  We develop an interior-point method for nonsmooth regularized
bound-constrained optimization problems. Our method consists of iteratively
solving a sequence of unconstrained nonsmooth barrier subproblems. We use a
variant of the proximal quasi-Newton trust-region algorithm TR of
arXiv:2103.15993v3 to solve the barrier subproblems, with additional
assumptions inspired from well-known smooth interior-point trust-region
methods. We show global convergence of our algorithm with respect to the
criticality measure of arXiv:2103.15993v3. Under an additional assumption
linked to the convexity of the nonsmooth term in the objective, we present an
alternative interior-point algorithm with a slightly modified criticality
measure, which performs better in practice. Numerical experiments show that our
algorithm performs better than the trust-region method TR, the trust-region
method with diagonal hessian approximations TRDH of arXiv:2309.08433, and the
quadratic regularization method R2 of arXiv:2103.15993v3 for two out of four
tested bound-constrained problems. On those two problems, our algorithm obtains
smaller objective values than the other solvers using fewer objective and
gradient evaluations. On the two other problems, it performs similarly to TR,
R2 and TRDH.
