An efficient gradient method with approximate optimal stepsize for the strictly convex quadratic minimization problem

Abstract

In this paper, a new type of stepsize, approximate optimal stepsize, for gradient method is introduced to interpret the Barzilai–Borwein (BB) method, and an efficient gradient method with an approximate optimal stepsize for the strictly convex quadratic minimization problem is presented. Based on a multi-step quasi-Newton condition, we construct a new quadratic approximation model to generate an approximate optimal stepsize. We then use the two well-known BB stepsizes to truncate it for improving numerical effects and treat the resulted approximate optimal stepsize as the new stepsize for gradient method. We establish the global convergence and R-linear convergence of the proposed method. Numerical results show that the proposed method outperforms some well-known gradient methods.

Keywords:

• Barzilai–Borwein (BB) method
• BFGS update formula
• approximating optimal stepsize
• strictly convex quadratic minimization
• gradient method

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by National Science Foundation of China [grant number 11461021], [grant number 11601012]; National Science Foundation of Shangxi [grant number 2017JM1014]; National Science Foundation of Guangxi [grant number 2014GXNSFAA118028], [grant number 2015GXNSFAA139011]; Scientific Research Foundation of Guangxi Education Department [grant number 2013YB236]; Scientific Research Project of Hezhou University [grant number 2014YBZK06], [grant number 2016HZXYSX03]. Guangxi Colleges and Universities Key Laboratory of Symbolic Computation and Engineering Data Processing (FH201701).