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A numerical method for interval multi-objective mixed-integer optimal control problems based on quantum heuristic algorithm

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Abstract

This article is concerned with the numerical solution for a class of interval multi-objective mixed-integer optimal control problems (IMOMIOCPs). The IMOMIOCPs under investigation are typical NP-hard problems involve unknown-but-bounded interval parameters, multiple objectives, and mixed-integer dynamic controls. Accordingly, a new numerical method based on quantum heuristic algorithm is designed, which has the following modules: (i) Control vector parameterization and the fourth order Runge–Kutta method for model discretization, (ii) interval programming based on interval credibility for addressing interval parameters, (iii) coevolution of Quantum Annealing and Quantum Krill Herd for searching the optimal mixed-integer decisions, and (iv) the multiple populations for multi-objective technology for establishing the Pareto optimal front. The analyses on convergence and computational complexity of the proposed optimization mechanism are given. Moreover, simulation results on benchmark functions and a practical engineering IMOMIOCP verify that the proposed numerical method is more excel at achieving promising results than some classic algorithms and state-of-the-art algorithms.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China [Grant Number 61573378], and the BUPT Excellent Ph.D. Students Foundation [Grant Number CX2019113].

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Appendix

Appendix

A1. Parameters tuning

T, \(\varGamma \), \(\varDelta \varGamma \) for QA and \({{N}^{\max }}\), \({{V}_{f}}\), \({{D}^{\max }}\) for QKH are the parameters to be tuned. For each parameter, the value range is obtained by the analysis of algorithm mechanism and past optimization examples. Then several candidate values \(\Upsilon \) are selected within the parameter range. Five benchmark examples with different computational complexities (F03, F05, F12, F14 and F18) are carried out for testing each candidate value while other parameters are set as the mean value of the corresponding range. The standard deviation of the obtained solution \(\delta \) is adopted as the evaluation criterion. The maximum fitness evaluations for testing are set as \({4}\times {{10}^{3}}\) for F03, \({6}\times {{10}^{3}}\) for F05, \(2.5\times {{10}^{4}}\) for F12, \(3.5\times {{10}^{5}}\) for F14 and \(2.5\times {{10}^{4}}\) for F19. When an optimal candidate value of any parameter is obtained, then it is fixed in the following tuning. The research results of parameters tuning are shown in Table 15, where the bold characters are the optimal candidate values.

Table 15 Results of parameters tuning
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Table 16 The basic MOMINLP benchmark examples for Experiment study 3
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Table 17 Parameter nomenclatures and value settings for Experimental study 4
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Liu, Z., Li, S. A numerical method for interval multi-objective mixed-integer optimal control problems based on quantum heuristic algorithm. Ann Oper Res 311, 853–898 (2022). https://doi.org/10.1007/s10479-021-03998-1

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