  Gauge duality theory was originated by Freund [Math. Programming,
38(1):47-67, 1987] and was recently further investigated by Friedlander,
Mac{\^e}do and Pong [SIAM J. Optm., 24(4):1999-2022, 2014]. When solving some
matrix optimization problems via gauge dual, one is usually able to avoid full
matrix decompositions such as singular value and/or eigenvalue decompositions.
In such an approach, a gauge dual problem is solved in the first stage, and
then an optimal solution to the primal problem can be recovered from the dual
optimal solution obtained in the first stage. Recently, this theory has been
applied to a class of \emph{semidefinite programming} (SDP) problems with
promising numerical results [Friedlander and Mac{\^e}do, SIAM J. Sci. Comp., to
appear, 2016]. In this paper, we establish some theoretical results on applying
the gauge duality theory to robust \emph{principal component analysis} (PCA)
and general SDP. For each problem, we present its gauge dual problem,
characterize the optimality conditions for the primal-dual gauge pair, and
validate a way to recover a primal optimal solution from a dual one. These
results are extensions of [Friedlander and Mac{\^e}do, SIAM J. Sci. Comp., to
appear, 2016] from nuclear norm regularization to robust PCA and from a special
class of SDP which requires the coefficient matrix in the linear objective to
be positive definite to SDP problems without this restriction. Our results
provide further understanding in the potential advantages and disadvantages of
the gauge duality theory.
