  Minimization problems with respect to a one-parameter family of generalized
relative entropies are studied. These relative entropies, which we term
relative $\alpha$-entropies (denoted $\mathscr{I}_{\alpha}$), 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. Minimizers of these relative $\alpha$-entropies on closed
and convex sets are shown to exist. Such minimizations generalize the maximum
R\'{e}nyi or Tsallis entropy principle. The minimizing probability distribution
(termed forward $\mathscr{I}_{\alpha}$-projection) for a linear family is shown
to obey a power-law. Other results in connection with statistical inference,
namely subspace transitivity and iterated projections, are also established. In
a companion paper, a related minimization problem of interest in robust
statistics that leads to a reverse $\mathscr{I}_{\alpha}$-projection is
studied.
