Newton-type methods for non-convex optimization under inexact Hessian information

P Xu, F Roosta, MW Mahoney - Mathematical Programming, 2020 - Springer
We consider variants of trust-region and adaptive cubic regularization methods for non-
convex optimization, in which the Hessian matrix is approximated. Under certain condition …

Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results

C Cartis, NIM Gould, PL Toint - Mathematical Programming, 2011 - Springer
Abstract An Adaptive Regularisation algorithm using Cubics (ARC) is proposed for
unconstrained optimization, generalizing at the same time an unpublished method due to …

Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function-and derivative-evaluation complexity

C Cartis, NIM Gould, PL Toint - Mathematical programming, 2011 - Springer
Abstract An Adaptive Regularisation framework using Cubics (ARC) was proposed for
unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program …

[图书][B] Evaluation Complexity of Algorithms for Nonconvex Optimization: Theory, Computation and Perspectives

C Cartis, NIM Gould, PL Toint - 2022 - SIAM
Do you know the difference between an optimist and a pessimist? The former believes we
live in the best possible world, and the latter is afraid that the former might be right.… In that …

On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming

C Cartis, NIM Gould, PL Toint - SIAM Journal on Optimization, 2011 - SIAM
We estimate the worst-case complexity of minimizing an unconstrained, nonconvex
composite objective with a structured nonsmooth term by means of some first-order …

A recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture

A Kopaničáková, R Krause - Computer Methods in Applied Mechanics and …, 2020 - Elsevier
The simulation of crack initiation and propagation in an elastic material is difficult, as crack
paths with complex topologies have to be resolved. Phase-field approaches allow to …

Worst case complexity of direct search

LN Vicente - EURO Journal on Computational Optimization, 2013 - Springer
In this paper, we prove that the broad class of direct-search methods of directional type
based on imposing sufficient decrease to accept new iterates shares the worst case …

An adaptive cubic regularization algorithm for nonconvex optimization with convex constraints and its function-evaluation complexity

C Cartis, NIM Gould, PL Toint - IMA Journal of Numerical …, 2012 - ieeexplore.ieee.org
The adaptive cubic regularization algorithm described in Cartis et al.(2009, Adaptive cubic
regularisation methods for unconstrained optimization. Part I: motivation, convergence and …

A Newton-MR algorithm with complexity guarantees for nonconvex smooth unconstrained optimization

Y Liu, F Roosta - arXiv preprint arXiv:2208.07095, 2022 - arxiv.org
In this paper, we consider variants of Newton-MR algorithm for solving unconstrained,
smooth, but non-convex optimization problems. Unlike the overwhelming majority of Newton …

Adaptive multilevel inexact SQP methods for PDE-constrained optimization

JC Ziems, S Ulbrich - SIAM Journal on Optimization, 2011 - SIAM
We present a class of inexact adaptive multilevel trust-region SQP methods for the efficient
solution of optimization problems governed by nonlinear PDEs. The algorithm starts with a …