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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multiobjective Optimization

#### 2.1. Theory

**Remark**

**1.**

**Definition**

**1.**

**Definition**

**2.**

- 1.
- a point ${u}^{\ast}$ dominates a point u, if $J\left({u}^{\ast}\right)\le J\left(u\right)$ and $J\left({u}^{\ast}\right)\ne J\left(u\right)$.
- 2.
- a feasible point ${u}^{\ast}$ is called globally Pareto optimal if there exists no feasible point $u\in \mathcal{U}$ dominating ${u}^{\ast}$. The image $J\left({u}^{\ast}\right)$ of a globally Pareto optimal point ${u}^{\ast}$ is called a globally Pareto optimal value. If this property holds in a neighborhood $U\left({u}^{\ast}\right)\subset \mathcal{U}$, then ${u}^{\ast}$ is called locally Pareto optimal.
- 3.
- the set of non-dominated feasible points is called the Pareto set ${\mathcal{P}}_{S}$ and its image the Pareto front ${\mathcal{P}}_{F}$.

**Theorem**

**1**

#### 2.2. Solution Methods

## 3. Surrogate Models

#### 3.1. Inaccuracies and $\u03f5$-Dominance

**Definition**

**3**

- 1.
- a point ${u}^{\ast}$ confidently dominates a point u, if ${J}^{r}\left({u}^{\ast}\right)+\u03f5\le {J}^{r}\left(u\right)-\u03f5$ and ${J}_{i}^{r}\left({u}^{\ast}\right)+{\u03f5}_{i}<{J}_{i}^{r}\left(u\right)-{\u03f5}_{i}$ for at least one $i\in 1,\dots ,k$.
- 2.
- The set of almost non-dominated points, which is a superset of the Pareto set ${\mathcal{P}}_{S}$, is defined as:$$\begin{array}{c}\hfill {\mathcal{P}}_{S,\u03f5}=\left(\right)open="\{"\; close="\}">{u}^{\ast}\in \mathcal{U}|\nexists u\in \mathcal{U}\phantom{\rule{4.pt}{0ex}}\mathit{which}\phantom{\rule{4.pt}{0ex}}\mathit{confidently}\phantom{\rule{4.pt}{0ex}}\mathit{dominates}\phantom{\rule{4.pt}{0ex}}{u}^{\ast}.\end{array}$$

**Theorem**

**2**

- (a)
- If ${\sum}_{i=1}^{k}{\widehat{\alpha}}_{\mathsf{min},i}>1$ then there exists no direction $q\left(u\right)$ with:$$\langle q\left(u\right),\nabla {J}_{i}\left(u\right)\rangle <0\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}i=1,\dots ,k,$$
- (b)
- All points u with ${\sum}_{i=1}^{k}{\widehat{\alpha}}_{\mathsf{min},i}=1$ are contained in the set:$$\begin{array}{c}\hfill {\mathcal{P}}_{S,\kappa}=\left(\right)open="\{"\; close="\}">u\in {\mathbb{R}}^{n}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}||\sum _{i=1}^{k}{\widehat{\alpha}}_{i}\nabla {J}_{i}\left(u\right){||}_{2}\le 2{\parallel \kappa \parallel}_{\infty}.\end{array}$$

#### 3.2. Surrogate Models for the Objective Function

- Response Surface Models (RSM);
- Radial Basis Functions (RBF);
- statistical models such as Kriging or Gauss regression;
- machine learning methods such as artificial neural networks or support vector machines;

- How large does the set ${\mathcal{U}}_{\mathsf{ref}}$ have to be?
- How can we pick the correct elements for ${\mathcal{U}}_{\mathsf{ref}}$?
- Do we define ${\mathcal{U}}_{\mathsf{ref}}$ in advance or online during the model building process?

#### 3.3. Surrogate Models for the Dynamical System

#### 3.3.1. ROMS via Proper Orthogonal Decomposition or the Reduced Basis Method

**Example**

**1**(Heat equation).

#### 3.3.2. Optimal Control Using Surrogate Models

- build a model once,
- construction of regular updates in a trust region approach,
- construction of regular updates using error estimators.

#### 3.4. ROM-Based Multiobjective Optimal Control of PDEs

#### 3.4.1. Scalarization

#### 3.4.2. Set-Oriented Approaches with $\u03f5$-Dominance

#### 3.5. Summary

## 4. Feedback Control

#### 4.1. Online Multiobjective Optimization

- compute a single Pareto-optimal solution according to some predefined preference,
- compute only a rough approximation of the Pareto set,
- compute an arbitrary Pareto-optimal control that satisfies additional constraints (e.g., stability).

#### 4.2. Offline-Online Decomposition

#### 4.2.1. Example: Autonomous Driving

#### 4.3. Summary

## 5. Reduction Techniques for Many-Objective Optimization Problems

## 6. Future Directions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The red lines are the Pareto set (

**a**) and Pareto front (

**b**) of an exemplary multiobjective optimization problem (two paraboloids) of the form ${min}_{u\in \mathbb{R}}J\left(u\right)$, $J:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$. The point ${J}^{\ast}={(0,0)}^{\top}$ is called the utopian point.

