A subspace minimization conjugate gradient method based on conic model for unconstrained optimization
- 141 Downloads
In this paper, we present a new conjugate gradient method, in which the search direction is computed by minimizing a selected approximate model in a two-dimensional subspace. That is, if the objective function is not close to a quadratic, the search direction is generated by a conic model. Otherwise, a quadratic model is considered. The direction of the proposed method is proved to possess the sufficient descent property. With the modified nonmonotone line search, we establish a global convergence of the proposed method under appropriate assumptions. R-linear convergence of the proposed method is also analyzed. Numerical results using two different test function collections show that the proposed algorithm is efficient.
KeywordsConjugate gradient method Conic model Subspace minimization Nonmonotone line search Global convergence
Mathematics Subject Classification90C30 90C06 65K05
We would like to thank the anonymous referees for their useful suggestions and comments. We also would like to thank Professor Dai, Y. H. and Dr. Kou, C. X. for their CGOPT code, and thank Professor Hager, W. W. and Zhang, H. C. for their CG DESCENT (5.3) code. This research is supported by National Science Foundation of China (No. 11461021) and Shaanxi Science Foundation (No. 2017JM1014).
- Dai YH, Yuan YX (2000) Nonlinear conjugate gradient methods. Shanghai Scientific and Technical Publishers, ShanghaiGoogle Scholar
- Di S, Sun WY (1996) A trust region method for conic model to solve unconstrained optimization. Optim Methods Softw 6(4):237–263Google Scholar
- Hager WW, Zhang H (2006b) Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent. ACM Trans Math Softw 32(1):113–137Google Scholar
- Liu HW, Liu ZX (2018a) An efficient Barzilai–Borwein conjugate gradient method for unconstrained optimization. J Optim Theory Appl. https://doi.org/10.1007/s10957-018-1393-3
- Schnabel RB (1982) Conic methods for unconstrained minimization and tensor methods for nonlinear equation. In: Mathamatical Program, pp 417–438Google Scholar
- Sun WY (1996) On nonquadratic model optimization methods. Asia Pac J Oper Res 13:43–63Google Scholar
- Yuan YX (2014) A review on subspace methods for nonlinear optimization. In: Proceedings of the international congress of mathematics. Korea, pp 807–827Google Scholar
- Yuan YX, Sun WY (1997) Optimization theory and methods. Science Press, BeijingGoogle Scholar