Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization

P. Tseng () and S. Yun ()
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P. Tseng: University of Washington
S. Yun: National University of Singapore

Journal of Optimization Theory and Applications, 2009, vol. 140, issue 3, No 9, 513-535

Abstract: Abstract We consider the problem of minimizing the weighted sum of a smooth function f and a convex function P of n real variables subject to m linear equality constraints. We propose a block-coordinate gradient descent method for solving this problem, with the coordinate block chosen by a Gauss-Southwell-q rule based on sufficient predicted descent. We establish global convergence to first-order stationarity for this method and, under a local error bound assumption, linear rate of convergence. If f is convex with Lipschitz continuous gradient, then the method terminates in O(n 2/ε) iterations with an ε-optimal solution. If P is separable, then the Gauss-Southwell-q rule is implementable in O(n) operations when m=1 and in O(n 2) operations when m>1. In the special case of support vector machines training, for which f is convex quadratic, P is separable, and m=1, this complexity bound is comparable to the best known bound for decomposition methods. If f is convex, then, by gradually reducing the weight on P to zero, the method can be adapted to solve the bilevel problem of minimizing P over the set of minima of f+δ X , where X denotes the closure of the feasible set. This has application in the least 1-norm solution of maximum-likelihood estimation.

Keywords: Nonsmooth optimization; Linear constraints; Support vector machines; Bilevel optimization; ℓ 1-regularization; Coordinate gradient descent; Global convergence; Linear convergence rate; Complexity bound (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (54)

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DOI: 10.1007/s10957-008-9458-3

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