A new interior-point approach for large two-stage stochastic problems

Two-stage stochastic models give rise to very large optimization problems. Several approaches havebeen devised for efficiently solving them, including interior-point methods (IPMs). However, usingIPMs, the linking columns associated to first-stage decisions cause excessive fill-in for the solutionof the normal equations. This downside is usually alleviated if variable splitting is applied to first-stage variables. This work presents a specialized IPM that applies variable splitting and exploits thestructure of the deterministic equivalent of the stochastic problem. The specialized IPM combinesCholesky factorizations and preconditioned conjugate gradients for solving the normal equations.This specialized IPM outperforms other approaches when the number of first-stage variables is largeenough. This paper provides computational results for two stochastic problems: (1) a supply chainsystem and (2) capacity expansion in an electric system. Both linear and convex quadratic formu-lations were used, obtaining instances of up to 38 million variables and six million constraints. Thecomputational results show that our procedure is more efficient than alternative state-of-the-art IPMimplementations (e.g., CPLEX) and other specialized solvers for stochastic optimizationPeer ReviewedPreprin

Speeding up Energy System Models - a Best Practice Guide

Background Energy system models (ESM) are widely used in research and industry to analyze todays and future energy systems and potential pathways for the European energy transition. Current studies address future policy design, analysis of technology pathways and of future energy systems. To address these questions and support the transformation of today’s energy systems, ESM have to increase in complexity to provide valuable quantitative insights for policy makers and industry. Especially when dealing with uncertainty and in integrating large shares of renewable energies, ESM require a detailed implementation of the underlying electricity system. The increased complexity of the models makes the application of ESM more and more difficult, as the models are limited by the available computational power of today’s decentralized workstations. Severe simplifications of the models are common strategies to solve problems in a reasonable amount of time – naturally significantly influencing the validity of results and reliability of the models in general. Solutions for Energy-System Modelling Within BEAM-ME a consortium of researchers from different research fields (system analysis, mathematics, operations research and informatics) develop new strategies to increase the computational performance of energy system models and to transform energy system models for usage on high performance computing clusters. Within the project, an ESM will be applied on two of Germany’s fastest supercomputers. To further demonstrate the general application of named techniques on ESM, a model experiment is implemented as part of the project. Within this experiment up to six energy system models will jointly develop, implement and benchmark speed-up methods. Finally, continually collecting all experiences from the project and the experiment, identified efficient strategies will be documented and general standards for increasing computational performance and for applying ESM to high performance computing will be documented in a best-practice guide

Two-Stage Stochastic Mixed Integer Linear Optimization

The primary focus of this dissertation is on optimization problems that involve uncertainty unfolding over time. In many real-world decisions, the decision-maker has to decide in the face of uncertainty. After the outcome of the uncertainty is observed, she can correct her initial decision by taking some corrective actions at a later time stage. These problems are known as stochastic optimization problems with recourse. In the case that the number of time stages is limited to two, these problems are referred to as two-stage stochastic optimization problems. We focus on this class of optimization problems in this dissertation. The optimization problem that is solved before the realization of uncertainty is called the first-stage problem and the problem solved to make a corrective action on the initial decision is called the second-stage problem. The decisions made in the second- stage are affected by both the first-stage decisions and the realization of random variables. Consequently, the two-stage problem can be viewed as a parametric optimization problem which involves the so-called value function of the second-stage problem. The value function describes the change in optimal objective value as the right-hand side is varied and understanding it is crucial to developing solution methods for two-stage optimization problems.In the first part of this dissertation, we study the value function of a MILP. We review the structural properties of the value function and its construction methods. We con- tribute by proposing a discrete representation of the MILP value function. We show that the structure of the MILP value function arises from two other optimization problems that are constructed from its discrete and continuous components. We show that our representation can explain certain structural properties of the MILP value function such as the sets over which the value function is convex. We then provide a simplification of the Jeroslow Formula obtained by applying our results. Finally, we describe a cutting plane algorithm for its construction and determine the conditions under which the pro- posed algorithm is finite.Traditionally, the solution methods developed for two-stage optimization problems consider the problem where the second-stage problem involves only continuous variables. In the recent years, however, two-stage problems with integer variables in the second- stage have been visited in several studies. These problems are important in practice and arise in several applications in supply chain, finance, forestry and disaster management, among others. The second part of this dissertation concerns the development and implementation of a solution method for the two-stage optimization problem where both the first and second stage involve mixed integer variables. We describe a generalization of the classical Benders’ method for solving mixed integer two-stage stochastic linear optimization problems. We employ the strong dual functions encoded in the branch-and-bound trees resulting from solution of the second-stage problem. We show that these can be used effectively within a Benders’ framework and describe a method for obtaining all required dual functions from a single, continuously refined branch-and-bound tree that is used to warm start the solution procedure for each subproblem.Finally, we provide details on the implementation of our proposed algorithm. The implementation allows for construction of several approximations of the value function of the second-stage problem. We use different warm-starting strategies within our proposed algorithm to solve the second-stage problems, including solving all second-stage problems with a single tree. We provide computational results on applying these strategies to the stochastic server problems (SSLP) from the stochastic integer programming test problem library (SIPLIB)

Analysis of large scale linear programming problems with embedded network structures: Detection and solution algorithms

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Linear programming (LP) models that contain a (substantial) network structure frequently arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection algorithms perform very well computationally and make positive contributions to the known body of results for the embedded network detection. For computational solution a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution made by the research reported in this thesis.This study is funded by the Turkish Educational Council and Mugla University

Mitigating Uncertainty via Compromise Decisions in Two-stage Stochastic Linear Programming

Stochastic Programming (SP) has long been considered as a well-justified yet computationally challenging paradigm for practical applications. Computational studies in the literature often involve approximating a large number of scenarios by using a small number of scenarios to be processed via deterministic solvers, or running Sample Average Approximation on some genre of high performance machines so that statistically acceptable bounds can be obtained. In this paper we show that for a class of stochastic linear programming problems, an alternative approach known as Stochastic Decomposition (SD) can provide solutions of similar quality, in far less computational time using ordinary desktop or laptop machines of today. In addition to these compelling computational results, we also provide a stronger convergence result for SD, and introduce a new solution concept which we refer to as the compromise decision. This new concept is attractive for algorithms which call for multiple replications in sampling-based convex optimization algorithms. For such replicated optimization, we show that the difference between an average solution and a compromise decision provides a natural stopping rule. Finally our computational results cover a variety of instances from the literature, including a detailed study of SSN, a network planning instance which is known to be more challenging than other test instances in the literature

The L-shaped method for large-scale mixed-integer waste management decision making problems

It is without a doubt that deciding upon strategic issues requires us to somehow anticipate and consider possible variations of the future. Unfortunately, when it comes to the actual modelling, the sheer size of the problems that accurately describe the uncertainty is often extremely hard to work with. This paper aims to describe a possible way of dealing with the issue of large-scale mixed integer models (in term of the number of possible future scenarios it can handle) for the studied waste management decision making problem. The algorithm is based on the idea of decomposing the overall problem alongside the different scenarios and solving these smaller problems instead. The use of the algorithm is demonstrated on a strategic waste management problem of choosing the optimal sites to build new incineration plants, while minimizing the expected cost of waste transport and processing. The uncertainty was modelled by 5,000 scenarios and the problem was solved to high accuracy using relatively modest means (in terms of computational power and needed software)