Optimization Under Uncertainty

This article is part of a series: Optimization Under Uncertainty. The following aspects are covered

• Stochastic programming

• Optimization with chance constraints

• Optimization of risk measures

Mathematical programming or numerical optimization deals with the challenge of computing the optimum of an objective function and the corresponding decision variables [Locatelli and Schoen, 2013]. A general problem of the minimization of a function \(f(x)\) subject to the inequality constraints \(g(x)\) and the equality constraints \(h(x)\) for the determination of a minimizer \({x^*}\) would be posed as:

\[\begin{split} \begin{equation} \begin{aligned} x^*=\underset{x}{\text{arg min }} f(x)\\ s.t. \text{ } g(x)\leq 0 \\ h(x)= 0 \end{aligned} \end{equation} \end{split}\]

Figure 1: Optimization of a deterministic function \(f(x)\): Numerical optimization seeks the optimal solution, e.g., the point \(x^*\) corresponding the minimum value \(f(x^*)\). Each point on the deterministic function \(f(x)\) corresponds to a scalar value.

With an uncertain parameter \(p\) following a probability distribution, the functions \(f(x,p)\), \(g(x,p)\), and \(h(x,p)\) become uncertain and effectively represent a distribution of possible values, as exemplary visualized for \(f(x,p)\) in Figure 2. The optimization problem under uncertainty can be stated as

\[\begin{split} \begin{equation} \begin{aligned} x^*=\underset{x}{\text{arg min }} f(x,p)\\ s.t. \text{ } g(x,p)\leq 0 \\ h(x,p)=0 \end{aligned} \end{equation} \end{split}\]

Figure 2: Optimization of an uncertain function \(f(x)\): The optimization of an uncertain function returns a distribution \(f(\hat{x})\) for a point \(\hat{x}\). In contrast, optimization of a determinisitc function returns a scalar value as shown in Figure 1.

Methods

To deal with the uncertainty in the objective function and the constraints, various methods exists:

\(\bullet\) Stochastic programming

\(\bullet\) Robust optimization

\(\bullet\) Chance-constraints

\(\bullet\) Optimization of risk measures

References

LS13

Marco Locatelli and Fabio Schoen. Global optimization : theory, algorithms, and applications. MOS-SIAM series on optimization. SIAM, Philadelphia, PA, 2013. ISBN 9781611972665.

Authors

Sonja Germscheid

Contributors

Manuel Dahmen