Interior-point proximal methods for nonsmooth optimization in Hilbert spaces with cone-ordered constraints

Abstract

We study an inexact interior-point method for nonsmooth, possibly nonconvex optimization in a Hilbert space with inequality constraints ordered by a cone in a Banach lattice, with particular emphasis on infinite-dimensional state-constrained optimal control. The objective function is given by the sum of a smooth, possibly nonconvex term and a convex, possibly nonsmooth term with a computable proximal mapping. The constraints are formulated by means of an order cone in a Banach lattice. This setting covers finite-dimensional nonsmooth nonlinear problems with componentwise constraints as well as infinite-dimensional PDE-constrained optimization problems with pointwise state constraints. The method is based on barrier-regularized subproblems, which are solved inexactly by a proximal-gradient method. We consider logarithmic and power-type barriers and derive the differentiability and curvature estimates needed for the convergence and complexity analysis. Under suitable constraint qualifications and compactness assumptions, we establish approximate KKT conditions for the original problem and convergence of the inexact interior-point sequence. For logarithmic and power-type barriers, we derive complementarity estimates and outer iteration bounds; the corresponding power-barrier rates and total inner-outer complexity bounds are stated under explicit barrier-path and uniform smoothness assumptions. For convex problems, we obtain stronger convergence results. We apply the framework to state-constrained semilinear elliptic optimal control and sparse dictionary learning with nonlinear side constraints. Numerical experiments illustrate the proposed method.