Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Modified Interior-Point Method for Large-and-Sparse Low-Rank
  Semidefinite Programs   Semidefinite programs (SDPs) are powerful theoretical tools that have been
studied for over two decades, but their practical use remains limited due to
computational difficulties in solving large-scale, realistic-sized problems. In
this paper, we describe a modified interior-point method for the efficient
solution of large-and-sparse low-rank SDPs, which finds applications in graph
theory, approximation theory, control theory, sum-of-squares, etc. Given that
the problem data is large-and-sparse, conjugate gradients (CG) can be used to
avoid forming, storing, and factoring the large and fully-dense interior-point
Hessian matrix, but the resulting convergence rate is usually slow due to
ill-conditioning. Our central insight is that, for a rank-$k$, size-$n$ SDP,
the Hessian matrix is ill-conditioned only due to a rank-$nk$ perturbation,
which can be explicitly computed using a size-$n$ eigendecomposition. We
construct a preconditioner to "correct" the low-rank perturbation, thereby
allowing preconditioned CG to solve the Hessian equation in a few tens of
iterations. This modification is incorporated within SeDuMi, and used to reduce
the solution time and memory requirements of large-scale matrix-completion
problems by several orders of magnitude.

21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an