Journal Article10.1007/S10107-006-0706-8
Cubic regularization of Newton method and its global performance
Yurii Nesterov,Boris T. Polyak +1 more
1.2K
TL;DR: This paper provides theoretical analysis for a cubic regularization of Newton method as applied to unconstrained minimization problem and proves general local convergence results for this scheme.
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Abstract: 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.
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Citations
Backtracking New Q-Newton's method: a good algorithm for optimization and solving systems of equations
TL;DR: This paper resolves convergence issues encountered by Newton’s method while retaining the quick rate of convergence for Newton's method, and develops a family of such methods, for general second order methods, some of them having the favour of quasi-Newton's methods.
3
•Posted Content
A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints
TL;DR: A computationally efficient method to solve non-convex programming problems with linear equality constraints is presented, based on a recursively feasible and descending sequential convex programming procedure proven to converge to a locally optimal solution.
Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method.
Nikita Doikov,Yurii Nesterov +1 more
TL;DR: In this article, the authors study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter.
An Active Set Trust-Region Method for Bound-Constrained Optimization
TL;DR: An active set trust-region algorithm for bound-constrained optimization problems using a sufficient descent condition to identify whether the function value is reduced or not and a critical measure is used which is computationally better than the other known critical measures.
No Spurious Solutions in Non-convex Matrix Sensing: Structure Compensates for Isometry
Igor Molybog,Somayeh Sojoudi,Javad Lavaei +2 more
- 25 May 2021
TL;DR: In this article, the kernel structure property (KSPP) is used to explain the non-existence of spurious local minima in low-rank matrix sensing under an incoherence assumption.
3
References
A method for the solution of certain non – linear problems in least squares
TL;DR: In this article, the problem of least square problems with non-linear normal equations is solved by an extension of the standard method which insures improvement of the initial solution, which can also be considered an extension to Newton's method.
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Roger Fletcher
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TL;DR: The aim of this book is to provide a Discussion of Constrained Optimization and its Applications to Linear Programming and Other Optimization Problems.
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Numerical methods for unconstrained optimization and nonlinear equations
John E. Dennis,Robert B. Schnabel +1 more
- 01 Mar 1983
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
8.2K