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A lower bound on the number of iterations of long-step primal-dual linear programming algorithms

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Recently, Todd has analyzed in detail the primal-dual affine-scaling method for linear programming, which is close to what is implemented in practice, and proved that it may take at leastn 1/3 iterations to improve the initial duality gap by a constant factor. He also showed that this lower bound holds for some polynomial variants of primal-dual interior-point methods, which restrict all iterates to certain neighborhoods of the central path. In this paper, we further extend his result to long-step primal-dual variants that restrict the iterates to a wider neighborhood. This neigh-borhood seems the least restrictive one to guarantee polynomiality for primal-dual path-following methods, and the variants are also even closer to what is implemented in practice.
... See also refs. [3,4]. ...
... Each main iteration consists of one so-called feasibility step, a μ-update, and a few centering steps, respectively. First, we find new iterates x f , y f and s f that satisfy equations (4) and (5), with ν replaced by ν + . As we will see, by taking θ small enough, this can be realized by one feasibility step, as discussed subsequently. ...
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In Roos [Roos, C., 2006, A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM Journal on Optimization, 16(4), 1110–1136.] presented a new primal-dual infeasible interior-point algorithm that uses full-Newton steps and whose iteration bound coincides with the best-known bound for infeasible interior-point algorithms. Each iteration consists of a step that restores the feasibility for an intermediate problem (the so-called feasibility step) and a few (usual) centering steps. No more than O(nlog(n/ϵ)) iterations are required for getting an ϵ-solution of the problem at hand, which coincides with the best-known bound for infeasible interior-point algorithms. In this article, we introduce a different feasibility step and show that the same complexity result can be obtained with a relatively simpler analysis.
... Sonnevend et al. [13] showed that a variant of MTY predictor-corrector algorithm requires Ω(n 1 3 ) iterations to reduce the duality gap by log n for certain LO problems. A similar result has been obtained by Todd et al. [15] for the primal-dual affine scaling algorithm and has been later extended by Todd and Ye [16] for long step primal-dual IPMs; they showed that these algorithms take Ω(n 1 3 ) iterations to reduce the duality gap by a constant. ...
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Chapter
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