Optimized Dimensionality Reduction for Moment-based Distributionally Robust Optimization

Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems can be reformulated as semidefinite programming (SDP) problems that can be solved in polynomial time, solving high-dimensional SDPs is still time-consuming. Unlike existing approximation approaches that first reduce the dimensionality of random parameters and then solve the approximated SDPs, we propose an optimized dimensionality reduction (ODR) approach. We first show that the ranks of the matrices in the SDP reformulations are small, by which we are then motivated to integrate the dimensionality reduction of random parameters with the subsequent optimization problems. Such integration enables two outer and one inner approximations of the original problem, all of which are low-dimensional SDPs that can be solved efficiently. More importantly, these approximations can theoretically achieve the optimal value of the original high-dimensional SDPs. As these approximations are nonconvex SDPs, we develop modified Alternating Direction Method of Multipliers (ADMM) algorithms to solve them efficiently. We demonstrate the effectiveness of our proposed ODR approach and algorithm in solving two practical problems. Numerical results show significant advantages of our approach on the computational time and solution quality over the three best possible benchmark approaches. Our approach can obtain an optimal or near-optimal (mostly within 0.1%) solution and reduce the computational time by up to three orders of magnitude

Distributionally Robust Optimal Power Flow with Strengthened Ambiguity Sets

Uncertainties that result from renewable generation and load consumption can complicate the optimal power flow problem. These uncertainties normally influence the physical constraints stochastically and require special methodologies to solve. Hence, a variety of stochastic optimal power flow formulations using chance constraints have been proposed to reduce the risk of physical constraint violations and ensure a reliable dispatch solution under uncertainty. The true uncertainty distribution is required to exactly reformulate the problem, but it is generally difficult to obtain. Conventional approaches include randomized techniques (such as scenario-based methods) that provide a priori guarantees of the probability of constraint violations but generally require many scenarios and produce high-cost solutions. Another approach is to use an analytical reformulation, which assumes that the uncertainties follow specific distributions such as Gaussian distributions. However, if the actual uncertainty distributions do not follow the assumed distributions, the results often suffer from case-dependent reliability. Recently, researchers have also explored distributionally robust optimization, which requires probabilistic constraints to be satisfied at chosen probability levels for any uncertainty distributions within a pre-defined ambiguity set. The set is constructed based on the statistical information that is extracted from historical data. Existing literature applying distributionally robust optimization to the optimal power flow problem indicates that the approach has promising performance with low objective costs as well as high reliability compared with the randomized techniques and analytical reformulation. In this dissertation, we aim to analyze the conventional approaches and further improve the current distributionally robust methods. In Chapter II, we derive the analytical reformulation of a multi-period optimal power flow problem with uncertain renewable generation and load-based reserve. It is assumed that the capacities of the load-based reserves are affected by outdoor temperatures through non-linear relationships. Case studies compare the analytical reformulation with the scenario-based method and demonstrate that the scenario-based method generates overly-conservative results and the analytical reformulation results in lower cost solutions but it suffers from reliability issues. In Chapters III, IV, and V, we develop new methodologies in distributionally robust optimization by strengthening the moment-based ambiguity set by including a combination of the moment, support, and structural property information. Specifically, we consider unimodality and log-concavity as most practical uncertainties exhibit these properties. The strengthened ambiguity sets are used to develop tractable reformulations, approximations, and efficient algorithms for the optimal power flow problem. Case studies indicate that these strengthened ambiguity sets reduce the conservativeness of the solutions and result in sufficiently reliable solutions. In Chapter VI, we compare the performance of the conventional approaches and distributionally robust approaches including moment and unimodality information on large-scale systems with high uncertainty dimensions. Through case studies, we evaluate each approach's performance by exploring its objective cost, computational scalability, and reliability. Simulation results suggest that distributionally robust optimal power flow including unimodality information produces solutions with better trade-offs between objective cost and reliability as compared to the conventional approaches or the distributionally robust approaches that do not include unimodality assumptions. However, considering unimodality also leads to longer computational times.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/150051/1/libowen_1.pd

Convex Nonlinear and Integer Programming Approaches for Distributionally Robust Optimization of Complex Systems

