joint work with Mituhiro Fukuda
We develop a first-order method for convex composite optimization problem under the Hölderian Error Bound (HEB) condition which is closely related to the Kurdyka-Łojasiewicz inequality and arises in many applications. The proposed method does not require the prior knowledge on the HEB condition, due to an adaptive proximal point approach. We prove the near optimality of the proposed method in view of the iteration complexity with respect to two kinds of approximation error: the objective function value and the norm of the (generalized) gradient.
joint work with Mikael Johansson
This paper introduces a novel technique for nonlinear acceleration of 1st-order methods for constrained convex optimization. Previous studies of nonlinear acceleration have only been able to provide convergence guarantees for unconstrained problems. We focus on Anderson acceleration of the projected gradient descent method, but our techniques can easily be extended to more sophisticated algorithms, such as mirror descent. We show that the convergence results for Anderson acceleration of smooth fixed-point iterations can be extended to the non-smooth case under certain technical conditions.
joint work with Christian Clason, Stanislav Mazurenko
We demonstrate that difficult non-convex non-smooth optimization problems, including Nash equilibrium problems and Potts segmentation, can be written using generalized conjugates of convex functionals. We then show through detailed convergence analysis that a conceptually straightforward extension of the primal-dual method of Chambolle and Pock is applicable to such problems. Under sufficient growth conditions we even demonstrate local linear convergence of the method. We illustrate these theoretical results numerically on the aforementioned example problems. (arXiv:1901.02746)