Cubic regularization of Newton method and its global performance
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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.
KeywordsGeneral nonlinear optimization Unconstrained optimization Newton method Trust-region methods Global complexity bounds Global rate of convergence
Mathematics Subject Classification (1991)49M15 49M37 58C15 90C25 90C30
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- 1.Bennet, A.A.: Newton's method in general analysis. Proc. Nat. Ac. Sci. USA. 2 (10), 592–598 (1916)Google Scholar
- 2.Conn, A.B., Gould, N.I.M., Toint, Ph.L.: Trust Region Methods. SIAM, Philadelphia, 2000Google Scholar
- 3.Dennis, J.E., Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, 1996Google Scholar
- 4.Fletcher, R.: Practical Methods of Optimization, Vol. 1, Unconstrained Minimization. John Wiley, NY, 1980Google Scholar
- 9.Nemirovsky, A., Yudin, D.: Informational complexity and efficient methods for solution of convex extremal problems. Wiley, New York, 1983Google Scholar
- 10.Nesterov, Yu.: Introductory lectures on convex programming: a basic course. Kluwer, Boston, 2004Google Scholar
- 11.Nesterov, Yu., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia, 1994Google Scholar
- 12.Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, NY, 1970Google Scholar
- 13.Polyak, B.T.: Gradient methods for minimization of functionals. USSR Comp. Math. Math. Phys. 3 (3), 643–653 (1963)Google Scholar