  We study minimization of a parametric family of relative entropies, termed
relative $\alpha$-entropies (denoted $\mathscr{I}_{\alpha}(P,Q)$). These arise
as redundancies under mismatched compression when cumulants of compressed
lengths are considered instead of expected compressed lengths. These parametric
relative entropies are a generalization of the usual relative entropy
(Kullback-Leibler divergence). Just like relative entropy, these relative
$\alpha$-entropies behave like squared Euclidean distance and satisfy the
Pythagorean property. Minimization of $\mathscr{I}_{\alpha}(P,Q)$ over the
first argument on a set of probability distributions that constitutes a linear
family is studied. Such a minimization generalizes the maximum R\'{e}nyi or
Tsallis entropy principle. The minimizing probability distribution (termed
$\mathscr{I}_{\alpha}$-projection) for a linear family is shown to have a
power-law.
