設計一個系統時，如果沒有考慮現實世界中所包含的不確定性而造成的劇烈變化，在單純的實驗環境中所設計出來的最佳系統就不一定能適用於現實世界中。為了解決不確定性的問題，設計者應該要設法獲得最穩解；而最穩解可以用制限最小最大最佳化演算法來獲得。本論文提出了一個產生逃脫向量的機制，用來解決差分進化演算法中，因人口多樣性不足而早熟的問題。此方法使用於制限最小最大最佳化演算法後，可以增進演算法的性能。為了評估各種制限最小最大最佳化演算法的性能，本篇論文提出了多種更複雜的測試問題。經實驗證明，本論文所改良的制限最小最大最佳化演算法，在有限的精確度之下，能夠在更複雜的測試問題上達到100%的成功率。

In system design, the best system designed under a simple experimental environment may not be suitable for application in real world if dramatic changes caused by uncertainties contained in the real world are considered. To deal with the problem caused by uncertainties, designers should try their best to get the most robust solution. The most robust solution can be obtained by constrained min-max optimization algorithms. In this thesis, the scheme of generating escape vectors has been proposed to solve the problem of premature convergence of differential evolution. After applying the proposed scheme to the constrained min-max optimization algorithm, the performance of the algorithm could be greatly improved. To evaluate the performance of constrained min-max optimization algorithms, more complex test problems have also been proposed in this thesis. Experimental results show that the improved constrained min-max optimization algorithm is able to achieve a 100% success rate on all considered test problems under limited accuracy.

[1] S. M. Guo, L. S. Shieh, G. Chen and N. P. Coleman, “Observer-type Kalman innovation filter for uncertain linear systems,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 37, pp. 1406-1418, 2001.
[2] S. M. Guo, L. S. Shieh, C. F. Lin and N. P. Coleman, “Evolutionary-programming-based Kalman filter for discrete time nonlinear uncertain systems,” Asian Journal of Control, vol. 3, pp. 319-333, 2001.
[3] A. Rapoport and R. B. Boebel, “Mixed strategies in strictly competitive games: A further test of the minimax hypothesis,” Games and Economic Behavior, vol. 4, no. 2, pp. 261-283, 1992.
[4] S. M. Guo, B. W. Lai, Y. C. Chou and C. C. Yang, “Novel wavelet-based image interpolations in lifting structures for image resolution enhancement,” Journal of Electronic Imaging, vol. 20, no. 3, pp. 033007-033007-22, 2011.
[5] S. M. Guo and C. C. Yang, “Fuzzy similarity measure-based hybrid image filter for color image restoration: multimethodology evolutionary computation,” Journal of Electronic Imaging, vol. 20, no. 3, pp. 033015-033015-18, 2011.
[6] R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” Journal of global optimization, vol. 11, no. 4, pp. 341-359, 1997.
[7] R. Storn and K. Price, Differential evolution–a simple and efficient adaptive scheme for global optimization over continuous spaces, Berkeley: ICSI, 1995.
[8] K. Price, R. M. Storn and J. A. Lampinen, Differential evolution–a practical approach to global optimization, Springer Science & Business Media, 2006.
[9] J. Brest, S. Greiner, B. Bošković, M. Mernik and V. Žumer, “Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems,” Evolutionary Computation, IEEE Transactions on, vol. 10, no. 6, pp. 646-657, 2006.
[10] J. Zhang and A. Sanderson, “JADE: Adaptive differential evolution with optional external archive,” Evolutionary Computation, IEEE Transactions on, vol. 13, no. 5, pp. 945-958, 2009.
[11] S. M. Islam, S. Das, S. Ghosh, S. Roy and P. N. Suganthan, “An adaptive differential evolution algorithm with novel mutation and crossover strategies for global numerical optimization,” Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, vol. 42, no. 2, pp. 482-500, 2012.
[12] K. Qin, V. L. Huang and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” Evolutionary Computation, IEEE Transactions on, vol. 13, no. 2, pp. 398-417, 2009.
[13] R. Storn and K. V. Price, “Minimizing the real functions of the ICEC’96 contest by differential evolution,” in International Conference on Evolutionary Computation, pp. 842-844, 1996.
[14] P. N. Suganthan, N. Hansen, J. J. Liang, K. Deb, Y. P. Chen, A. Auger and S. Tiwari, “Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization,” KanGAL report 2005005, 2005.
[15] K. Tang, X. Yao, P. N. Suganthan, C. MacNish, Y. P. Chen, C. M. Chen and Z. Yang, “Benchmark functions for the CEC’2008 special session and competition on large scale global optimization,” Nature Inspired Computation and Applications Laboratory, USTC, China, 2007.
[16] J. Derrac, S. García, D. Molina and F. Herrera, “A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms,” Swarm and Evolutionary Computation, vol. 1, no. 1, pp. 3-18, 2011.
[17] S. Das and P. N. Suganthan, “Problem definitions and evaluation criteria for CEC 2011 competition on testing evolutionary algorithms on real world optimization problems,” Jadavpur University, Nanyang Technological University, Kolkata, 2010.
[18] J. J. Liang, B. Y. Qu, P. N. Suganthan and A. G. Hernandez-Diaz, “Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization,” Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, 2013.
[19] J. J. Liang, B. Y. Qu and P. N. Suganthan, “Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization,” Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 2013.
[20] T. Basar, G. J. Olsder, G. J. Clsder, T. Basar, T. Baser and G. J. Olsder, Dynamic noncooperative game theory, vol. 200, London: Academic press, 1995.
[21] J. W. Herrmann, “A genetic algorithm for minimax optimization problems,” in Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on, IEEE, 1999.
[22] Y. Shi and R. A. Krohling, “Co-evolutionary particle swarm optimization to solve min-max problems,” in Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on, IEEE, 2002.
[23] J. Hur, H. Lee and M. J. Tahk, “Parameter robust control design using bimatrix co-evolution algorithms,” Engineering Optimization, vol. 35, no. 4, pp. 417-426, 2003.
[24] M. T. Jensen, “A new look at solving minimax problems with coevolutionary genetic algorithms,” in Metaheuristics: Computer Decision-Making, Springer US, pp. 369-384, 2004.
[25] M. Cramer, S. D. Sudhoff and E. L. Zivi, “Evolutionary algorithms for minimax problems in robust design,” Evolutionary Computation, IEEE Transactions on, vol. 13, no. 2, pp. 444-453, 2009.
[26] B. Lu, Y. Cao, M. jie Yuan and J. Zhou, “Reference variable methods of solving min–max optimization problems,” Journal of Global Optimization, vol. 42, no. 1, pp. 1-21, 2008.
[27] G. A. Segundo, R. A. Krohling and R. C. Cosme, “A differential evolution approach for solving constrained min–max optimization problems,” Expert Systems with Applications, vol. 39, no. 11, pp. 13440-13450, 2012.
[28] S. M. Guo, C. C. Yang, H. Y. Chang, and J. S. H. Tsai, “Constraint-activated differential evolution for constrained min–max optimization problems: Theory and methodology,” Expert Systems with Applications, vol. 42, no. 3, pp. 1626-1636, 2015.
[29] E. Mezura-Montes, J. Velázquez-Reyes and C. A. C. Coello, “A comparative study of differential evolution variants for global optimization,” in Proceedings of the 8th annual conference on Genetic and evolutionary computation, ACM, pp. 485-492, 2006.
[30] T. Takahama and S. Sakai, “Constrained optimization by the ε constrained differential evolution with an archive and gradient-based mutation,” in Evolutionary Computation (CEC), 2010 IEEE Congress on, IEEE, 2010.
[31] S. Das, R. Mallipeddi, and D. Maity, “Adaptive evolutionary programming with p-best mutation strategy,” Swarm and Evolutionary Computation, vol. 9, pp. 58-68, 2013.
[32] M. Á. Sainz, P. Herrero, J. Armengol and J. Vehí, “Continuous minimax optimization using modal intervals,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 18-30, 2008.