The Tikhonov identical regularized total least squares (TI) is to deal with the ill-conditioned system of linear equations where the data are contaminated by noise. A standard approach for (TI) is to reformulate it as a problem of finding a zero point of some decreasing concave non-smooth univariate function such that the classical bisection search and Dinkelbach’s method can be applied. In this paper, by exploring the hidden convexity of (TI), we reformulate it as a new problem of finding a zero point of a strictly decreasing, smooth and concave univariate function. This allows us to apply the classical Newton’s method to the reformulated problem, which converges globally to the unique root with an asymptotic quadratic convergence rate. Moreover, in every iteration of Newton’s method, no optimization subproblem such as the extended trust-region subproblem is needed to evaluate the new univariate function value as it has an explicit expression. Promising numerical results based on the new algorithm are reported.
Fractional programming Quadratic programming Total least square Tikhonov regularization Bisection method Newton’s method Trust-region subproblem S-lemma
Mathematics Subject Classification
90C26 90C20 90C32
This is a preview of subscription content, log in to check access
Beck, A., Teboulle, M.: A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid. Math. Program. Ser. A. 118(1), 13–15 (2009)MathSciNetCrossRefMATHGoogle Scholar
Beck, A., Ben-Tal, A., Teboulle, M.: Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares. SIAM J. Matrix Anal. Appl. 28(2), 425–445 (2006)MathSciNetCrossRefMATHGoogle Scholar
Zhang, A., Hayashi, S.: Celis–Dennis–Tapia based approach to quadratic fractional programming problems with two quadratic constraints. Numer. Algebra Control Optim. 1(1), 83–98 (2011)MathSciNetCrossRefMATHGoogle Scholar