**Figure 2.**Example for the $\u03f5$-dominance property. A point-wise comparison is illustrated in (

**a**)–(

**d**). The uncertainties are marked by the dashed boxes. Only in case (

**d**), the lower left point confidently dominates the other point. In (

**e**), the entirety of Pareto fronts for the exact problem (${\mathcal{P}}_{F}$) and the inexact problem (${\mathcal{P}}_{F,\u03f5}$) are shown in red and orange, respectively.

**Figure 3.**Example for an Multiobjective Optimization Problem (MOP) from production [32] where inexactness is introduced due to uncertainties in pricing. The Pareto set ${\mathcal{P}}_{S}$ for the exact problem is shown in (

**a**), and the inexact set ${\mathcal{P}}_{S,\u03f5}$ is shown in (

**b**).

**Figure 4.**Exact and inexact solutions (${\mathcal{P}}_{S}$ and ${\mathcal{P}}_{S,\kappa}$) for a simple example with $J:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$, cf. [66] for details. (

**a**) The sets ${\mathcal{P}}_{S}$ and ${\mathcal{P}}_{S,\kappa}$ (for a random error with $\kappa ={(0.01,0.01)}^{\top}$) are shown in red and green, respectively. The background is colored according to the optimality condition $\parallel q\left(u\right)\parallel $, which has to be zero for all substationary points. The dashed white line shows the error bound as derived in Theorem 2. (

**b**) The corresponding Pareto fronts.

**Figure 5.**Trust region method. (

**a**) The Reduced Order Model (ROM)-based optimal control problem is solved within the trust region ${\delta}^{0}$. (

**b**) If the improvement is poor for the full system (i.e., $\rho $ is small), then the trust region radius is reduced, and we repeat the computation with the same problem. (

**c**) If the improvement is acceptable (intermediate values of $\rho $), then we compute a new model and proceed with a smaller trust region ${\delta}^{1}<{\delta}^{0}$. (

**d**) If the improvement is good (i.e., $\rho \approx 1$), then the trust region radius is increased.

**Figure 6.**Reference point method. (

**a**) Determination of a Pareto-optimal solution by solving (13). (

**b**) Determination of the consecutive point on the Pareto front by adjusting the target and solving the next scalar problem.

**Figure 7.**(

**a**) Pareto front for an MOCP involving the Navier–Stokes equations (flow stabilization vs. cost), solved by coupling of an ROM (created once in advance) with the reference point method. Although we observe acceptable agreement, convergence cannot be guaranteed. (

**b**) Pareto front for a heat flow MOCP (reference tracking vs. cost), solved by the TR-POD approach coupled with the reference point method. Convergence is achieved while reducing the number of expensive finite element (FEM) evaluations by a factor of $\approx 22$, cf. (

**c**).

**Figure 8.**(

**a**) Pareto set of a semilinear heat flow MOP with four controls (coloring according to ${u}_{4}$), solved directly including an FEM model in the subdivision algorithm. (

**b**) Pareto set of the same problem, solved with localized ROMs. (

**c**) The reference controls for which the local ROMs have been computed are shown in black, and the colored dots are sample points at which the objective function was evaluated. The colorings denote assignments to a specific ROM. (

**d**) The corresponding Pareto fronts, where the FEM solution is shown in green and the ROM solution in red.

**Figure 9.**Sketch of the MPC method. Due to the real-time constraints, the optimization problem has to be solved faster than the sample time h.

**Figure 10.**Results for the offline-online MPC approach from [48]. (

**a**) Different constraint scenarios ((I)–(VI)), i.e., constant velocity, acceleration, deceleration and stopping. (

**b**) Example track driven with the MPC algorithm. The red lines define the velocity bounds; the black dashed lines are trajectories corresponding to a constant weight $\alpha $; and the green line is a trajectory where the weight is changed from 0 (energy efficient) over 0.5 (average) to 1 (fast). (

**c**) Comparison between the MPC algorithm (coupled with a simple heuristic for the weighting) and the global optimum obtained via dynamic programming.

**Figure 11.**Visualization of the hierarchical structure of Pareto sets. (

**a**) Pareto set of an example problem with $J:{\mathbb{R}}^{3}\to {\mathbb{R}}^{4}$. (

**b**) The four Pareto sets taking only three objectives into account form the boundary of the original Pareto set. (

**c**) The Pareto sets in (b) are again bounded by the respective bi-objective subproblems.

**Table 1.**Overview of publications (in chronological order) where surrogate modeling and multiobjective optimization are combined. MOEA, Multiobjective Evolutionary Algorithm; RSM, Response Surface Model; POD, Proper Orthogonal Decomposition; TR, Trust Region.

