  In this paper, we propose a general algorithmic framework to solve a class of
optimization problems on the product of complex Stiefel manifolds based on the
matrix polar decomposition. We establish the weak convergence, global
convergence and linear convergence rate of this general algorithmic approach
using the \L{}ojasiewicz gradient inequality and the Morse-Bott property. This
general algorithm and its convergence results are applied to the simultaneous
approximate tensor diagonalization and simultaneous approximate tensor
compression, which include as special cases the low rank orthogonal
approximation, best rank-1 approximation and low multilinear rank approximation
for higher order complex tensors. We also present a symmetric variant of this
general algorithm to solve a symmetric variant of this class of optimization
models, which essentially optimizes over a single Stiefel manifold. We
establish its weak convergence, global convergence and linear convergence rate
in a similar way. This symmetric variant and its convergence results are
applied to the simultaneous approximate symmetric tensor diagonalization, which
includes as special cases the low rank symmetric orthogonal approximation and
best symmetric rank-1 approximation for higher order complex symmetric tensors.
It turns out that well-known algorithms such as LROAT, S-LROAT, HOPM, S-HOPM
are all special cases of this general algorithmic framework and its symmetric
variant, and our convergence results subsume the results found in the
literature designed for those special cases. All the algorithms and convergence
results in this paper also apply to the real case.
