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arXiv · 1509.00777

A Note on the Convexity of $\log \det ( I + KX^{-1} )$ and its Constrained Optimization Representation

Kwang-Ki K. Kim
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Abstract

This note provides another proof for the {\em convexity} ({\em strict convexity}) of $\log \det ( I + KX^{-1} )$ over the positive definite cone for any given positive semidefinite matrix $K \succeq 0$ (positive definite matrix $K \succ 0$) and the {\em strictly convexity} of $\log \det (K + X^{-1})$ over the positive definite cone for any given $K \succeq 0$. Equivalent optimization representation with linear matrix inequalities (LMIs) for the functions $\log \det ( I + KX^{-1} )$ and $\log \det (K + X^{-1})$ are presented. Their optimization representations with LMI constraints can be particularly useful for some related synthetic design problems.

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Kwang-Ki K. Kim. 2015-09-02. A Note on the Convexity of $\log \det ( I + KX^{-1} )$ and its Constrained Optimization Representation. https://doi.org/10.1007/s12555-018-0263-y

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