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New Conjugate Gradient Methods for Unconstrained Optimization

| 2010

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 40752 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezamoddin
  7. Abstract:
  8. We discuss conjugate gradient methods for which both the gradient and func-tion values are considered in computing the conjugate gradient parameter. We pro-pose new conjugate gradient methods as members of Dai-Liao’s family of conjugate gradient methods and Andrei’s family of hybrid conjugate gradient methods. For computing the conjugate gradient parameter in our methods, three modified secant equations proposed by Zhang, Deng and Chen, Li and Fukushima, and Yuan are used. It is shown that under proper conditions, three of the proposed methods are globally convergent for uniformly convex functions and two other methods are glob-ally convergent for general functions. It is also shown that under proper conditions on strictly convex functions and using a nonmonotone line seach, three of our pro-posed methods are globally convergent. To enhance the performance of the line search procedure for conjugate gradient methods, we propose a new approach for computing the initial value of the step-length in initiating the line search procedure. Using the Kruscal-Wallis and Friedman’s non-parametric stochastical tests and the performance profile proposed by Dolan and Mor´e, we make an extensive comparative analysis of the outputs obtained by the implementations of our algorithms and other efficient representative conjugate gradient algorithms on a set of unconstrained op-timization test problems from the CUTEr collection. Comparison results show that our proposed methods in the family of Dai-Liao’s methods, specially the proposed method based on Yuan’s modified secant equation, are more efficient than the other considered methods in this family. In the family of Andrei’s hybrid methods, our proposed method based on the Li-Fukushima’s modified secant equation outperforms the other considered methods in this family, and another proposed method, that is based on Yuan’s modified secant equation, is also competitive and at times even more efficient than the other considered methods in this family
  9. Keywords:
  10. Unconstrainted Optimization ; Line Search ; Global Convergence ; Conjuagate Gradient Method ; Modified Secant Condition

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