The primary focus of the dissertation is to develop distributionally robust optimization (DRO) models and related solution approaches for decision making in energy and healthcare service systems with uncertainties, which often involves nonlinear constraints and discrete decision variables. Without assuming specific distributions, DRO techniques solve for solutions against the worst-case distribution of system uncertainties. In the DRO framework, we consider both risk-neutral (e.g., expectation) and risk-averse (e.g., chance constraint and Conditional Value-at-Risk (CVaR)) measures. The aim is twofold: i) developing efficient solution algorithms for DRO models with integer and/or binary variables, sometimes nonlinear structures and ii) revealing managerial insights of DRO models for specific applications. We mainly focus on DRO models of power system operations, appointment scheduling, and resource allocation in healthcare. Specifically, we first study stochastic optimal power flow (OPF), where (uncertain) renewable integration and load control are implemented to balance supply and (uncertain) demand in power grids. We propose a chance-constrained OPF (CC-OPF) model and investigate its DRO variant which is reformulated as a semidefinite programming (SDP) problem. We compare the DRO model with two benchmark models, in the IEEE 9-bus, 39-bus, and 118-bus systems with different flow congestion levels. The DRO approach yields a higher probability of satisfying the chance constraints and shorter solution time. It also better utilizes reserves at both generators and loads when the system has congested flows. Then we consider appointment scheduling under random service durations with given (fixed) appointment arrival order. We propose a DRO formulation and derive a conservative SDP reformulation. Furthermore, we study a scheduling variant under random no-shows of appointments and derive tractable reformulations for certain beliefs of no-show patterns. One preceding problem of appointment scheduling in the healthcare service operations is the surgery block allocation problem that assigns surgeries to operating rooms. We derive an equivalent 0-1 SDP reformulation and a less conservative 0-1 second-order cone programming (SOCP) reformulation for its DRO model. Finally, we study distributionally robust chance-constrained binary programs (DCBP) for limiting the probability of undesirable events, under mean-covariance information. We reformulate DCBPs as equivalent 0-1 SOCP formulations under two moment-based ambiguity sets. We further exploit the submodularity of the 0-1 SOCP reformulations under diagonal and non-diagonal matrices. We derive extended polymatroid inequalities via submodularity and lifting, which are incorporated into a branch-and-cut algorithm incorporated for efficiently solving DCBPs. We demonstrate the computational efficacy and solution performance with diverse instances of a chance-constrained bin packing problem.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149946/1/zyiling_1.pd

Stochastic Methods in Optimization and Machine Learning

Stochastic methods are indispensable to the modeling, analysis and design of complex systems involving randomness. In this thesis, we show how simulation techniques and simulation-based computational methods can be applied to a wide spectrum of applied domains including engineering, optimization and machine learning. Moreover, we show how analytical tools in statistics and computer science including empirical processes, probably approximately correct learning, and hypothesis testing can be used in these contexts to provide new theoretical results. In particular, we apply these techniques and present how our results can create new methodologies or improve upon existing state-of-the-art in three areas: decision making under uncertainty (chance-constrained programming, stochastic programming), machine learning (covariate shift, reinforcement learning) and estimation problems arising from optimization (gradient estimate of composite functions) or stochastic systems (solution of stochastic PDE). The work in the above three areas will be organized into six chapters, where each area contains two chapters. In Chapter 2, we study how to obtain feasible solutions for chance-constrained programming using data-driven, sampling-based scenario optimization (SO) approach. When the data size is insufficient to statistically support a desired level of feasibility guarantee, we explore how to leverage parametric information, distributionally robust optimization and Monte Carlo simulation to obtain a feasible solution of chance-constrained programming in small-sample situations. In Chapter 3, We investigate the feasibility of sample average approximation (SAA) for general stochastic optimization problems, including two-stage stochastic programming without the relatively complete recourse. We utilize results from the Vapnik-Chervonenkis (VC) dimension and Probably Approximately Correct learning to provide a general framework. In Chapter 4, we design a robust importance re-weighting method for estimation/learning problem in the covariate shift setting that improves the best-know rate. In Chapter 5, we develop a model-free reinforcement learning approach to solve constrained Markov decision processes (MDP). We propose a two-stage procedure that generates policies with simultaneous guarantees on near-optimality and feasibility. In Chapter 6, we use multilevel Monte Carlo to construct unbiased estimators for expectations of random parabolic PDE. We obtain estimators with finite variance and finite expected computational cost, but bypassing the curse of dimensionality. In Chapter 7, we introduce unbiased gradient simulation algorithms for solving stochastic composition optimization (SCO) problems. We show that the unbiased gradients generated by our algorithms have finite variance and finite expected computational cost

Distributionally Robust Stochastic Knapsack Problem

International audienceThis paper considers a distributionally robust version of a quadratic knapsack prob-lem. In this model, a subsets of items is selected to maximizes the total profit while requiring thata set of knapsack constraints be satisfied with high probability. In contrast to the stochastic pro-gramming version of this problem, we assume that only part of the information on random datais known, i.e., the first and second moment of the random variables, their joint support, and pos-sibly an independence assumption. As for the binary constraints, special interest is given to thecorresponding semidefinite programming (SDP) relaxation. While in the case that the model onlyhas a single knapsack constraint we present an SDP reformulation for this relaxation, the case ofmultiple knapsack constraints is more challenging. Instead, two tractable methods are presented forproviding upper and lower bounds (with its associated conservative solution) on the SDP relaxation.An extensive computational study is given to illustrate the tightness of these bounds and the valueof the proposed distributionally robust approach

Distributionally Robust Stochastic Knapsack Problem

International audienc