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Gradient Methods for Large Scale Convex Quadratic Functions

In: Optimization and Regularization for Computational Inverse Problems and Applications

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  • Yaxiang Yuan

    (Chinese Academy of Sciences, State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science)

Abstract

The gradient method is one of the most simple methods for solving unconstrained optimization, it has the advantages of being easy to program and suitable for large scale problems. Different step-lengths give different gradient algorithms. In 1988, Barzilai and Borwein gave two interesting choices for the step-length and established superlinearly convergence results for two-dimensional convex quadratic problems. Barzilai and Borwein’s work triggered much research on the gradient method in the past two decades. In this chapter we investigate how the BB method can be further improved. We generalize the convergence result for the gradient method with retards. Our generalization allows more choices for the step-lengths. An intuitive analysis is given on the impact of the step-length for the speed of convergence of the gradient method. We propose a short BB step-length method. Numerical results on random generated problems are given to show that our short step technique can improve the BB method for large scale and ill-conditioned problems, particularly when high accurate solutions are needed.

Suggested Citation

  • Yaxiang Yuan, 2010. "Gradient Methods for Large Scale Convex Quadratic Functions," Springer Books, in: Yanfei Wang & Changchun Yang & Anatoly G. Yagola (ed.), Optimization and Regularization for Computational Inverse Problems and Applications, chapter 0, pages 141-155, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-13742-6_7
    DOI: 10.1007/978-3-642-13742-6_7
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