二阶微分方程方法求解带不等式约束的优化问题
Second-Order Differential Equation Method for Solving Optimization Problems with Inequality Constraints
摘要: 针对只含有不等式约束的优化问题，本文首先给出了其Karush-Kuhn-Tucker (KKT)条件，并利用光滑互补函数将KKT系统转化为一类光滑的方程组问题；其次，将光滑方程组问题转化为无约束优化问题；最后，本文提出一类二阶微分方程系统求解无约束优化问题，并讨论了二阶微分方程系统的解的稳定性及收敛速度。
Abstract: For optimization problems with only inequality constraints, this paper first presents their Karush-Kuhn-Tucker (KKT) conditions, and uses smooth complementarity functions to transform the KKT system into a class of smooth system of equations problems. Secondly, this article transforms the problem of smooth equation systems into an unconstrained optimization problem. Finally, this article proposes a class of second-order differential equation systems for solving unconstrained optimization problems, and discusses the stability and convergence speed of the solutions of second-order differential equation systems.
文章引用：李思怡, 姜莹, 宁文琪, 任泓燃. 二阶微分方程方法求解带不等式约束的优化问题[J]. 理论数学, 2024, 14(12): 1-6. https://doi.org/10.12677/pm.2024.1412399

参考文献

[1] 蓝倜恩. 优化技术在土建结构设计中的应用[J]. 建筑结构学报, 1981, 2(6): 12-19.
[2] 孙芳垂. 建筑结构优化设计案例分析[M]. 北京: 中国建筑工业出版社, 2011.
[3] 常剑峰, 钟约先, 韩赞东. 制造系统中能力约束下的生产批量计划优化方法[J]. 清华大学学报: 自然科学版, 2004, 44(5): 605-608.
[4] 肖依永, 常文兵, 张人千. 能力与资源双重约束下的启发式组合生产计划研究[J]. 中国管理科学, 2008, 16(6): 33-40.
[5] 周珩. RICU异常行为患者身体约束优化流程应用效果观察[J]. 护理学, 2022, 11(2): 213-217.
[6] 钱立泽. 树种算法改进及其求解实际约束优化问题应用[D]: [硕士学位论文]. 长春: 吉林财经大学, 2022.
[7] 黄文, 陈建恒, 郭媛媛, 等. 黎曼流形上的无约束优化及其应用[J]. 厦门大学学报(自然科学版), 2023, 62(6): 1012-1038.
[8] Arrow, K.J. and Hurwicz, L. (1956) Reduction of Constrained Maxima to Saddle-Point Problems. Berkeley Symposium on Mathematical Statistics and Probability, 3, 1-20.
[9] Antipin, A.S. (2000) Solving Variational Inequalities with Coupling Constraints with the Use of Differential Equations. Differential Equations, 36, 1587-1596. [Google Scholar] [CrossRef] 
[10] Attouch, H., Chbani, Z. and Riahi, H. (2019) Fast Proximal Methods via Time Scaling of Damped Inertial Dynamics. SIAM Journal on Optimization, 29, 2227-2256. [Google Scholar] [CrossRef] 
[11] Attouch, H. and Cabot, A. (2017) Asymptotic Stabilization of Inertial Gradient Dynamics with Time-Dependent Viscosity. Journal of Differential Equations, 263, 5412-5458. [Google Scholar] [CrossRef] 
[12] Extushenko, Y. (1974) Two Numerical Methods of Solving Nonlinear Programming. Doklady Mathematics, 15, 420-423.
[13] Evtushenko, Y.G. and Zhadan, V.G. (1973) Numerical Methods of Solving Some Operational Research Problems. USSR Computational Mathematics and Mathematical Physics, 13, 56-77. [Google Scholar] [CrossRef] 
[14] Evtushenko, Y.G. and Zhadan, V.G. (1977) A Relaxation Method for Solving Problems of Non-Linear Programming. USSR Computational Mathematics and Mathematical Physics, 17, 73-87. [Google Scholar] [CrossRef] 
[15] Sun, J., Fu, W., Alcantara, J.H. and Chen, J. (2021) A Neural Network Based on the Metric Projector for Solving SOCCVI Problem. IEEE Transactions on Neural Networks and Learning Systems, 32, 2886-2900. [Google Scholar] [CrossRef] [PubMed]

为你推荐