  In this paper, we propose two algorithms for nonlinear semi-infinite
semi-definite programs with infinitely many convex inequality constraints,
called SISDP for short. A straightforward approach to the SISDP is to use
classical methods for semi-infinite programs such as discretization and
exchange methods and solve a sequence of (nonlinear) semi-definite programs
(SDPs). However, it is often too demanding to find exact solutions of SDPs.
  Our first approach does not rely on solving SDPs but on approximately
following {a path leading to a solution, which is formed on the intersection of
the semi-infinite region and the interior of the semi-definite region. We show
weak* convergence of this method to a Karush-Kuhn-Tucker point of the SISDP
under some mild assumptions and further provide with sufficient conditions for
strong convergence. Moreover, as the second method, to achieve fast local
convergence, we integrate a two-step sequential quadratic programming method
equipped with Monteiro-Zhang scaling technique into the first method. We
particularly prove two-step superlinear convergence of the second method using
Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and
Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling
directions. Finally, we conduct some numerical experiments to demonstrate the
efficiency of the proposed method through comparison with a discretization
method that solves SDPs obtained by finite relaxation of the SISDP.
