Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares

Amir Beck, Aharon Ben-Tal, Marc Teboulle

Research output: Contribution to journalArticlepeer-review

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 optimally 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 ε-global optimal solution in a computational effort of O(n3 log ε-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.

Original languageEnglish
Pages (from-to)425-445
Number of pages21
JournalSIAM Journal on Matrix Analysis and Applications
Volume28
Issue number2
DOIs
StatePublished - 2006

Keywords

  • Convex programming
  • Fractional programming
  • Nonconvex quadratic optimization
  • Regularized total least squares

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