Free-energy calculations based on atomistic Hamiltonians and sampling are key to a first-principles understanding of biomolecular processes, material properties, and macromolecular chemistry. Here, we generalize the free-energy perturbation method and derive nonlinear Hamiltonian transformation sequences yielding free-energy estimates with minimal mean squared error with respect to the exact values. Our variational approach applies to finite sampling and holds for any finite number of

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... te states. We show that our sequences are also optimal for the Bennett acceptance ratio (BAR) method, thereby generalizing BAR to small sampling sizes and non-Gaussian error distributions.