Abstract
The problem of minimizing the sum of a convex quadratic function and the ratio of two quadratic functions can be reformulated as a Celis–Dennis–Tapia (CDT) problem and, thus, according to some recent results, can be polynomially solved. However, the degree of the known polynomial approaches for these problems is fairly large and that justifies the search for efficient local search procedures. In this paper the CDT reformulation of the problem is exploited to define a local search algorithm. On the theoretical side, its convergence to a stationary point is proved. On the practical side it is shown, through different numerical experiments, that the main cost of the algorithm is a single Schur decomposition to be performed during the initialization phase. The theoretical and practical results for this algorithm are further strengthened in a special case.
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References
Beck, A., Ben-Tal, A.: On the solution of the Tikhonov regularization of the total least squares problem. SIAM J. Optim. 17(1), 98–118 (2006)
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)
Beck, A., Teboulle, M.: A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid. Math. Program. 118, 13–35 (2009)
Beck, A., Teboulle, M.: On minimizing quadratically constrained ratio of two quadratic functions. J. Convex Anal. 17, 789–804 (2010)
Benson, H.P.: Global optimization of nonlinear sums of ratios. J. Math. Anal. Appl. 263, 301–315 (2001)
Benson, H.P.: Global optimization algorithm for the nonlinear sum of ratios problem. J. Optim. Theory Appl. 112, 1–29 (2002)
Benson, H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problems. J. Glob. Optim. 22, 343–364 (2002)
Benson, H.P.: Solving sum of ratios fractional programs via concave minimization. J. Optim. Theory Appl. 135, 1–17 (2007)
Ben-Tal, A., den Hertog, D.: Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math. Program. 143, 1–29 (2014)
Bienstock, D.: A note on polynomial solvability of the CDT problem. SIAM J. Optim. 26(1), 488–498 (2016)
Celis, M.R., Dennis, J.E., Tapia, R.A.: A trust region strategy for nonlinear equality constrained optimization. In: Boggs, P.T., Byrd, R.H., Schnabel, R.B. (eds.) Numerical Optimization, pp. 71–82. SIAM, Philadelphia (1985)
Consolini, L., Locatelli, M.: On the complexity of quadratic programming with two quadratic constraints. Math. Program. 164(1–2), 91–128 (2017)
Depetrini, D., Locatelli, M.: Approximation of linear fractional/multiplicative problems. Math. Program. 128, 437–443 (2011)
Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)
Fierro, R.D., Golub, G.H., Hansen, P.C., O’Leary, D.P.: Regularization by truncated total least squares. SIAM J. Sci. Comput. 18(4), 1223–1241 (1997)
Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. J. Glob. Optim. 19, 83–102 (2001)
Golub, G.H., Hansen, P.C., O’Leary, D.P.: Tikhonov regularization and total least squares. SIAM J. Numer. Anal. 21, 185–194 (1999)
Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993)
Hansen, P.C.: Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6(1), 1–35 (1994)
Hansen, P.C.: Deconvolution and regularization with Toeplitz matrices. Numer. Algorithms 29, 323–378 (2002)
Konno, H., Fukaishi, K.: A branch and bound algorithm for solving low rank linear multiplicative and fractional programming problems. J. Glob. Optim. 18, 283–299 (2000)
Kuno, T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. J. Glob. Optim. 22, 155–174 (2002)
Lampe, J., Voss, H.: Large-scale Tikhonov regularization of total least squares. J. Comput. Appl. Math. 238, 95–108 (2013)
Locatelli, M.: Alternative branching rules for some nonconvex problems. Optim. Methods Softw. 30(2), 365–378 (2015)
Lu, S., Pereverzev, S.V., Tautenhahn, U.: Regularized total least squares: computational aspects and error bounds. SIAM J. Matrix Anal. Appl. 31(3), 918–941 (2009)
Matsui, T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9, 113–119 (2006)
Mittal, S., Schulz, A.S.: An FPTAS for optimizing a class of low-rank functions over a polytope. Math. Program. 141, 103–120 (2013)
Nguyen, V.-B., Sheu, R.-L., Xia, Y.: An SDP approach for quadratic fractional problems with a two-sided quadratic constraint. Optim. Methods Softw. 31(4), 701–719 (2016)
Sakaue, S., Nakatsukasa, Y., Takeda, A., Iwata, S.: Solving generalized CDT problems via two-parameter eigenvalues. SIAM J. Optim. 26(3), 1669–1694 (2016)
Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, vol. 1. Kluwer Academic Publishers, Berlin (1995)
Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18, 219–229 (2003)
Sima, D.M., Van Huffel, S., Golub, G.H.: Regularied total least squares based on quadratic eigenvalue problem solvers. BIT Numer. Math. 44, 793–812 (2004)
Uhlig, F.: Definite and semidefinite matrices in a real symmetric matrix pencil. Pac. J. Math. 49, 561–568 (1973)
Wang, Y.-J., Zhang, K.-C.: Global optimization of nonlinear sum of ratios problem. Appl. Math. Comput. 158(2), 319–330 (2004)
Yang, M., Xia, Y., Wang, J., Peng, J.: Efficiently solving total least squares with Tikhonov identical regularization. Comput. Optim. Appl. 70, 571–592 (2018)
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This research was supported by NSFC under Grants 11571029, 11471325 and 11771056.
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Consolini, L., Locatelli, M., Wang, J. et al. Efficient local search procedures for quadratic fractional programming problems. Comput Optim Appl 76, 201–232 (2020). https://doi.org/10.1007/s10589-020-00175-1
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DOI: https://doi.org/10.1007/s10589-020-00175-1