  Sparse principal component analysis addresses the problem of finding a linear
combination of the variables in a given data set with a sparse coefficients
vector that maximizes the variability of the data. This model enhances the
ability to interpret the principal components, and is applicable in a wide
variety of fields including genetics and finance, just to name a few.
  We suggest a necessary coordinate-wise-based optimality condition, and show
its superiority over the stationarity-based condition that is commonly used in
the literature, and which is the basis for many of the algorithms designed to
solve the problem. We devise algorithms that are based on the new optimality
condition, and provide numerical experiments that support our assertion that
algorithms, which are guaranteed to converge to stronger optimality conditions,
perform better than algorithms that converge to points satisfying weaker
optimality conditions.
