We investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to … Read more
This paper presents an acceleration of the optimal subgradient algorithm OSGA \cite{NeuO} for solving convex optimization problems, where the objective function involves costly affine and cheap nonlinear terms. We combine OSGA with a multidimensional subspace search technique, which leads to low-dimensional problem that can be solved efficiently. Numerical results concerning some applications are reported. A … Read more
Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate … Read more
In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with nonconstant relaxation parameters and stepsizes of the resolvents. Besides proving existence and uniqueness of strong global solutions … Read more
It has been an open question whether the Linear Programming (LP) problem can be solved in strong polynomial time. The simplex algorithm does not offer a polynomial bound, and polynomial algorithms by Khachiyan and Karmarkar don’t have the strong characteristic. The curious fact that non-linear algorithms would be needed to deliver the strongest complexity result … Read more
We present a flexible Alternating Direction Method of Multipliers (F-ADMM) algorithm for solving optimization problems involving a strongly convex objective function that is separable into $n \geq 2$ blocks, subject to (non-separable) linear equality constraints. The F-ADMM algorithm uses a \emph{Gauss-Seidel} scheme to update blocks of variables, and a regularization term is added to each … Read more
Large scale nonsmooth convex optimization is a common problem for a range of computational areas including machine learning and computer vision. Problems in these areas contain special domain structures and characteristics. Special treatment of such problem domains, exploiting their structures, can significantly improve the the computational burden. We present a weighted Mirror Descent method to … Read more
A framework is presented whereby a general convex conic optimization problem is transformed into an equivalent convex optimization problem whose only constraints are linear equations and whose objective function is Lipschitz continuous. Virtually any subgradient method can be applied to solve the equivalent problem. Two methods are analyzed. (In version 2, the development of algorithms … Read more
We begin by considering second order dynamical systems of the from $\ddot x(t) + \Gamma (\dot x(t)) + \lambda(t)B(x(t))=0$, where $\Gamma: {\cal H}\rightarrow{\cal H}$ is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space ${\cal H}$, $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator and $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function depending … Read more
The proximal multiplier method with proximal distances (PMAPD) proposed by O. Sarmiento C., E. A. Papa Quiroz and P. R. Oliveira, applied to solve a convex program with separable structure unified the works of Chen and Teboulle (PCPM method), Kyono and Fukushima (NPCPMM) and Auslender and Teboulle (EPDM) and extended the convergence properties for the … Read more