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Conjugate Residual Method for Large Scale Unconstrained Nonlinear Optimization

Siyadati, Maryam | 2023

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 56302 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezam
  7. Abstract:
  8. Nowadays, solving large-scale unconstrained optimization problems has wide applications in data science and machine learning. Therefore, the development and analysis of efficient algorithms for solving unconstrained optimization problems is of great interest. Line search and trust region are two general frameworks for guaranteeing the convergence of algorithms for solving unconstrained optimization problems. Conjugate gradient (CG) methods and the conjugate residual (CR) balance by Hestenes and Stiefel, have been presented for solving linear systems with symmetric and positive definite coefficient matrices. The basic feature of CR, that is, residual minimization, is important and can be used in imprecise Newton methods and in the working framework of line search to solve unconstrained optimization problems. On the other hand, one can use CR for the imprecise solution of subproblems of the trust region. Actually, CR leads to a uniform reduction of the convex quadratic models. In order to apply CR, even in the presence of negative curvature, some corrections and changes are necessary to calculate the corrective direction of the iterative methods for solving unconstrained optimization problems. Recently, Dahito and Urban showed how to apply CR in algorithms for solving large-scale unconstrained optimization problems. Additionally, using CR, they presented an efficient confidence region algorithm for solving large-scale unconstrained nonlinear least squares problems. Here, we study, analyze and implement the algorithms presented by Dahito and Urban and report the test results obtained by execution of our written programs
  9. Keywords:
  10. Unconstrainted Optimization ; Line Search ; Trust Region ; Conjuagate Gradient Method ; Inexact Newton Method ; Trust-Region Newton Methods ; Conjugate Residual Method

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