We propose a simple {\em projection and algorithm} to solve the feasibility problem \[ \text{ find } x \in L \cap \Omega, \] where $L$ and $\Omega$ are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space $V$. This projection and rescaling algorithm is inspired by previous work … Read more
It is known that the gradient descent algorithm converges linearly when applied to a strongly convex function with Lipschitz gradient. In this case the algorithm’s rate of convergence is determined by condition number of the function. In a similar vein, it has been shown that a variant of the Frank-Wolfe algorithm with away steps converges … Read more
In this paper we study new algorithmic structures with Douglas- Rachford (DR) operators to solve convex feasibility problems. We propose to embed the basic two-set-DR algorithmic operator into the String-Averaging Projections (SAP) and into the Block-Iterative Pro- jection (BIP) algorithmic structures, thereby creating new DR algo- rithmic schemes that include the recently proposed cyclic Douglas- … Read more
The conjugate gradient (CG) method is an efficient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper we propose some generalized CG (GCG) methods for solving the $\ell_1$-regularized (possibly not strongly) convex QP that terminate at an optimal solution in a finite number of iterations. At each iteration, our methods first … Read more
We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to the dual sequence, and the residual part of the primal … Read more
We present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. We present computational guarantees for the method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply our method to the low-rank … Read more
The {\it forward-backward algorithm} is a powerful tool for solving optimization problems with a {\it additively separable} and {\it smooth} + {\it nonsmooth} structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov has been proved useful to improve the theoretical rate of convergence for the function values from the standard … Read more
In a Hilbert space setting $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t), \end{equation*} where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, … Read more
This is a review paper on some of the physics, modeling, and iterative algorithms in proton computed tomography (pCT) image reconstruction. The primary challenge in pCT image reconstruction lies in the degraded spatial resolution resulting from multiple Coulomb scattering within the imaged object. Analytical models such as the most likely path (MLP) have been proposed … Read more
This paper describes an accelerated HPE-type method based on general Bregman distances for solving monotone saddle-point (SP) problems. The algorithm is a special instance of a non-Euclidean hybrid proximal extragradient framework introduced by Svaiter and Solodov [28] where the prox sub-inclusions are solved using an accelerated gradient method. It generalizes the accelerated HPE algorithm presented … Read more