Optimization under uncertainty: state-of-the-art and opportunities

Abstract

A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large-scale optimization models. Decision-making under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multi-period or multi-stage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recourse-based stochastic programming, robust stochastic programming, probabilistic (chance-constraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the state-of-the-art in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomial-time approximation schemes for multi-stage stochastic programs and the application of global optimization algorithms to two-stage and chance-constraint formulations.

A systematic framework for enterprise-wide optimization: Synthesis and design of processing networks under uncertainty

“…When dealing with large industrial problems, the solution of these optimization problems may require large computational time and resources, especially when a large number of uncertain data is considered (Sahinidis, 2004).…”

“…Often, the nature of the problem requires the formulation of large scale non-linear and nonconvex problems (Karuppiah & Grossmann, 2006) whose solution to global optimality is still an open problem. Finally, the inclusion of data uncertainty in the decision-making problem causes a significant increase in problem size and complexity (Dua & Pistikopoulos, 1998;Karuppiah & Grossmann, 2008;Paules IV & Floudas, 1992;Sahinidis, 2004). Because of this complexity, formulation and solution of real industrial problems require considerable time and resources investment, as well as deep knowledge of optimization theory and algorithms.…”

Proposition of methodology for optimization of energy system design under uncertainty

“…Despite this may be solved by discretizing the uncertain variables space and solve it as a multi-period problem (Sahinidis (2004)), number of model evaluation grows exponentially with n u . It is then crucial to apply uncertainty efficient propagation method on the model.…”

Hybrid simulation–optimization methods: A taxonomy and discussion

“…Stochastic dynamic programming makes also use of simulation, when solving large complex models with the so-called reinforcement learning algorithms. A good overview of stochastic, fuzzy and stochastic dynamic programming is given by Sahinidis [1].…”