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

Abstract

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

References in corpus (2)

• CVXPY: A Python-Embedded Modeling Language for Convex Optimization
• Semidefinite Programming Approach to Gaussian Sequential Rate-Distortion Trade-offs

Cited by in corpus (1)

• Sharp moment-entropy inequalities and capacity bounds for log-concave distributions