### Coralia Cartis (University of Edinburgh)

#### Adaptive cubic overestimation methods for unconstrained optimization

*Joint work with Nick Gould and Philippe Toint.*

*Wednesday 31 October 2007 at 15.30, JCMB 5326*

##### Abstract

An Adaptive Cubic Overestimation (ACO) algorithm for unconstrained
optimization, generalizing a method due to Nesterov & Polyak
(Math. Programming 108, 2006, pp 177-205), is proposed. At
each iteration of Nesterov & Polyak's approach, the global minimizer of a
local cubic overestimator of the objective function is determined, and this
ensures a significant improvement in the objective so long as the
Hessian of the objective is Lipschitz continuous and its Lipschitz
constant is available. The twin requirements of global model optimality
and the availability of Lipschitz constants somewhat limit the
applicability of such an approach, particularly for large-scale
problems. However the promised powerful worst-case theoretical guarantees
prompt us to investigate variants in which estimates of the required
Lipschitz constant are refined and in which computationally-viable
approximations to the global model-minimizer are sought. We show that
the excellent global and local convergence properties and worst-case
iteration complexity bounds obtained by Nesterov & Polyak are retained,
and sometimes extended to a wider class of problems, by our ACO approach.
Numerical experiments with small-scale test problems from the CUTEr set show
superior performance of the ACO algorithm when compared to a trust-region
implementation.

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