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

Solving the bifurcated and nonbifurcated robust network loading problem with k-adaptive routing

International audienceWe experiment with an alternative routing scheme for the robust network loading problem with demand uncertainty. Named k‐adaptive, it is based on the fact that the decision‐maker chooses k second‐stage solutions and then commits to one of them only after realization of the uncertainty. This routing scheme, with its corresponding k‐partition of the uncertainty set, is dynamically defined under an iterative method to sequentially improve the solution. The method has an inherent characteristic of multiplying the number of variables and constraints after each iteration, so that additional measures are introduced in the solution strategy in order to control time performance. We compare our k‐adaptive results with the ones obtained through other routing schemes and also verify the effectiveness of the methods utilized using several realistic networks from SNDlib and other sources

A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming

Despite methodological advances for modeling decision problems under uncertainty, faithfully representing endogenous uncertainty still proves challenging, both in terms of modeling capabilities and computational requirements. A novel framework called Decision Programming provides an approach for solving such decision problems using off-the-shelf mathematical optimization solvers. This is made possible by using influence diagrams to represent a given decision problem, which is then formulated as a mixed-integer linear programming problem. In this paper, we focus on the type of endogenous uncertainty that received less attention in the introduction of Decision Programming: conditionally observed information. Multi-stage stochastic programming (MSSP) models use conditional non-anticipativity constraints (C-NACs) to represent such uncertainties, and we show how such constraints can be incorporated into Decision Programming models. This allows us to consider the two main types of endogenous uncertainty simultaneously, namely decision-dependent information structure and decision-dependent probability distribution. Additionally, we present a decomposition approach that provides significant computational savings and also enables considering continuous decision variables in certain parts of the problem, whereas the original formulation was restricted to discrete variables only. The extended framework is illustrated with two example problems. The first considers an illustrative multiperiod game and the second is a large-scale cost-benefit problem regarding climate change mitigation. Neither of these example problems could be solved with existing frameworks.Comment: 26 pages, 10 figure

Extending Distance-based Ranking Models In Estimation of Distribution Algorithms

Recently, probability models on rankings have been proposed in the field of estimation of distribution algorithms in order to solve permutation-based combinatorial optimisation problems. Particularly, distance-based ranking models, such as Mallows and Generalized Mallows under the Kendall’s-t distance, have demonstrated their validity when solving this type of problems. Nevertheless, there are still many trends that deserve further study. In this paper, we extend the use of distance-based ranking models in the framework of EDAs by introducing new distance metrics such as Cayley and Ulam. In order to analyse the performance of the Mallows and Generalized Mallows EDAs under the Kendall, Cayley and Ulam distances, we run them on a benchmark of 120 instances from four well known permutation problems. The conducted experiments showed that there is not just one metric that performs the best in all the problems. However, the statistical test pointed out that Mallows-Ulam EDA is the most stable algorithm among the studied proposals

Stochastic Constraint Programming with And-Or Branch-and-Bound

Complex multi-stage decision making problems often involve uncertainty, for example, regarding demand or processing times. Stochastic constraint programming was proposed as a way to formulate and solve such decision problems, involving arbitrary constraints over both decision and random variables. What stochastic constraint programming currently lacks is support for the use of factorized probabilistic models that are popular in the graphical model community. We show how a state-ofthe-art probabilistic inference engine can be integrated into standard constraint solvers. The resulting approach searches over the And-Or search tree directly, and we investigate tight bounds on the expected utility objective. This significantly improves search efficiency and outperforms scenario-based methods that ground out the possible worlds.status: publishe

Distributed algorithms for nonlinear tree-sparse problems

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