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一元DC复合优化问题的最优性条件
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Abstract:
约束优化问题在自动控制、图像处理、水处理、网络分析、工程设计中有着十分重要的应用，实际生活中的许多问题在一定条件下都可以看作或者转化为一个约束优化问题，因此约束优化问题的研究具有非常重要的意义。本文将在函数不一定具有连续性，集合不一定是闭集的情形下，利用函数上图、次微分性质和凸化技巧，引入新的约束规范条件，对一元DC复合约束优化问题进行研究。从而建立了一元DC复合优化问题的局部和全局最优性条件的充分和必要条件，推广了前人的结论。
Constrained optimization problems have very important applications in automatic control, image processing, water treatment, network analysis, and engineering design. Many problems in real life can be seen as or transformed into a constrained optimization problem under certain conditions, so the study of constrained optimization problems is of great significance. This article will study the univariate DC composite constrained optimization problem by utilizing the epigraph of functions, subdifferential properties, and convexification techniques in situations where functions may not be continuous and sets may not be closed. By introducing new constrained standard conditions, this research establishes the sufficient and necessary conditions for local and global optimality of the univariate DC composite optimization problem, thus extending the conclusions of predecessors.
| [1] | Fang, D.H., Li, C. and Ng, K.F. (2010) Constraint Qualifications for Optimality Conditions and Total Lagrange Dualities in Convex Infinite Programming. Nonlinear Analysis: Theory, Methods & Applications, 73, 1143-1159. https://doi.org/10.1016/j.na.2010.04.020 |
| [2] | Bo？, R.I., Grad, S. and Wanka, G. (2008) New Regularity Conditions for Strong and Total Fenchel-Lagrange Duality in Infinite Dimensional Spaces. Nonlinear Analysis: Theory, Methods & Applications, 69, 323-336. https://doi.org/10.1016/j.na.2007.05.021 |
| [3] | Long, X., Sun, X. and Peng, Z. (2016) Approximate Optimality Conditions for Composite Convex Optimization Problems. Journal of the Operations Research Society of China, 5, 469-485. https://doi.org/10.1007/s40305-016-0140-4 |
| [4] | Fang, D. and Zhang, Y. (2019) Optimality Conditions and Total Dualities for Conic Programming Involving Composite Function. Optimization, 69, 305-327. https://doi.org/10.1080/02331934.2018.1561695 |
| [5] | Li, G. and Zhou, Y. (2015) The Stable Farkas Lemma for Composite Convex Functions in Infinite Dimensional Spaces. Acta Mathematicae Applicatae Sinica, English Series, 31, 677-692. https://doi.org/10.1007/s10255-015-0493-1 |
| [6] | Fang, D.H., Ansari, Q.H. and Zhao, X.P. (2018) Constraint Qualifications and Zeroduality Gap Properties in Conical Programming Involving Composite Functions. Journal of Nonlinear and Convex Analysis, 19, 53-69. |
| [7] | Dinh, N., Nghia, T.T.A. and Vallet, G. (2010) A Closedness Condition and Its Applications to DC Programs with Convex Constraints. Optimization, 59, 541-560. https://doi.org/10.1080/02331930801951348 |
| [8] | Fang, D.H. and Zhao, X.P. (2014) Local and Global Optimality Conditions for DC Infinite Optimization Problems. Taiwanese Journal of Mathematics, 18, 817-834. https://doi.org/10.11650/tjm.18.2014.3888 |
| [9] | Mordukhovich, B.S. (2006) Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer-Verlag. |
| [10] | Zalinescu, C. (2002) Convex Analysis in General Vector Spaces. World Scientific. |
| [11] | Li, C. and Ng, K.F. (2003) Constraint Qualification, the Strong CHIP, and Best Approximation with Convex Constraints in Banach Spaces. SIAM Journal on Optimization, 14, 584-607. https://doi.org/10.1137/s1052623402415846 |