Surveys | |

Tabatabei et al. [2], Chugh et al. [3] | Extensive surveys on meta modeling for MOEAs |

Voutchkov and Keane [74], Knowles and Nakayama [1], Jin [75] | Surveys on meta modeling approaches from statistics (RSM, RBF) and machine learning in combination with MOEAs |

Benner et al. [83], Taira et al. [85], Peherstorfer et al. [84] | Surveys on reduced order modeling of dynamical systems |

Algorithms Using Meta Models for the Objective Function | |

Ong et al. [112], Ray et al. [113] | Combination of RBF and MOEA |

Chung and Alonso [114], Keane [115] | Combination of kriging models and MOEA |

Karakasis and Giannakoglou [116] | RBF as an inexpensive pre-processing step in a MOEA |

Knowles [117] | Combination of DoE and an interactive method |

Zhang et al. [118] | Combination of Gaussian process models and scalarization |

Telen et al. [79] | Combination of DoE and scalarization and MOEA |

Chugh et al. [119] | Kriging model in combination with reference vector approach for MOPs |

Meta Models Specifically Tailored to Multiobjective Optimization | |

Shimoyama et al. [120] | Kriging surrogate for hypervolume approximation (MOEA) |

Pan et al. [121] | Surrogate model for dominance relations with uncertainties |

Algorithms Using Surrogate Models for the System Dynamics | |

Iapichino et al. [105] | Combination of POD and weighted sum |

Banholzer et al. [108,109] | Combination of POD and reference point method |

Iapichino et al. [106] | Combination of RB and weighted sum |

Peitz [73] | Combination of TR-POD and reference point method |

Beermann et al. [110,111] | Combination of POD and set-oriented method |

Applications | |

Albunni et al. [53] | POD and MOEA applied to the Maxwell equation |

Ma and Qu [80] | MO of a switched reluctance motor by coupling RSM and MOEA (particle swarm optimization) |

Peitz et al. [107] | POD-based multiobjective optimal control of the Navier–Stokes equations via scalarization and set-oriented methods |

Wang et al. [122] | MOEA with multi-fidelity surrogate-management and offline-online decomposition applied to a trauma system |

Algorithms without Offline Phase: Computation of Single Points | |

Kerrigan et al. [131], Wojsznis et al. [154] | Scalarization via Weighted Sum (WS) |

Kerrigan and Maciejowski [155], He et al. [137] | Scalarization via lexicographic ordering |

Bemporad and Muñoz de la Peña [132,133] | Scalarization via WS for convex objectives, guaranteed stability for large gain vs. noise robust stabilizing objectives |

Geisler and Trächtler [134] | WS, online adaptation of weights using gradient information |

Maestre et al. [143] | Scalarization via game-theoretic approach |

Zavala and Flores-Tlacuahuac [135] | Scalarization via reference point approach |

Hackl et al. [129] | Scalarization via WS for Linear Time-Invariant (LTI) systems |

Zavala [136] | Scalarization via $\u03f5$-constraint: economic objective, stability as constraint |

Grüne and Stieler [130] | Economic objectives, performance bounds via selection criterion |

Algorithms without Offline Phase: Approximation of the Entire Pareto Set | |

Laabidi et al. [138,140], Garcìa et al. [141] | ANN for state prediction, optimization via MOEA, selection of Pareto point via WS |

Bouani et al. [139] | ANN for state prediction, comparison of two MOEAs and WS for MOP |

Nakayama et al. [142] | Few MOEA iterations online, selection via satisficing trade-off method |

Algorithms with Offline Phase | |

Fonseca [145], Herreros et al. [146] | Offline computation of Pareto optimal controller parameters using MOEA |

Scherer et al. [156] | Robust control using a common Lyapunov function for multiple stability criteria |

Ben Aicha et al. [147] | Offline computation of Pareto optimal controllers parameters via EA and WS, online selection according to geometric mean of objectives |

Krüger et al. [148] | Offline computation of Pareto optimal controllers parameters via Set oriented methods, parametric model reduction for increased efficiency |

Hernández et al. [149], Xiong et al. [150] | Offline computation of Pareto-optimal controllers parameters via simple cell mapping |

Peitz et al. [48] | Offline-online decomposition similar to explicit MPC |

Applications | |

Zambrano and Camacho [157] | MOMPC of a solar refrigeration plant via scalarization |

Porfírio et al. [158] | MOMPC of an industrial splitter using a min-max reformulation |

Pedersen and Yang [159] | MO PID controller design for magnetic levitation systems via MOEA |

Li et al. [160] | Multiobjective adaptive cruise control for vehicles |

Hu et al. [161] | MOMPC of high-power converters via WS |

Núñez et al. [162] | MOMPC of dynamic pickup and delivery problems using MOEA |

Peitz et al. [163] | MOMPC of an industrial laundry, scalarization of a traveling salesman problem via WS |

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**MDPI and ACS Style**

Peitz, S.; Dellnitz, M.
A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction. *Math. Comput. Appl.* **2018**, *23*, 30.
https://doi.org/10.3390/mca23020030

**AMA Style**

Peitz S, Dellnitz M.
A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction. *Mathematical and Computational Applications*. 2018; 23(2):30.
https://doi.org/10.3390/mca23020030

**Chicago/Turabian Style**

Peitz, Sebastian, and Michael Dellnitz.
2018. "A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction" *Mathematical and Computational Applications* 23, no. 2: 30.
https://doi.org/10.3390/mca23020030