Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Infeasibility detection with primal-dual hybrid gradient for large-scale
  linear programming   We study the problem of detecting infeasibility of large-scale linear
programming problems using the primal-dual hybrid gradient method (PDHG) of
Chambolle and Pock (2011). The literature on PDHG has mostly focused on
settings where the problem at hand is assumed to be feasible. When the problem
is not feasible, the iterates of the algorithm do not converge. In this
scenario, we show that the iterates diverge at a controlled rate towards a
well-defined ray. The direction of this ray is known as the infimal
displacement vector $v$. The first contribution of our work is to prove that
this vector recovers certificates of primal and dual infeasibility whenever
they exist. Based on this fact, we propose a simple way to extract approximate
infeasibility certificates from the iterates of PDHG. We study three different
sequences that converge to the infimal displacement vector: the difference of
iterates, the normalized iterates, and the normalized average. All of them are
easy to compute, and thus the approach is suitable for large-scale problems.
Our second contribution is to establish tight convergence rates for these
sequences. We demonstrate that the normalized iterates and the normalized
average achieve a convergence rate of $O(1/k)$, improving over the known rate
of $O(1/\sqrt{k})$. This rate is general and applies to any fixed-point
iteration of a nonexpansive operator. Thus, it is a result of independent
interest since it covers a broad family of algorithms, including, for example,
ADMM, and can be applied settings beyond linear programming, such as quadratic
and semidefinite programming. Further, in the case of linear programming we
show that, under nondegeneracy assumptions, the iterates of PDHG identify the
active set of an auxiliary feasible problem in finite time, which ensures that
the difference of iterates exhibits eventual linear convergence to the infimal
displacement vector.

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The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas