Approximation in stochastic integer programming

Approximation algorithms are the prevalent solution methods in the field of stochastic programming. Problems in this field are very hard to solve. Indeed, most of the research in this field has concentrated on designing solution methods that approximate the optimal solutions. However, efficiency in the complexity theoretical sense is usually not taken into account. Quality statements mostly remain restricted to convergence to an optimal solution without accompanying implications on the running time of the algorithms for attaining more and more accurate solutions. However, over the last twenty years also some studies on performance analysis of approximation algorithms for stochastic programming have appeared. In this direction we find both probabilistic analysis and worst-case analysis. There have been studies on performance ratios and on absolute divergence from optimality. Only recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most combinatorial optimization problems.

Convex approximations for a class of mixed-integer recourse models

We consider mixed-integer recourse (MIR) models with a single recourse constraint.We relate the secondstage value function of such problems to the expected simple integer recourse (SIR) shortage function. This allows to construct convex approximations for MIR problems by the same approach used for SIR models.

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

Output analysis for approximated stochastic programs

Because of incomplete information and also for the sake of numerical tractability one mostly solves an approximated stochastic program instead of the underlying ''true'' decision problem. However, without an additional analysis, the obtained output (the optimal value and optimal solutions of the approximated stochastic program) should not be used to replace the sought solution of the ''true'' problem. Methods of output analysis have to be tailored to the structure of the problem and they should also reflect the source, character and precision of the input data. The scope of various approaches based on results of asymptotic and robust statistics, of the moment problem and on general results of parametric programming will be discussed from the point of view of their applicability and possible extensions

Supply chain network design under uncertainty and risk

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We consider the research problem of quantitative support for decision making in supply chain network design (SCND). We first identify the requirements for a comprehensive SCND as (i) a methodology to select uncertainties, (ii) a stochastic optimisation model, and (iii) an appropriate solution algorithm. We propose a process to select a manageable number of uncertainties to be included in a stochastic program for SCND. We develop a comprehensive two-stage stochastic program for SCND that includes uncertainty in demand, currency exchange rates, labour costs, productivity, supplier costs, and transport costs. Also, we consider conditional value at risk (CV@R) to explore the trade-off between risk and return. We use a scenario generator based on moment matching to represent the multivariate uncertainty. The resulting stochastic integer program is computationally challenging and we propose a novel iterative solution algorithm called adaptive scenario refinement (ASR) to process the problem. We describe the rationale underlying ASR, validate it for a set of benchmark problems, and discuss the benefits of the algorithm applied to our SCND problem. Finally, we demonstrate the benefits of the proposed model in a case study and show that multiple sources of uncertainty and risk are important to consider in the SCND. Whereas in the literature most research is on demand uncertainty, our study suggests that exchange rate uncertainty is more important for the choice of optimal supply chain strategies in international production networks. The SCND model and the use of the coherent downside risk measure in the stochastic program are innovative and novel; these and the ASR solution algorithm taken together make contributions to knowledge

Design and architecture of a stochastic programming modelling system

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Decision making under uncertainty is an important yet challenging task; a number of alternative paradigms which address this problem have been proposed. Stochastic Programming (SP) and Robust Optimization (RO) are two such modelling ap-proaches, which we consider; these are natural extensions of Mathematical Pro-gramming modelling. The process that goes from the conceptualization of an SP model to its solution and the use of the optimization results is complex in respect to its deterministic counterpart. Many factors contribute to this complexity: (i) the representation of the random behaviour of the model parameters, (ii) the interfac-ing of the decision model with the model of randomness, (iii) the difficulty in solving (very) large model instances, (iv) the requirements for result analysis and perfor-mance evaluation through simulation techniques. An overview of the software tools which support stochastic programming modelling is given, and a conceptual struc-ture and the architecture of such tools are presented. This conceptualization is pre-sented as various interacting modules, namely (i) scenario generators, (ii) model generators, (iii) solvers and (iv) performance evaluation. Reflecting this research, we have redesigned and extended an established modelling system to support modelling under uncertainty. The collective system which integrates these other-wise disparate set of model formulations within a common framework is innovative and makes the resulting system a powerful modelling tool. The introduction of sce-nario generation in the ex-ante decision model and the integration with simulation and evaluation for the purpose of ex-post analysis by the use of workflows is novel and makes a contribution to knowledge

Modelling and solution methods for stochastic optimisation

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis we consider two research problems, namely, (i) language constructs for modelling stochastic programming (SP) problems and (ii) solution methods for processing instances of different classes of SP problems. We first describe a new design of an SP modelling system which provides greater extensibility and reuse. We implement this enhanced system and develop solver connections. We also investigate in detail the following important classes of SP problems: singlestage SP with risk constraints, two-stage linear and stochastic integer programming problems. We report improvements to solution methods for single-stage problems with second-order stochastic dominance constraints and two-stage SP problems. In both cases we use the level method as a regularisation mechanism. We also develop novel heuristic methods for stochastic integer programming based on variable neighbourhood search. We describe an algorithmic framework for implementing decomposition methods such as the L-shaped method within our SP solver system. Based on this framework we implement a number of established solution algorithms as well as a new regularisation method for stochastic linear programming. We compare the performance of these methods and their scale-up properties on an extensive set of benchmark problems. We also implement several solution methods for stochastic integer programming and report a computational study comparing their performance. The three solution methods, (a) processing of a single-stage problem with second-order stochastic dominance constraints, (b) regularisation by the level method for two-stage SP and (c) method for solving integer SP problems, are novel approaches and each of these makes a contribution to knowledge.Financial support was obtained from OptiRisk Systems

An approximation framework for two-stage ambiguous stochastic integer programs under mean-MAD information

We consider two-stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model. If all decision variables are continuous, then we are able to derive the worst-case and best-case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known. Contrary to most existing results in the literature, these probability distributions are the same for every first-stage decision. The ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables. For such problems, we develop a general approximation framework and derive error bounds for using these approximatons. We apply this approximation framework to mixed-ambiguous mixed-integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous. To illustrate these results we carry out numerical experiments on a surgery block allocation problem. (C) 2018 Elsevier B.V. All rights reserved