Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossib...

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Bibliographic Details
Published in:Computational optimization and applications 2023-12, Vol.86 (3), p.1247-1274
Main Authors: Kaya, C. Yalçın, Maurer, Helmut
Format: Article
Language:English
Subjects:
Algorithms
Chebyshev approximation
Circuits
Convex and Discrete Geometry
Management Science
Mathematics
Mathematics and Statistics
Multiple objective analysis
Numerical methods
Operations Research
Operations Research/Decision Theory
Optimal control
Optimization
Statistics
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Summary:Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-023-00535-7