Jean-Paul Watson

Scheduling Space–Ground Communications for the Air Force Satellite Control Network

We present the first coupled formal and empirical analysis of the Satellite Range Scheduling application. We structure our study as a progression; we start by studying a simplified version of the problem in which only one resource is present. We show that the simplified version of the problem is equivalent to a well-known machine scheduling problem and use this result to prove that Satellite Range Scheduling is NP-complete. We also show that for the one-resource version of the problem, algorithms from the machine scheduling domain outperform a genetic algorithm previously identified as one of the best algorithms for Satellite Range Scheduling. Next, we investigate if these performance results generalize for the problem with multiple resources. We exploit two sources of data: actual request data from the U.S. Air Force Satellite Control Network (AFSCN) circa 1992 and data created by our problem generator, which is designed to produce problems similar to the ones currently solved by AFSCN. Three main results emerge from our empirical study of algorithm performance for multiple-resource problems. First, the performance results obtained for the single-resource version of the problem do not generalize: the algorithms from the machine scheduling domain perform poorly for the multiple-resource problems. Second, a simple heuristic is shown to perform well on the old problems from 1992; however it fails to scale to larger, more complex generated problems. Finally, a genetic algorithm is found to yield the best overall performance on the larger, more difficult problems produced by our generator.

Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems

Numerous planning problems can be formulated as multi-stage stochastic programs and many possess key discrete (integer) decision variables in one or more of the stages. Progressive hedging (PH) is a scenario-based decomposition technique that can be leveraged to solve such problems. Originally devised for problems possessing only continuous variables, PH has been successfully applied as a heuristic to solve multi-stage stochastic programs with integer variables. However, a variety of critical issues arise in practice when implementing PH for the discrete case, especially in the context of very difficult or large-scale mixed-integer problems. Failure to address these issues properly results in either non-convergence of the heuristic or unacceptably long run-times. We investigate these issues and describe algorithmic innovations in the context of a broad class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce the need for sufficient combinations of resources. The necessity and efficacy of our techniques is empirically assessed on a two-stage stochastic network flow problem with integer variables in both stages.

Pyomo: modeling and solving mathematical programs in Python

We describe Pyomo, an open source software package for modeling and solving mathematical programs in Python. Pyomo can be used to define abstract and concrete problems, create problem instances, and solve these instances with standard open-source and commercial solvers. Pyomo provides a capability that is commonly associated with algebraic modeling languages such as AMPL, AIMMS, and GAMS. In contrast, Pyomo’s modeling objects are embedded within a full-featured high-level programming language with a rich set of supporting libraries. Pyomo leverages the capabilities of the Coopr software library, which together with Pyomo is part of IBM’s COIN-OR open-source initiative for operations research software. Coopr integrates Python packages for defining optimizers, modeling optimization applications, and managing computational experiments. Numerous examples illustrating advanced scripting applications are provided.

A Multiple-Objective Analysis of Sensor Placement Optimization in Water Networks

Terrorism concerns have recently led to increased interest in the potential use of sensors to detect malicious attacks on municipal water systems. A key deployment issue is identifying where the sensors should be placed in order to maximize the level of protection. Researchers have proposed several algorithms for constructing such sensor placements, each optimizing with respect to a different design objective. The use of disparate objectives raises several questions, in particular (1) What is the relationship between optimal placements obtained under different design objectives? and (2) Is there any risk in focusing on speci?c design objectives? To answer these questions, we develop mixed-integer linear programming models for the sensor placement problem over a range of design objectives. Using two real-world water systems, we show that optimal solutions with respect to one design objective are typically highly sub-optimal with respect to other design objectives. The implication is that robust algorithms for the sensor placement problem must carefully and simultaneously consider multiple, disparate design objectives.

Contrasting Structured and Random Permutation Flow-Shop Scheduling Problems: Search-Space Topology and Algorithm Performance

The use of random test problems to evaluate algorithm performance raises an important, and generally unanswered, question: Are the results generalizable to more realistic problems? Researchers generally assume that algorithms with superior performance on difficult, random test problems will also perform well on more realistic, structured problems. Our research explores this assumption for the permutation flow-shop scheduling problem. We introduce a method for generating structured flow-shop problems, which are modeled after features found in some real-world manufacturing environments. We perform experiments that indicate significant differences exist between the search-space topologies of random and structured flow-shop problems, and demonstrate that these differencescan affect the performance of certain algorithms. Yet despite these differences, and in contrast to difficult random problems, the majority of structured flow-shop problems were easily solved to optimality by most algorithms. For the problems not optimally solved, differences in performance were minor. We conclude that more realistic, structured permutation flow-shop problems are actually relatively easy to solve. Our results also raise doubts as to whether superior performance on difficult random scheduling problems translates into superior performance on more realistic kinds of scheduling problems.

