  In this paper, we consider a nonlinear semi-infinite program that minimizes a
function including a log-determinant (logdet) function over positive definite
matrix constraints and infinitely many convex inequality constraints, called
SIPLOG for short. The main purpose of the paper is to develop an algorithm for
computing a Karush-Kuhn-Tucker (KKT) point for the SIPLOG efficiently. More
specifically, we propose an interior point sequential quadratic
programming-type method that inexactly solves a sequence of semi-infinite
quadratic programs approximating the SIPLOG. Furthermore, to generate a search
direction in the dual matrix space associated with the semi-definite
constraint, we solve scaled Newton equations {that yield} the family of
Monteiro-Zhang directions. We prove that the proposed method weakly* converges
to a KKT point under some mild assumptions. Finally, we conduct some numerical
experiments to demonstrate the efficiency of the proposed method.
