NDLI: Efficient Inverse Maintenance and Faster Algorithms for Linear Programming

Content Provider	IEEE Xplore Digital Library
Author	Yin Tat Lee Sidford, A.
Copyright Year	2015
Description	Author affiliation: Dept. of EECS, MIT, Cambridge, MA, USA (Sidford, A.) \|\| Dept. of Math., MIT, Cambridge, MA, USA (Yin Tat Lee)
Abstract	In this paper, we consider the following inverse maintenance problem: given A ∈ $R^{n×d}$ and a number of rounds r, at round k, we receive a n x n diagonal matrix $D^{(k)}$ and we wish to maintain an efficient linear system solver for $A^{T}D^{(k)}A$ under the assumption $D^{(k)}$ does not change too rapidly. This inverse maintenance problem is the computational bottleneck in solving multiple optimization problems. We show how to solve this problem with Õ (nnz(A) + $d^{ω})$ preprocessing time and amortized Õ(nnz(A) + $d^{2})$ time per round, improving upon previous running times. Consequently, we obtain the fastest known running times for solving multiple problems including, linear programming and computing a rounding of a polytope. In particular given a feasible point in a linear program with n variables, d constraints, and constraint matrix A ∈ $R^{d×n},$ we show how to solve the linear program in time Õ((nnz(A) + $d^{2})√$ d $log(∈^{-1})).$ We achieve our results through a novel combination of classic numerical techniques of low rank update, preconditioning, and fast matrix multiplication as well as recent work on subspace embeddings and spectral sparsification that we hope will be of independent interest.
Starting Page	230
Ending Page	249
File Size	368163
Page Count	20
File Format	PDF
ISBN	9781467381918
ISSN	02725428
DOI	10.1109/FOCS.2015.23
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-10-17
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Maintenance engineering Linear systems Linear programming Stability analysis Approximation algorithms Approximation methods Optimization Linear Programming Inverse Maintenance Problem Linear Systems
Content Type	Text
Resource Type	Article

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