Approximations of Semicontinuous Functions with Applications to Stochastic Optimization and Statistical Estimation

Upper semicontinuous (usc) functions arise in the analysis of maximization problems, distributionally robust optimization, and function identification, which includes many problems of nonparametric statistics. We establish that every usc function is the limit of a hypo-converging sequence of piecewise affine functions of the difference-of-max type and illustrate resulting algorithmic possibilities in the context of approximate solution of infinite-dimensional optimization problems. In an effort to quantify the ease with which classes of usc functions can be approximated by finite collections, we provide upper and lower bounds on covering numbers for bounded sets of usc functions under the Attouch-Wets distance. The result is applied in the context of stochastic optimization problems defined over spaces of usc functions. We establish confidence regions for optimal solutions based on sample average approximations and examine the accompanying rates of convergence. Examples from nonparametric statistics illustrate the results

Volumetric center method for stochastic convex programs using sampling

We develop an algorithm for solving the stochastic convex program (SCP) by combining Vaidya's volumetric center interior point method (VCM) for solving non-smooth convex programming problems with the Monte-Carlo sampling technique to compute a subgradient. A near-central cut variant of VCM is developed, and for this method an approach to perform bulk cut translation, and adding multiple cuts is given. We show that by using near-central VCM the SCP can be solved to a desirable accuracy with any given probability. For the two-stage SCP the solution time is independent of the number of scenarios

A Stochastic Benders Decomposition Scheme for Large-Scale Data-Driven Network Design

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a data-driven version of network design where operational costs are uncertain and estimated using historical data. This problem is notoriously computationally challenging, and instances with as few as fifty nodes cannot be solved to optimality by current decomposition techniques. Accordingly, we propose a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless generates deterministically valid cuts (as opposed to the probabilistically valid cuts frequently proposed in the stochastic optimization literature) via a dual averaging technique. We implement both single-cut and multi-cut variants of this Benders decomposition algorithm, as well as a k-cut variant that uses clustering of the historical scenarios. On instances with 100-200 nodes, our algorithm achieves 4-5% optimality gaps, compared with 13-16% for deterministic Benders schemes, and scales to instances with 700 nodes and 50 commodities within hours. Beyond network design, our strategy could be adapted to generic two-stage stochastic mixed-integer optimization problems where second-stage costs are estimated via a sample average

Assessing the quality of convex approximations for two-stage totally unimodular integer recourse models

We consider two types of convex approximations of two-stage totally unimodular integer recourse models. Although worst-case error bounds are available for these approximations, their actual performance has not yet been investigated, mainly because this requires solving the original recourse model. In this paper we assess the quality of the approximating solutions using Monte Carlo sampling, or more specifically, using the so-called multiple replications procedure. Based on numerical experiments for an integer newsvendor problem, a fleet allocation and routing problem, and a stochastic activity network investment problem, we conclude that the error bounds are reasonably sharp if the variability of the random parameters in the model is either small or large; otherwise, the actual error of using the convex approximations is much smaller than the error bounds suggest. Moreover, we conclude that the solutions obtained using the convex approximations are good only if the variability of the random parameters is medium to large. In case this variability is small, however, typically sampling methods perform best, even with modest sample sizes. In this sense, the convex approximations and sampling methods can be considered as complementary solution methods. Moreover, as required for our applications, we extend our approach to derive new error bounds dealing with deterministic second-stage side constraints and relatively complete recourse, and perfect dependencies in the right-hand side vector

New approaches to Risk Management and Scenario Approximation in Financial Optimization

The first part of the thesis addresses the problem of risk management in financial optimization modeling. Motivation for constructing a new concept of risk measurement is given through the history of development: utility theory, risk/return tradeoff, and coherent risk measures. The process of describing investor\u27s preferences is presented through the proposed collection of Rational Level Sets (RLS). Based on RLS, a new concept termed Rational Risk Measures (RRM) for nancial optimization models is defined. The advantages of RRM over coherent risk measures are discussed. Approximation of a given set of scenarios using tail information is addressed in the second part of the thesis. Motivation for the scenario approximation problem, as a way of reducing computation time and preserving solution accuracy, is given through examples of financial optimization and asset allocation models. Using the basic ideas of Conditional Value at Risk (CVaR), this thesis develops a new methodology for scenario approximation for stochastic portfolio optimization. First, the concepts termed Scenarios-at-Risk (SaR) and Scenarios-at-Gain (SaG) are proposed as for the purpose of partitioning the underlying multivariate domain for a xed investment portfolio and a fixed probability level of CVaR. Then, under a given set of CVaR values, a twostage method is developed for determining a smaller, and discrete, set of scenarios over which CVaR risk control is satisfied for all portfolios of interest. Convergence of the method is shown and numerical results are presented to validate the proposed technique

Modeling and Analysis of Scheduling Problems Containing Renewable Energy Decisions

With globally increasing energy demands, world citizens are facing one of society\u27s most critical issues: protecting the environment. To reduce the emission of greenhouse gases (GHG), which are by-products of conventional energy resources, people are reducing the consumption of oil, gas, and coal collectively. In the meanwhile, interest in renewable energy resources has grown in recent years. Renewable generators can be installed both on the power grid side and end-use customer side of power systems. Energy management in power systems with multiple microgrids containing renewable energy resources has been a focus of industry and researchers as of late. Further, on-site renewable energy provides great opportunities for manufacturing plants to reduce energy costs when faced with time-varying electricity prices. To efficiently utilize on-site renewable energy generation, production schedules and energy supply decisions need to be coordinated. As renewable energy resources like solar and wind energy typically fluctuate with weather variations, the inherent stochastic nature of renewable energy resources makes the decision making of utilizing renewable generation complex. In this dissertation, we study a power system with one main grid (arbiter) and multiple microgrids (agents). The microgrids (MGs) are equipped to control their local generation and demand in the presence of uncertain renewable generation and heterogeneous energy management settings. We propose an extension to the classical two-stage stochastic programming model to capture these interactions by modeling the arbiter\u27s problem as the first-stage master problem and the agent decision problems as second-stage subproblems. To tackle this problem formulation, we propose a sequential sampling-based optimization algorithm that does not require a priori knowledge of probability distribution functions or selection of samples for renewable generation. The subproblems capture the details of different energy management settings employed at the agent MGs to control heating, ventilation and air conditioning systems; home appliances; industrial production; plug-in electrical vehicles; and storage devices. Computational experiments conducted on the US western interconnect (WECC-240) data set illustrate that the proposed algorithm is scalable and our solutions are statistically verifiable. Our results also show that the proposed framework can be used as a systematic tool to gauge (a) the impact of energy management settings in efficiently utilizing renewable generation and (b) the role of flexible demands in reducing system costs. Next, we present a two-stage, multi-objective stochastic program for flow shops with sequence-dependent setups in order to meet production schedules while managing energy costs. The first stage provides optimal schedules to minimize the total completion time, while the second stage makes energy supply decisions to minimize energy costs under a time-of-use electricity pricing scheme. Power demand for production is met by on-site renewable generation, supply from the main grid, and an energy storage system. An ε-constraint algorithm integrated with an L-shaped method is proposed to analyze the problem. Sets of Pareto optimal solutions are provided for decision-makers and our results show that the energy cost of setup operations is relatively high such that it cannot be ignored. Further, using solar or wind energy can save significant energy costs with solar energy being the more viable option of the two for reducing costs. Finally, we extend the flow shop scheduling problem to a job shop environment under hour-ahead real-time electricity pricing schemes. The objectives of interest are to minimize total weighted completion time and energy costs simultaneously. Besides renewable generation, hour-ahead real-time electricity pricing is another source of uncertainty in this study as electricity prices are released to customers only hours in advance of consumption. A mathematical model is presented and an ε-constraint algorithm is used to tackle the bi-objective problem. Further, to improve computational efficiency and generate solutions in a practically acceptable amount of time, a hybrid multi-objective evolutionary algorithm based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) is developed. Five methods are developed to calculate chromosome fitness values. Computational tests show that both mathematical modeling and our proposed algorithm are comparable, while our algorithm produces solutions much quicker. Using a single method (rather than five) to generate schedules can further reduce computational time without significantly degrading solution quality