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Cubic regularization of Newton method and its global performance. (English) Zbl 1142.90500
Summary: In this paper, we provide theoretical analysis for a cubic regularization of Newton method as applied to unconstrained minimization problem. For this scheme, we prove general local convergence results. However, the main contribution of the paper is related to global worst-case complexity bounds for different problem classes including some nonconvex cases. It is shown that the search direction can be computed by standard linear algebra technique.

90C53Methods of quasi-Newton type
90C30Nonlinear programming
[1]Bennet, A.A.: Newton’s method in general analysis. Proc. Nat. Ac. Sci. USA. 2 (10), 592–598 (1916) · doi:10.1073/pnas.2.10.592
[2]Conn, A.B., Gould, N.I.M., Toint, Ph.L.: Trust Region Methods. SIAM, Philadelphia, 2000
[3]Dennis, J.E., Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, 1996
[4]Fletcher, R.: Practical Methods of Optimization, Vol. 1, Unconstrained Minimization. John Wiley, NY, 1980
[5]Goldfeld, S., Quandt, R., Trotter, H.: Maximization by quadratic hill climbing. Econometrica. 34, 541–551 (1966) · Zbl 0145.40901 · doi:10.2307/1909768
[6]Kantorovich, L.V.: Functional analysis and applied mathematics. Uspehi Matem. Nauk. 3 (1), 89–185 (1948), (in Russian). Translated as N.B.S. Report 1509, Washington D.C. (1952)
[7]Levenberg, K.: A method for the solution of certain problems in least squares. Quart. Appl. Math. 2, 164–168 (1944)
[8]Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431–441 (1963) · Zbl 0112.10505 · doi:10.1137/0111030
[9]Nemirovsky, A., Yudin, D.: Informational complexity and efficient methods for solution of convex extremal problems. Wiley, New York, 1983
[10]Nesterov, Yu.: Introductory lectures on convex programming: a basic course. Kluwer, Boston, 2004
[11]Nesterov, Yu., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia, 1994
[12]Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, NY, 1970
[13]Polyak, B.T.: Gradient methods for minimization of functionals. USSR Comp. Math. Math. Phys. 3 (3), 643–653 (1963)
[14]Polyak, B.T.: Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theory and Appl. 99 (3), 553–583 (1998) · Zbl 0961.90074 · doi:10.1023/A:1021798932766