  We consider the multi-objective mean-variance-skewness-kurtosis (MVSK)
problem in portfolio selection, with and without shorting and leverage.
Additionally, we define a sparse variant of MVSK where feasible portfolios have
supports contained in a chosen class of sets. To find the MVSK problem's Pareto
front, we linearly scalarize the four objectives of MVSK into a scalar-valued
degree four polynomial $F_{\lambda}$ depending on some hyper-parameter $\lambda
\in \Delta^4$. As one of our main results, we identify a set of
hyper-parameters for which $F_{\lambda}$ is convex over the probability simplex
(or over the cube). By exploiting the convexity and neatness of the
scalarization, we can compute part of the Pareto front. We compute an optimizer
of the scalarization $F_{\lambda}$ for each $\lambda$ in a grid sampling of
$\Delta^4$. To see each optimizer's quality, we plot scaled portfolio objective
values against hyper-parameters. Doing so, we reveal a sub-set of optimizers
that provide a superior trade-off among the four objectives in MVSK.