Problem difficulty for tabu search in job-shop scheduling

Tabu search algorithms are among the most effective approaches for solving the job-shop scheduling problem (JSP). Yet, we have little understanding of why these algorithms work so well, and under what conditions. We develop a model of problem difficulty for tabu search in the JSP, borrowing from similar models developed for SAT and other NP-complete problems. We show that the mean distance between random local optima and the nearest optimal solution is highly correlated with the cost of locating optimal solutions to typical, random JSPs. Additionally, this model accounts for the cost of locating sub-optimal solutions, and provides an explanation for differences in the relative difficulty of square versus rectangular JSPs. We also identify two important limitations of our model. First, model accuracy is inversely correlated with problem difficulty, and is exceptionally poor for rare, very high-cost problem instances. Second, the model is significantly less accurate for structured, non-random JSPs. Our results are also likely to be useful in future research on difficulty models of local search in SAT, as local search cost in both SAT and the JSP is largely dictated by the same search space features. Similarly, our research represents the first attempt to quantitatively model the cost of tabu search for any NP-complete problem, and may possibly be leveraged in an effort to understand tabu search in problems other than job-shop scheduling.

Two-stage robust optimization for N-k contingency-constrained unit commitment

This paper proposes a two-stage robust optimization approach to solve the N- k contingency-constrained unit commitment (CCUC) problem. In our approach, both generator and transmission line contingencies are considered. Compared to the traditional approach using a given set of components as candidates for possible failures, our approach considers all possible component failure scenarios. We consider the objectives of minimizing the total generation cost under the worst-case contingency scenario and/or the total pre-contingency cost. We formulate CCUC as a two-stage robust optimization problem and develop a decomposition framework to enable tractable computation. In our framework, the master problem makes unit commitment decisions and the subproblem discovers the worst-case contingency scenarios. By using linearization techniques and duality theory, we transform the subproblem into a mixed-integer linear program (MILP). The most violated inequalities generated from the subproblem are fed back into the master problem during each iteration. Our approach guarantees a globally optimal solution in a finite number of iterations. In reported computational experiments, we test both primal and dual decomposition approaches. Our computational results verify the effectiveness of our proposed approach.

The Traveling Salesrep Problem, Edge Assembly Crossover, and 2-opt

Optimal results for the Traveling Salesrep Problem have been reported on problems with up to 3038 cities using a GA with Edge Assembly Crossover (EAX). This paper first attempts to independently replicate these results on Padbergs 532 city problem. We then evaluate the performance contribution of the various algorithm components. The incorporation of 2-opt into the EAX GA is also explored. Finally, comparative results are presented for a population-based form of 2-opt that uses partial restarts.

PySP: modeling and solving stochastic programs in Python

Although stochastic programming is a powerful tool for modeling decision-making under uncertainty, various impediments have historically prevented its wide-spread use. One factor involves the ability of non-specialists to easily express stochastic programming problems as extensions of their deterministic counterparts, which are typically formulated first. A second factor relates to the difficulty of solving stochastic programming models, particularly in the mixed-integer, non-linear, and/or multi-stage cases. Intricate, configurable, and parallel decomposition strategies are frequently required to achieve tractable run-times on large-scale problems. We simultaneously address both of these factors in our PySP software package, which is part of the Coopr open-source Python repository for optimization; the latter is distributed as part of IBM’s COIN-OR repository. To formulate a stochastic program in PySP, the user specifies both the deterministic base model (supporting linear, non-linear, and mixed-integer components) and the scenario tree model (defining the problem stages and the nature of uncertain parameters) in the Pyomo open-source algebraic modeling language. Given these two models, PySP provides two paths for solution of the corresponding stochastic program. The first alternative involves passing an extensive form to a standard deterministic solver. For more complex stochastic programs, we provide an implementation of Rockafellar and Wets’ Progressive Hedging algorithm. Our particular focus is on the use of Progressive Hedging as an effective heuristic for obtaining approximate solutions to multi-stage stochastic programs. By leveraging the combination of a high-level programming language (Python) and the embedding of the base deterministic model in that language (Pyomo), we are able to provide completely generic and highly configurable solver implementations. PySP has been used by a number of research groups, including our own, to rapidly prototype and solve difficult stochastic programming problems.

Toward scalable, parallel progressive hedging for stochastic unit commitment

Given increasing penetration of variable generation units, there is significant interest in the power systems research community concerning the development of solution techniques that directly address the stochasticity of these sources in the unit commitment problem. Unfortunately, despite significant attention from the research community, stochastic unit commitment solvers have not made their way into practice, due in large part to the computational difficulty of the problem. In this paper, we address this issue, and focus on the development of a decomposition scheme based on the progressive hedging algorithm of Rockafellar and Wets. Our focus is on achieving solve times that are consistent with the requirements of ISO and utilities, on modest-scale instances, using reasonable numbers of scenarios. Further, we make use of modest-scale parallel computing, representing capabilities either presently deployed, or easily deployed in the near future. We demonstrate our progress to date on a test instance representing a simplified version of the US western interconnect (WECC-240).