Steepest Descent Method with New Step Sizes

Steepest descent method is a simple gradient method for optimization. This method has a slow convergence in heading to the optimal solution, which occurs because of the zigzag form of the steps. Barzilai and Borwein modified this algorithm so that it performs well for problems with large dimensions. Barzilai and Borwein method results have sparked a lot of research on the method of steepest descent, including alternate minimization gradient method and Yuan method. Inspired by previous works, we modified the step size of the steepest descent method. We then compare the modification results against the Barzilai and Borwein method, alternate minimization gradient method and Yuan method for quadratic function cases in terms of the iterations number and the running time. The average results indicate that the steepest descent method with the new step sizes provide good results for small dimensions and able to compete with the results of Barzilai and Borwein method and the alternate minimization gradient method for large dimensions. The new step sizes have faster convergence compared to the other methods, especially for cases with large dimensions.

Authors:

• Bib Paruhum Silalahi
• Djihad Wungguli
• Sugi Guritman

Keywords:

• Convergence
• iteration
• line search
• running time
• steepest descent
• unconstrained optimization.

References:

[1] Barzilai J, Borwein JM. 1988. Two point step size gradient methods.
IMA Journal of Numerical Analysis, 8: 141-148.
[2] Bazara MS, Sherali HD, Shetty CM. 2006. Nonlinear Programming:
Theory and Algorithms. USA: Wiley-Interescience.
[3] Cauchy A. 1847. General method for solving simultaneous equations
Systems, Comp. Rend. Sci. Paris, 25: 46-89
[4] Dai YH, Yuan Y. 2003. Alternate minimization gradient method. IMA
Journal of Numerical Analysis, 23: 377-393.
[5] Griva I, Nash SG, Sofer A. 2009. Linear and Nonlinear Optimization.
USA: Society for Industrial and Applied Mathematics.
[6] Sun Wenyu, Yuan Y. 2006. Optimization Theory and Methods:
Nonlinear Programming. New York: Spinger Science, Business Media.
[7] Yuan Y. 2006. A new stepsize for the steepest descent method. Journal
of Computational Mathematics, 24:149-156.