joint work with Stephen Wright
It has been known for many years that both gradient descent and stochastic coordinate descent achieve a global convergence rate of $O(1/k)$ in the objective value, when applied to a scheme for minimizing a Lipschitz-continuously differentiable, unconstrained convex function. In this work, we improve this rate to $o(1/k)$. We extend the result to proximal gradient and proximal stochastic coordinate descent with arbitrary samplings for the coordinates on regularized problems to show similar $o(1/k)$ convergence rates.
joint work with Nobuo Yamashita
We consider a special proximal alternating direction method of multipliers (ADMM) for the structured convex quadratic optimization problem. In this work, we propose a proximal ADMM whose regularized matrix in the proximal term is generated by the BFGS update (or limited memory BFGS) at every iteration. These types of matrices use second-order information of the objective function. The convergence of the proposed method is proved under certain assumptions. Numerical results are presented to show the effectiveness.
joint work with Shuai Liu, Yousong Luo
Under the assumption of prox-regularity and the presence of a tilt stable local minimum we are able to show that a VU like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function. We will also consider the inverse problem. If a fast track exists around a strict local minimum does this imply the existence of a tilt stable local minimum? We investigate conditions under which this is so by studying the closely related notions of fast track and of partial smoothness and their equivalence.