A FAMILY OF THE LOCAL CONVERGENCE OF THE IMPROVED SECANT METHODS FOR NONLINEAR EQUALITY CONSTRAINED OPTIMIZATION SUBJECT TO BOUNDS ON VARIABLES

• Published in: Journal of systems science and complexity Vol. 27; no. 2; pp. 307 - 326
• Main Authors: Zhang, Yong, Zhu, Detong
• Format: Journal Article
• Language: English
• Published: Beijing Academy of Mathematics and Systems Science, Chinese Academy of Sciences 01.04.2014
• Subjects: BFGS; Complex Systems; Control; Hessian矩阵; Mathematics; Mathematics and Statistics; Mathematics of Computing; Operations Research/Decision Theory; Statistics; Systems Theory; 变量范围; 可行性试验; 局部收敛性; 等式约束优化; 约束优化问题; 非线性等式; local convergence; secant methods; second order correction; Affine scaling
• ISSN: 1009-6124; 1559-7067
• DOI: 10.1007/s11424-013-0169-y

This paper studies a family of the local convergence of the improved secant methods for solving the nonlinear equality constrained optimization subject to bounds on variables. The Hessian of the Lagrangian is approximated using the DFP or the BFGS secant updates. The improved secant methods are used to generate a search direction. Combining with a suitable step size, each iterate switches to trial step of strict interior feasibility. When the Hessian is only positive definite in an affine null subspace, one shows that the algorithms generate the sequences converging q-linearly and two-step q-superlinearly. Yhrthermore, under some suitable assumptions, some sequences generated by the algorithms converge locally one-step q-superlinearly. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithms.