  In this paper, we consider optimizing a smooth, convex, lower semicontinuous
function in Riemannian space with constraints. To solve the problem, we first
convert it to a dual problem and then propose a general primal-dual algorithm
to optimize the primal and dual variables iteratively. In each optimization
iteration, we employ a proximal operator to search optimal solution in the
primal space. We prove convergence of the proposed algorithm and show its
non-asymptotic convergence rate. By utilizing the proposed primal-dual
optimization technique, we propose a novel metric learning algorithm which
learns an optimal feature transformation matrix in the Riemannian space of
positive definite matrices. Preliminary experimental results on an optimal fund
selection problem in fund of funds (FOF) management for quantitative investment
showed its efficacy.
