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Minimization problems with respect to a one-parameter family of generalizedrelative entropies are studied. These relative entropies, which we termrelative $\alpha$-entropies (denoted $\mathscr{I}_{\alpha}$), arise asredundancies under mismatched compression when cumulants of compressed lengthsare considered instead of expected compressed lengths. These parametricrelative 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 thePythagorean property. Minimizers of these relative $\alpha$-entropies on closedand convex sets are shown to exist. Such minimizations generalize the maximumR\'{e}nyi or Tsallis entropy principle. The minimizing probability distribution(termed forward $\mathscr{I}_{\alpha}$-projection) for a linear family is shownto obey a power-law. Other results in connection with statistical inference,namely subspace transitivity and iterated projections, are also established. Ina companion paper, a related minimization problem of interest in robuststatistics that leads to a reverse $\mathscr{I}_{\alpha}$-projection isstudied.

Minimization Problems Based on Relative $\alpha$-Entropy I: Forward Projection