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

RESUMO

We study an inexact interior-point method for nonsmooth, nonconvex optimization problems with conic inequality constraints. 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 standard constraint qualifications, we establish KKT-type optimality conditions for the original problem and show that the primal iterates and barrier-induced multipliers satisfy approximate KKT conditions. For the interior-point scheme, we prove convergence to stationary points and derive bounds on the total number of proximal-gradient iterations needed to reach approximate stationarity. In particular, we show that the total complexity is dominated by the final outer iterations because of the growth of the barrier curvature. In the convex setting, we obtain stronger convergence results: the full sequence converges weakly to a global solution and strongly under a quadratic growth condition. We apply the abstract results to state-constrained semilinear elliptic optimal control problems, for which all assumptions are verified, and to finite-dimensional sparse dictionary-learning problems with nonlinear side constraints. Numerical experiments support the theoretical findings.