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

• Published in: Computational optimization and applications Vol. 86; no. 3; pp. 1247 - 1274
• Main Authors: Kaya, C. Yalçın, Maurer, Helmut
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2023; Springer Nature B.V
• 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; Rayleigh problem; Tuberculosis; Optimization over efficient set; Optimal control; Numerical methods; Multi-objective optimization; Optimization over Pareto front; Time-delay problems
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-023-00535-7

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.