A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

Abstract

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

Flood Management of Lake Toke: MPC Operation under Uncertainty

“…Figure 9 shows both the stochastic behavior of the water flowing through the flood gatesV g and the given flow through the turbines,V t . As Figure 6 indicates, the inflow forecasts are updated every 24 h. Stochastic MPC algorithms can be posed in different ways, e.g., as scenario tree based algorithms (Raso et al, 2014), (Krishnamoorthy et al, 2018), or as multi objective based algorithms (Peitz and Dellnitz, 2018). Here, the focus is on a multi objective based algorithm.…”

Using optimal control to understand complex metabolic pathways

“…the set of the best compromises, which is also called the Pareto set). Many methods have been developed for multicriteria optimal control, and they are usually classified in three categories [129]: scalarization techniques, continuation methods and set-oriented approaches. Here we use the -constraint scalarization method, which transforms the original multiobjective problem into a finite set of single-objective optimal control problems.…”

The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review

“…First, we briefly present some basic aspects of multi-objective optimization problems (MOPs) required for the understanding of this paper. For a more thorough discussion, we refer the interested reader, e.g., to References [12,29,30].…”

“…−→ p = 10 −→ p = 5 −→ p = 1 −→ p = −1 Figure 11. Optimal ∆ p,−1 one-point archives A for the connected Pareto front P 1 given by Equation (12) with q = −1 and different values of p: In all cases, the archives are located in the line x = y.…”

“…The Hausdorff distance d H (e.g., Reference [1]) measures how far two subsets of a metric space are from each other. Due to its properties, it is frequently used in many research areas such as computer vision [2][3][4], fractal geometry [5], the numerical computation of attractors in dynamical systems [6][7][8], or convergence of multi-objective algorithms to the Pareto set/front of a given multi-objective optimization problem [9][10][11][12][13][14][15]. One possible drawback of the classical Hausdorff distance, however, is that it punishes single outliers which leads to inequitable performance evaluations in some cases.…”