Computational Study of Mean-Risk Stochastic Programs

Mean-risk stochastic programs model uncertainty by including risk measures in the objective function. This allows for modeling risk averseness for many problems in science and engineering. This dissertation addresses gaps in the literature on stochastic programs with mean-risk objectives. This includes a need for a computational study of the few available algorithms for this class of problems. The study was aimed at implementing and performing an empirical investigation of decomposition algorithms for stochastic linear programs with absolute semideviation (ASD) and quantile deviation (QDEV) as mean-risk measures. Specifically, the goals of the study were to analyze for specific instances how algorithms perform across different levels of risk, investigate the effect of using ASD and QDEV as risk measures, and understand when it is appropriate to use the risk-averse approach over the risk-neutral one. We derive two new subgradient based algorithms for the ASD and QDEV models, respectively. These algorithms are based on decomposing the stochastic program stage-wise and using a single (aggregated) cut in the master program to approximate the mean and deviation terms of the mean-risk objective function. We also consider a variant of each of the algorithms from the literature in which the mean-risk objective function is approximated by separate optimality cuts, one for the mean and one for the deviation term. These algorithms are implemented and applied to standard stochastic programming test instances to study their comparative performance. Both the aggregated cut and separate cut algorithms have comparable computational performance for ASD, while the separate cut algorithm outperforms its aggregate counterpart for QDEV. The computational study also reveals several insights on mean-risk stochastic linear programs. For example, the results show that for most standard test instances the risk-neutral approach is still appropriate. We show that this is the case due to the test instances having random variables with uniform marginal distributions. In contrast, when these distributions are changed to be non-uniform, the risk-averse approach is preferred. The results also show that the QDEV mean-risk measure has broader flexibility than ASD in modeling risk

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Risk Minimization, Regret Minimization and Progressive Hedging Algorithms

This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. A relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. Such a relationship opens a door to a decomposition scheme, called progressive hedging, for solving multistage risk minimization and regret minimization problems. In particular, the classical progressive hedging algorithm is modified in order to handle a new class of linkage constraints that arises from reformulations and other applications of risk and regret minimization problems. Numerical results are provided to show the efficiency of the progressive hedging algorithms.Comment: 21 pages, 2 figure

A mixed integer linear programming model for optimal sovereign debt issuance

Copyright @ 2011, Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in the European Journal of Operational Research. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version is available at the link below.Governments borrow funds to finance the excess of cash payments or interest payments over receipts, usually by issuing fixed income debt and index-linked debt. The goal of this work is to propose a stochastic optimization-based approach to determine the composition of the portfolio issued over a series of government auctions for the fixed income debt, to minimize the cost of servicing debt while controlling risk and maintaining market liquidity. We show that this debt issuance problem can be modeled as a mixed integer linear programming problem with a receding horizon. The stochastic model for the interest rates is calibrated using a Kalman filter and the future interest rates are represented using a recombining trinomial lattice for the purpose of scenario-based optimization. The use of a latent factor interest rate model and a recombining lattice provides us with a realistic, yet very tractable scenario generator and allows us to do a multi-stage stochastic optimization involving integer variables on an ordinary desktop in a matter of seconds. This, in turn, facilitates frequent re-calibration of the interest rate model and re-optimization of the issuance throughout the budgetary year allows us to respond to the changes in the interest rate environment. We successfully demonstrate the utility of our approach by out-of-sample back-testing on the UK debt issuance data

Incorporating statistical model error into the calculation of acceptability prices of contingent claims

The determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics. Given this model, optimal bid and ask prices can be found by stochastic optimization. However, the model for the underlying asset price process is typically based on data and found by a statistical estimation procedure. We define a confidence set of possible estimated models by a nonparametric neighborhood of a baseline model. This neighborhood serves as ambiguity set for a multi-stage stochastic optimization problem under model uncertainty. We obtain distributionally robust solutions of the acceptability pricing problem and derive the dual problem formulation. Moreover, we prove a general large deviations result for the nested distance, which allows to relate the bid and ask prices under model ambiguity to the quality of the observed data.Comment: 27 pages, 2 figure

Software tools for stochastic programming: A Stochastic Programming Integrated Environment (SPInE)

SP models combine the paradigm of dynamic linear programming with modelling of random parameters, providing optimal decisions which hedge against future uncertainties. Advances in hardware as well as software techniques and solution methods have made SP a viable optimisation tool. We identify a growing need for modelling systems which support the creation and investigation of SP problems. Our SPInE system integrates a number of components which include a flexible modelling tool (based on stochastic extensions of the algebraic modelling languages AMPL and MPL), stochastic solvers, as well as special purpose scenario generators and database tools. We introduce an asset/liability management model and illustrate how SPInE can be used to create and process this model as a multistage SP application