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Finding a Global Optimal Solution for a Quadratically Constrained Fractional Quadratic Problem with Applications to the Regularized Total Least Squares

Published: 01 June 2006 Publication History

Abstract

We consider the problem of minimizing a fractional quadratic problem involving the ratio of two indefinite quadratic functions, subject to a two-sided quadratic form constraint. This formulation is motivated by the so-called regularized total least squares (RTLS) problem. A key difficulty with this problem is its nonconvexity, and all current known methods to solve it are guaranteed only to converge to a point satisfying first order necessary optimality conditions. We prove that a global optimal solution to this problem can be found by solving a sequence of very simple convex minimization problems parameterized by a single parameter. As a result, we derive an efficient algorithm that produces an $\epsilon$-global optimal solution in a computational effort of $O(n^3 \log \epsilon^{-1})$. The algorithm is tested on problems arising from the inverse Laplace transform and image deblurring. Comparison to other well-known RTLS solvers illustrates the attractiveness of our new method.

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  • (2023)Coordinate descent methods for fractional minimizationProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3620104(40488-40518)Online publication date: 23-Jul-2023
  • (2022)Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applicationsJournal of Global Optimization10.1007/s10898-022-01225-085:3(743-756)Online publication date: 27-Aug-2022
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Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications  Volume 28, Issue 2
June 2006
275 pages

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 June 2006

Author Tags

  1. convex programming
  2. fractional programming
  3. nonconvex quadratic optimization
  4. regularized total least squares

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Cited By

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  • (2024)Robust battery lifetime prediction with noisy measurements via total-least-squares regressionIntegration, the VLSI Journal10.1016/j.vlsi.2023.10213696:COnline publication date: 1-May-2024
  • (2023)Coordinate descent methods for fractional minimizationProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3620104(40488-40518)Online publication date: 23-Jul-2023
  • (2022)Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applicationsJournal of Global Optimization10.1007/s10898-022-01225-085:3(743-756)Online publication date: 27-Aug-2022
  • (2021)Optimal Feature Manipulation Attacks Against Linear RegressionIEEE Transactions on Signal Processing10.1109/TSP.2021.311595169(5580-5594)Online publication date: 1-Jan-2021
  • (2021)An efficient Gauss–Newton algorithm for solving regularized total least squares problemsNumerical Algorithms10.1007/s11075-021-01145-289:3(1049-1073)Online publication date: 18-Jun-2021
  • (2019)A fast algorithm for globally solving Tikhonov regularized total least squares problemJournal of Global Optimization10.1007/s10898-018-0719-x73:2(311-330)Online publication date: 1-Feb-2019
  • (2018)Efficiently solving total least squares with Tikhonov identical regularizationComputational Optimization and Applications10.1007/s10589-018-0004-470:2(571-592)Online publication date: 1-Jun-2018
  • (2016)An Improved TR Prefilter for SSR Maximization in Indoor Wireless Communication SystemWireless Personal Communications: An International Journal10.1007/s11277-016-3407-590:3(1519-1532)Online publication date: 1-Oct-2016
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  • (2014)Copositivity and constrained fractional quadratic problemsMathematical Programming: Series A and B10.1007/s10107-013-0690-8146:1-2(325-350)Online publication date: 1-Aug-2014
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