一类求解非凸无约束优化问题的改进L-BFGS方法  

A Modified L-BFGS Method for Nonconvex Unconstrained Optimization

作　　者：杨博 邢宇航 刘粉干 鲁娅妮 YANG Bo;XING Yuhang;LIU Fengan;LU Yani(Troops No.95841,Jiuquan 735018,Gansu;Rocket Force University of Engineering,Xi’an 710025,Shaanxi)

机构地区：[1]95841部队,甘肃酒泉735018 [2]火箭军工程大学,陕西西安710025

出　　处：《火箭军工程大学学报》2024年第6期67-72,共6页Journal of Rocket Force University of Engineering

摘　　要：针对传统有限记忆BFGS(Limited-memory BFGS,L-BFGS)方法在求解非凸函数极小值问题时不一定全局收敛的问题,从非凸目标函数的曲率信息和方法的全局收敛性考虑,提出了一种新的带参迭代方程,并用该方程修正了L-BFGS算法;最后,对改进算法(L-MBFGS)进行了收敛性证明及数值实验验证。理论分析表明:该方法对于一般函数(可能非凸)既保证了Hessian矩阵的正定性,又具有充分下降性和全局收敛性。数值实验结果表明:相同数据规模下,L-MBFGS方法在Wood测试函数中的最优值可优于标准L-BFGS方法 1个数量级,优于ML-BFGS相似方法 2个数量级;在Dixon测试函数中的计算效率也明显优于L-BFGS方法和ML-BFGS相似方法。To solve the problem that the traditional limited-memory BFGS(L-BFGS)does not always complete global convergence in nonconvex function minimum solution,a noval param-eterized iterative equation was proposed with the consideration of the curvature information of nonconvex objective functions and the global convergence of the method.Furthermore,the pro-posed equation was used to modify the L-BFGS algorithm.Additionally,the improved algorithm(L-MBFGS)was verified by convergence and numerical experiments.Theoretical analysis indi-cated that L-MBFGS not only guarantees the posistive definiteness of the Hessian matrix for general functions(which may be non-convex),but also has sufficient descent and glob-al convergence.Numerical experimental results showed that under the same data size,the opti-mal value of the L-MBFGS in the Wood test function can be one order of magnitude better than the standard L-BFGS.Simultaneously,its computing efficiency in the Dixon test function signifi-cantly better than the standard L-BFGS and ML-BFGS similar method.

关 键 词：非凸无约束优化 改进的L-BFGS方法 全局收敛性 Wolfe线搜索准则 

分 类 号：O224[理学—运筹学与控制论]