  Regularized generalized canonical correlation analysis (RGCCA) is a
generalization of regularized canonical correlation analysis to three or more
sets of variables, which is a component-based approach aiming to study the
relationships between several sets of variables. Sparse generalized canonical
correlation analysis (SGCCA) (proposed in Tenenhaus et al. (2014)), combines
RGCCA with an `1-penalty, in which blocks are not necessarily fully connected,
makes SGCCA a flexible method for analyzing a wide variety of practical
problems, such as biology, chemistry, sensory analysis, marketing, food
research, etc. In Tenenhaus et al. (2014), an iterative algorithm for SGCCA was
designed based on the solution to the subproblem (LM-P1 for short) of
maximizing a linear function on the intersection of an `1-norm ball and a unit
`2-norm sphere proposed in Witten et al. (2009). However, the solution to the
subproblem (LM-P1) proposed in Witten et al. (2009) is not correct, which may
become the reason that the iterative algorithm for SGCCA is slow and not always
convergent. For this, we first characterize the solution to the subproblem
LM-P1, and the subproblems LM-P2 and LM-P3, which maximize a linear function on
the intersection of an `1-norm sphere and a unit `2-norm sphere, and an `1-norm
ball and a unit `2-norm sphere, respectively. Then we provide more efficient
block coordinate descent (BCD) algorithms for SGCCA and its two variants,
called SGCCA-BCD1, SGCCA-BCD2 and SGCCA-BCD3, corresponding to the subproblems
LM-P1, LM-P2 and LM-P3, respectively, prove that they all globally converge to
their stationary points. We further propose gradient projected (GP) methods for
SGCCA and its two variants when using the Horst scheme, called SGCCA-GP1,
SGCCA-GP2 and SGCCA-GP3, corresponding to the subproblems LM-P1, LM-P2 and
LM-P3, respectively, and prove that they all
