  Many problems in statistical learning, imaging, and computer vision involve
the optimization of a non-convex objective function with singularities at the
boundary of the feasible set. For such challenging instances, we develop a new
interior-point technique building on the Hessian-barrier algorithm recently
introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt.
2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a
generalized self-concordant function. This class of functions is sufficiently
general to include most of the commonly used barrier functions in the
literature of interior point methods. We prove global convergence to an
approximate stationary point of the method, and in cases where the feasible set
admits an easily computable self-concordant barrier, we verify worst-case
optimal iteration complexity of the method. Applications in non-convex
statistical estimation and $L^{p}$-minimization are discussed to given the
efficiency of the method.
