We study minimization of a parametric family of relative entropies, termed relative $\alpha$-entropies (denoted 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 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 I$\alpha$-projection) for a linear family is shown to have a power-law.