertain constraints. We show that such a Bonferroni approximation is exact when the uncertainties are separable across the individual constraints, i.e., each uncertain constraint involves a different set of uncertain parameters and corresponding distribution families. We show that, while in general the optimization over the Bonferroni approximation is NP-hard, there are various sufficient conditions under which it is convex and tractable. 3 - Distributionally Robust Contingency-Constrained Unit Commitment Chaoyue Zhao, Oklahoma State University, 4599 North Washington Street APT.32J, Stillwater, OK, 74075, United States, cherryzhao09@gmail.com In this work, we propose a distributionally robust optimization approach for the contingency-constrained UC problem. We consider a case where the true probability distribution of contingencies is difficult to accurately estimate. Instead of assigning a probability estimate for each contingency scenario, we consider a set of contingency probability distributions based on the N - k security criterion and moment information. Our approach considers all possible distributions in the ambiguity set, and is hence distributionally robust. Meanwhile, as this approach utilizes moment information, it can benefit from available data and become less conservative than robust optimization approaches. 4 - Ambiguous Risk Constraints with Moment and Unimodality Information Ruiwei Jiang, University of Michigan, 1205 Beal Ave., Ann Arbor, MI, 48109, United States, ruiwei@umich.edu, Bowen Li, Johanna Mathieu We study risk constraints based on probabilistic guarantee and conditional value- at-risk under distributional ambiguity. We find that these risk constraints can be recast as a set of second-order conic constraints if we characterize the ambiguity based on moment and unimodality information. We demonstrate the theoretical results via a computational case study on power system operations. 382B Parallelization and Evaluation of Stochastic Programming Algorithms Sponsored: Optimization, Optimization Under Uncertainty Sponsored Session Chair: Jean-Paul Watson, Sandia National Laboratories, Albuquerque, NM, 87113-2065, United States, jwatson@sandia.gov 1 - Parallel Decomposition and Analysis of Contingency-Constrained AC Optimal Power Flow Problems Jean-Paul Watson, Distinguished Member of Technical Staff, Sandia National Laboratories, 7305 Blue Cypress Avenue NE, TB82

Albuquerque, NM, 87113-2065, United States, jwatson@sandia.gov, Carl Laird, Anya Castillo

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