Abstract
This paper provides another proof for the convexity (strict convexity) of log det(I+KX-1) over the positive definite cone for any given positive semidefinite matrix K ⪰ 0 (positive definite matrix K ≻ 0) and the strict convexity of log det(K +X-1) over the positive definite cone for any given K ⪰ 0. Equivalent optimization representations with linear matrix inequalities (LMIs) for the functions log det(I+KX-1) and log det(K+X-1) are also presented. It was shown that these optimization representations with LMI constraints can be particularly useful for some related synthetic design problems. An iterative procedure based on the proposed LMI is presented to solve the minimax mutual information game with covariance and expected power constraints.
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Recommended by Associate Editor Young Ik Son under the direction of Editor PooGyeon Park. This work is supported by INHA UNIVERSITY Research Grant (INHA-55435). We thank the three anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions that have resulted in a stronger manuscript.
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Kim, KK.K. Optimization and Convexity of log det(I+KX−1). Int. J. Control Autom. Syst. 17, 1067–1070 (2019). https://doi.org/10.1007/s12555-018-0263-y
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DOI: https://doi.org/10.1007/s12555-018-0263-y