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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.