Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Euclidean Distance Bounds for LMI Analytic Centers using a Novel Bound
  on the Lambert Function   Linear matrix inequalities (LMIs) are ubiquitous in modern control theory, as
well as in a variety of other fields in science and engineering. Their analytic
centers, i.e. the maximum determinant elements of the feasible set spanned by
these LMIs, are the solution of many well-known problems in statistics,
communications, geometry, and control and can be approximated to arbitrary
precision by semidefinite programs (SDPs). The quality of these approximations
is measured with respect to the difference in log-determinant of both the exact
and the approximate solution to these SDPs, a quantity that follows directly
from the duality theory of semidefinite programming. However, in many
applications the relevant parameters are functions of the entries of the LMI
argument $X$. In these cases it is of interest to develop metrics that quantify
the quality of approximate solutions based on the error on these parameters,
something that the log-determinant error fails to capture due to the non-linear
interaction of all the matrix entries. In this work we develop upper bounds on
the Frobenius norm error between suboptimal solutions $X_f$ and the exact
optimizer $X_*$ of maximum determinant problems, a metric that provides a
direct translation to the entry-wise error of $X$ and thus to the relevant
parameters of the application. We show that these bounds can be expressed
through the use of the Lambert function $W(x)$, i.e. the solution of the
equation $W(x) e^{W(x)}= x$, and derive novel bounds for one of its branches to
generate efficient closed-form bounds on the Euclidean distance to the LMI
analytic center. Finally, we test the quality of these bounds numerically in
the context of interior point methods termination criteria.

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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph