求解区间优化问题的一种谱梯度法
A Kind of Spectral Gradient Method for Interval-Valued Optimization Problems
摘要: 本文主要研究区间约束优化问题的求解算法。基于广义Hukuhara导数的Karush-Kuhn-Tucker条件，结合互补结构，利用Fischer-Burmeister函数，将原区间约束优化问题转化为无约束优化问题，并给出一种谱梯度法求解转化后的问题。最后，数值实验验证了算法的有效性。
Abstract: This paper concentrates on designing an algorithm for solving interval constrained optimization problems. Based on the Karush-Kuhn-Tucker conditions utilizing generalized Hukuhara derivatives, and incorporating complementary structure, the Fischer-Burmeister function is employed to transform the original interval constrained optimization problem into an unconstrained one. A spectral gradient method for solving the transformed problem is presented. Numerical experiments verify the effectiveness of this algorithm.
文章引用：郭二威. 求解区间优化问题的一种谱梯度法[J]. 应用数学进展, 2025, 14(7): 258-268. https://doi.org/10.12677/aam.2025.147362

参考文献

[1] Pedroni, N. (2022) Computational Methods for the Robust Optimization of the Design of a Dynamic Aerospace System in the Presence of Aleatory and Epistemic Uncertainties. Mechanical Systems and Signal Processing, 164, Article 108206. [Google Scholar] [CrossRef] 
[2] Wu, Q., Liu, X., Qin, J. and Zhou, L. (2021) Multi-Criteria Group Decision-Making for Portfolio Allocation with Consensus Reaching Process under Interval Type-2 Fuzzy Environment. Information Sciences, 570, 668-688. [Google Scholar] [CrossRef] 
[3] Safi, M.R. and Razmjoo, A. (2013) Solving Fixed Charge Transportation Problem with Interval Parameters. Applied Mathematical Modelling, 37, 8341-8347. [Google Scholar] [CrossRef] 
[4] Inuiguchi, M. and Sakawa, M. (1995) Minimax Regret Solution to Linear Programming Problems with an Interval Objective Function. European Journal of Operational Research, 86, 526-536. [Google Scholar] [CrossRef] 
[5] Gong, D.W., Qin, N.N. and Sun, X.Y. (2010) Evolutionary Algorithms for Multi-Objective Optimization Problems with Interval Parameters. 2010 IEEE Fifth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA), Changsha, 23-26 September 2010, 411-420. [Google Scholar] [CrossRef] 
[6] Ishibuchi, H. and Tanaka, H. (1990) Multiobjective Programming in Optimization of the Interval Objective Function. European Journal of Operational Research, 48, 219-225. [Google Scholar] [CrossRef] 
[7] Wu, H. (2007) The Karush-Kuhn-Tucker Optimality Conditions in an Optimization Problem with Interval-Valued Objective Function. European Journal of Operational Research, 176, 46-59. [Google Scholar] [CrossRef] 
[8] Wu, H.C. (2008) Wolfe Duality for Interval-Valued Optimization. Journal of Optimization Theory and Applications, 138, 497-509. [Google Scholar] [CrossRef] 
[9] Wu, H. (2009) The Karush-Kuhn-Tucker Optimality Conditions in Multiobjective Programming Problems with Interval-Valued Objective Functions. European Journal of Operational Research, 196, 49-60. [Google Scholar] [CrossRef] 
[10] Sun, Y.H. and Wang, L.S. (2013) Optimality Conditions and Duality in Nondifferentiable Interval-Valued Programming. Journal of Industrial & Management Optimization, 9, 131-142. [Google Scholar] [CrossRef] 
[11] Villanueva, F.R., de Oliveira, V.A. and Costa, T.M. (2023) Optimality Conditions for Interval Valued Optimization Problems. Fuzzy Sets and Systems, 454, 38-55. [Google Scholar] [CrossRef] 
[12] Barzilai, J. and Borwein, J.M. (1988) Two-Point Step Size Gradient Methods. IMA Journal of Numerical Analysis, 8, 141-148. [Google Scholar] [CrossRef] 
[13] Raydan, M. (1993) On the Barzilai and Borwein Choice of Steplength for the Gradient Method. IMA Journal of Numerical Analysis, 13, 321-326. [Google Scholar] [CrossRef] 
[14] Dai, Y.H.J. (2002) R-Linear Convergence of the Barzilai and Borwein Gradient Method. IMA Journal of Numerical Analysis, 22, 1-10. [Google Scholar] [CrossRef] 
[15] Xiao, Y., Wang, Q. and Wang, D. (2010) Notes on the Dai-Yuan-Yuan Modified Spectral Gradient Method. Journal of Computational and Applied Mathematics, 234, 2986-2992. [Google Scholar] [CrossRef] 
[16] Wu, H. (2007) The Karush-Kuhn-Tucker Optimality Conditions for the Optimization Problem with Fuzzy-Valued Objective Function. Mathematical Methods of Operations Research, 66, 203-224. [Google Scholar] [CrossRef] 
[17] Stefanini, L. and Arana-Jiménez, M. (2019) Karush-Kuhn-Tucker Conditions for Interval and Fuzzy Optimization in Several Variables under Total and Directional Generalized Differentiability. Fuzzy Sets and Systems, 362, 1-34. [Google Scholar] [CrossRef] 
[18] Mangasarian, O.L. and Fromovitz, S. (1967) The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints. Journal of Mathematical Analysis and Applications, 17, 37-47. [Google Scholar] [CrossRef] 
[19] Fischer, A. (1992) A Special Newton-Type Optimization Method. Optimization, 24, 269-284. [Google Scholar] [CrossRef] 
[20] 周群艳. 解无约束优化问题的一种新的谱梯度方法(英文) [J]. 应用数学, 2020, 33(4): 857-868.
[21] Dennis Jr, J.E. and Wolkowicz, H. (1993) Sizing and Least-Change Secant Methods. SIAM Journal on Numerical Anal-ysis, 30, 1291-1314. [Google Scholar] [CrossRef] 
[22] Yuan, Y. (1991) A Modified BFGS Algorithm for Unconstrained Optimization. IMA Journal of Numerical Analysis, 11, 325-332.

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