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The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a $Oleft(1/k^{2}right)$ non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in practical scenarios. Experimental comparison with Algebraic Reconstruction Techniques are performed uncovering certain behaviors of accelerated Proximal Gradient algorithms that apparently have not yet been noticed when these are applied to tomographic image reconstruction.
Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds.
Based on an observation that additive Schwarz methods for general convex optimization can be interpreted as gradient methods, we propose an acceleration scheme for additive Schwarz methods. Adopting acceleration techniques developed for gradient meth
Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, num
This paper gives a unified convergence analysis of additive Schwarz methods for general convex optimization problems. Resembling to the fact that additive Schwarz methods for linear problems are preconditioned Richardson methods, we prove that additi
In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately solves a sequ