  Derivative-free optimization (DFO) consists in finding the best value of an
objective function without relying on derivatives. To tackle such problems, one
may build approximate derivatives, using for instance finite-difference
estimates. One may also design algorithmic strategies that perform space
exploration and seek improvement over the current point. The first type of
strategy often provides good performance on smooth problems but at the expense
of more function evaluations. The second type is cheaper and typically handles
non-smoothness or noise in the objective better. Recently, full-low evaluation
methods have been proposed as a hybrid class of DFO algorithms that combine
both strategies, respectively denoted as Full-Eval and Low-Eval. In the
unconstrained case, these methods showed promising numerical performance.
  In this paper, we extend the full-low evaluation framework to bound and
linearly constrained derivative-free optimization. We derive convergence
results for an instance of this framework, that combines finite-difference
quasi-Newton steps with probabilistic direct-search steps. The former are
projected onto the feasible set, while the latter are defined within tangent
cones identified by nearby active constraints. We illustrate the practical
performance of our instance on standard linearly constrained problems, that we
adapt to introduce noisy evaluations as well as non-smoothness. In all cases,
our method performs favorably compared to algorithms that rely solely on
Full-eval or Low-eval iterations.
