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Efficiency of coordinate descent methods on huge-scale optimization problems. (English) Zbl 1257.90073
For the unconstrained minimization of a differentiable convex function \(f(x_{1},\dots,x_{n})\) (\(x_{i}\in \mathbb{R}^{n_{i}}\), \(i=1,\dots,n\)) with a globally Lipschitz gradient, the random coordinate descent method consists in randomly choosing, at iteration \(k\), some \(i_{k}\in \{1,\dots,n\}\) and then updating the current iterate by making a step in the direction of the negative of the partial gradient with respect to \(x_{i_{k}}\), the step size being equal to the reciprocal of the Lipschitz constant of this partial gradient. The expected objective function value is shown to converge to the infimum of \(f\); for strongly convex functions, the rate of convergence is linear and an accelerated version is presented. A modification of the method for constrained problems is also introduced. Implementation issues are discussed, and some preliminary numerical experiments are reported.

90C25 Convex programming
65K05 Numerical mathematical programming methods
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