74 research outputs found
Convergence of the steepest descent method for minimizing quasiconvex functions
To minimize a continuously differentiable quasiconvex function f : ℝ n →ℝ, Armijo's steepest descent method generates a sequence x k +1 = x k − t k ∇ f ( x k ), where t k >0. We establish strong convergence properties of this classic method: either , s.t. ; or arg min f = ∅, ∥ x k ∥ ↓ ∞ and f(x k )↓ inf f . We also discuss extensions to other line searches.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45245/1/10957_2005_Article_BF02192649.pd
Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function
In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function . Instead of the original objective function f , we employ a convex approximation f k + 1 at the k th iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate even it the iteration points are calculated approximately, where are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45249/1/10957_2004_Article_417694.pd
Large-scale optimization with the primal-dual column generation method
The primal-dual column generation method (PDCGM) is a general-purpose column
generation technique that relies on the primal-dual interior point method to
solve the restricted master problems. The use of this interior point method
variant allows to obtain suboptimal and well-centered dual solutions which
naturally stabilizes the column generation. As recently presented in the
literature, reductions in the number of calls to the oracle and in the CPU
times are typically observed when compared to the standard column generation,
which relies on extreme optimal dual solutions. However, these results are
based on relatively small problems obtained from linear relaxations of
combinatorial applications. In this paper, we investigate the behaviour of the
PDCGM in a broader context, namely when solving large-scale convex optimization
problems. We have selected applications that arise in important real-life
contexts such as data analysis (multiple kernel learning problem),
decision-making under uncertainty (two-stage stochastic programming problems)
and telecommunication and transportation networks (multicommodity network flow
problem). In the numerical experiments, we use publicly available benchmark
instances to compare the performance of the PDCGM against recent results for
different methods presented in the literature, which were the best available
results to date. The analysis of these results suggests that the PDCGM offers
an attractive alternative over specialized methods since it remains competitive
in terms of number of iterations and CPU times even for large-scale
optimization problems.Comment: 28 pages, 1 figure, minor revision, scaled CPU time
An inexact bundle variant suited to column generation
International audienceWe give a bundle method for constrained convex optimization. Instead of using penalty functions, it shifts iterates towards feasibility, by way of a Slater point, assumed to be known. Besides, the method accepts an oracle delivering function and subgradient values with unknown accuracy. Our approach is motivated by a number of applications in column generation, in which constraints are positively homogeneous--so that zero is a natural Slater point--and an exact oracle may be time consuming. Finally, our convergence analysis employs arguments which have been little used so far in the bundle community. The method is illustrated on a number of cutting-stock problems
- …