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A new limited memory method for unconstrained nonlinear least squares.

Morteza Kimiaei1, Arnold Neumaier1

  • 1Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria.

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Summary

A new algorithm, LMLS, offers a robust and efficient solution for large-scale least squares problems. It outperforms traditional methods by employing novel techniques for Jacobian estimation and adaptive strategies.

Keywords:
Black box least squaresLimited memory methodNon-monotone techniqueTrust region method

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Area of Science:

  • Numerical analysis
  • Optimization algorithms
  • Computational mathematics

Background:

  • Large-scale unconstrained black box least squares problems pose significant computational challenges.
  • Existing limited memory and quasi-Newton methods often struggle with efficiency and robustness for these problems.

Purpose of the Study:

  • To introduce a novel limited memory trust region algorithm, LMLS, designed for large unconstrained black box least squares problems.
  • To enhance the efficiency and robustness of solving such optimization problems.

Main Methods:

  • Development of a new limited memory trust region algorithm (LMLS).
  • Incorporation of a non-monotone technique and an adaptive radius strategy.
  • Implementation of a Broyden-like algorithm utilizing previous good points.
  • Utilizing heuristic Jacobian matrix estimation in a random subspace.

Main Results:

  • LMLS demonstrates robustness and efficiency in numerical experiments.
  • The algorithm shows superior performance compared to traditional limited memory solvers.
  • LMLS outperforms standard quasi-Newton approximation methods.

Conclusions:

  • The proposed LMLS algorithm is a viable and effective tool for large-scale least squares problems.
  • The novel features of LMLS contribute to its improved performance.
  • LMLS represents a significant advancement in optimization techniques for black box problems.