Groupe d’études et de recherche en analyse des décisions

# A proximal quasi-Newton trust-region method for nonsmooth regularized optimization

## Aleksandr Aravkin, Robert Baraldi et Dominique Orban

We develop a trust-region method for minimizing the sum of a smooth term $$f$$ and a nonsmooth term $$h$$, both of which can be nonconvex. Each iteration of our method minimizes a possibly nonconvex model of $$f + h$$ in a trust region. The model coincides with $$f + h$$ in value and subdifferential at the center. We establish global convergence to a first-order stationary point when $$f$$ satisfies a smoothness condition that holds, in particular, when it has Lipschitz-continuous gradient, and $$h$$ is proper and lower semi-continuous. The model of $$h$$ is required to be proper, lower-semi-continuous and prox-bounded. Under these weak assumptions, we establish a worst-case $$O(1/\epsilon^2)$$ iteration complexity bound that matches the best known complexity bound of standard trust-region methods for smooth optimization. We detail a special instance in which we use a limited-memory quasi-Newton model of $$f$$ and compute a step with the proximal gradient method, resulting in a practical proximal quasi-Newton method. We establish similar convergence properties and complexity bound for a quadratic regularization variant, and provide an interpretation as a proximal gradient method with adaptive step size for nonconvex problems. We describe our Julia implementations and report numerical results on inverse problems from sparse optimization and signal processing. Our trust-region algorithm exhibits promising performance and compares favorably with linesearch proximal quasi-Newton methods based on convex models.

, 30 pages