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Superlinearly Convergent Exact Penalty Projected Structured Schemes for Constrained Nonlinear Least Squares

Ansari, Mohammad Reza | 2012

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 43141 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezamoddin
  7. Abstract:
  8. We present two projected structured algorithms for solving nonlinearly constrained nonlinear least squares problems. The first algorithm makes use of a line search scheme and the second algorithm utilizes a combined trust region-line search scheme. These approaches are based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method of Coleman and Conn for nonlinearly constrained optimization problems. The structured adaptation also makes use of the ideas of Nocedal and Overton for handling quasi-Newton updates of projected structured Hessians and appropriates the structuring scheme of Dennis, Martinez and Tapia. For robustness of the first algorithm, we present a new specific nonsmooth line search strategy, taking account of the least squares objective. We stablish its global and local two-step Q-superlinear convergence. For robustness of the second algorithm, we propose a new penalty parameter updating strategy and a specific line search strategy within the trust region, taking account of the least squares objective and special structured considerations for the approximate projected least squares Hessians. Our added usage of a second order correction step in the global phase, combined trust region-line search strategy,and, of course, the new adaptive penalty parameter updating strategy help in speeding up the global iterations in reaching the asymptotic phase confidently. For both algorithms, we discuss the details of our implementation and, using the performance profile proposed by Dolan and Moré, provide comparative results of the testing of our programs and three nonlinear programming codes from KNITRO on test problems (both small and large residuals) from Hock and Schittkowski, Lukšan and Vlček, Biegler, Nocedal, Schmid and Ternet and some randomly generated ones due to Bartels and Mahdavi-Amiri. The numerical esults obtained by our proposed algorithms show a clear local two-step superlinear convergence rate for both algorithms. Also, the results obtained by our algorithms in comparison with the algorithm proposed by Mahdavi-Amiri and Bartels, the best methods tested in the collection of Hock and Schittkowski and three methods in KNITRO, indeed confirm the practical significance of our special considerations for the inherent structure of the least squares in the first algorithm and affirm the practical significance of our adaptive penalty update scheme, combined trust region-line search strategy, and special structured considerations for the approximate projected least squares Hessians of the second algorithm
  9. Keywords:
  10. Constrained Nonlinear Programming ; Projected Structured Hessian Update ; Global Convergence ; Exact Penalty Function ; Nonlinear Least Squares

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