Simultaneous orthogonal collocation decomposition method for extended Lion and Man problems

Abstract

Lion and Man problems are classical examples of pursuit and evasion games. However, the traditional analytic methods and indirect numerical methods only can handle the generalization of Lion and Man problems in games with small scales and simple scenarios. In this study, we first extend the original Lion and Man problems to a more complicated and time-varying game environment. Then we propose the simultaneous orthogonal collocation decomposition (SOCD) method, which is a direct method for exploring solutions of Lion and Man problems in a complicated game environment. Compared to indirect methods, SOCD method is much easier to apply. The max-minimization problem in Lion and Man problems is decomposed into two subproblems of optimal control, which are discretized by using the orthogonal collocation method. Local solutions of the resulting nonlinear programming problems lead to the optimal control problems. We also develop the receding horizon optimization method based on SOCD method to solve Lion and Man problems online in a time-varying game environment. In this method, the whole optimization time domain is divided into several short optimization cycles, and Lion and Man problems in each cycle are based on real-time observations of the game environment. The validity of these two methods is tested by conducting numerical simulations, and the results demonstrate that these methods provide a unified framework for solving extended Lion and Man problems.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Number 61773341, the State Key Laboratory Project of China under Grant Number ICT1804, the Joint Innovation Fund of the China Academy of Launch Vehicle Technology and Universities under Grant Number CALT201603, the Equipment Pre-Research Project of China under Grant Number 30506030302, and the Fundamental Research Funds for the Central Universities under Grant Number 2019XZZX005-1-05.

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Correspondence to Zhijiang Shao.

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Zhu, Q., Wang, K., Shao, Z. et al. Simultaneous orthogonal collocation decomposition method for extended Lion and Man problems. Optim Eng (2020). https://doi.org/10.1007/s11081-020-09513-y

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Keywords

  • Complicated game environment
  • Extended Lion and Man problems
  • Receding horizon optimization
  • Simultaneous orthogonal collocation decomposition
  • Time-varying game environment