## SEMINARIO: Recent Progress on Derivative-Free Trust-Region Methods

Trust-region methods are a broad class of methods for continuous

optimization that finds application in a variety of problems

and contexts. The basic principle consists of iteratively optimizing

a model of the objective function in a restricted region. In particular,

they have been studied and applied for problems without using derivatives,

where models are built solely by sampled function values.

Trust-region derivative-free methods are guaranteed to

converge for deterministic smooth functions, in the sense

of generating a sequence or run of iterates converging to criticality

(of first and second order type). Such a convergence is called

global as there is no assumption on the starting point.

It has also been proved that the order of complexity, or the

global rate in which criticality decays, matches the

derivative-based case.

In the deterministic non-smooth case, and given some knowledge

of the non-smooth structure, it is possible to design globally

convergent approaches with appropriated complexity, either by

smoothing the original function in some parametrized way or by

moving a compositive non-smooth structure directly to the

trust-region subproblem.

Trust-region methods can be based as well on probabilistic models

of the objective function, thus considering the derivative-free

case where sample points are randomly generated. Such methods

exhibit similar properties of convergence and complexity as those

using deterministic models, now not for all runs but for a

a set of those that occurs with probability one.

Finally, we are observing now the first attempts for the

stochastic case, where the objective function can only be

observed, and possibly approximated by sample averaging.

This talk will attempt to overview all such developments and

point out what still remains unknown.