Yara Elshiaty

We present the Multilevel Bregman Proximal Gradient Descent (ML-BPGD) method, a novel multilevel optimization framework tailored to constrained convex problems with relative Lipschitz smoothness. Our approach extends the classical multilevel optimization framework (MGOPT) to Bregman-based geometries and constrained domains. We provide a rigorous analysis of ML-BPGD for multiple levels and establish well-posedness together with a global linear convergence rate. We demonstrate the effectiveness of ML-BPGD in the context of image reconstruction and validate its performance through numerical experiments.

We study multilevel techniques, commonly used in PDE multigrid literature, to solve structured optimization problems. For a given hierarchy of levels, we formulate a coarse model that approximates the problem at each level and provides a descent direction for the fine-grid objective using fewer variables. Unlike common algebraic approaches, we assume the objective function and its gradient can be evaluated at each level. Under the assumptions of strong convexity and gradient L-smoothness, we analyze convergence and extend the method to box-constrained optimization. Large-scale numerical experiments on a discrete tomography problem show that the multilevel approach converges rapidly when far from the solution and performs competitively with state-of-the-art methods.

We study the minimization of smooth, possibly nonconvex functions over the positive orthant, a key setting in Poisson inverse problems, using the exponentiated gradient (EG) method. Interpreting EG as Riemannian gradient descent (RGD) with the $e$-Exp map from information geometry as a retraction, we prove global convergence under weak assumptions -- without the need for $L$-smoothness -- and finite termination of Riemannian Armijo line search. Numerical experiments, including an accelerated variant, highlight EG's practical advantages, such as faster convergence compared to RGD based on interior-point geometry.