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The primal problem of multinomial likelihood maximization restricted to aconvex closed subset of the probability simplex is studied. Contrary to widelyheld belief, a solution of this problem may assign a positive mass to anoutcome with zero count. Related flaws in the simplified Lagrange and Fencheldual problems, which arise because the recession directions are ignored, areidentified and corrected.  A solution of the primal problem can be obtained by the PP (perturbed primal)algorithm, that is, as the limit of a sequence of solutions of perturbed primalproblems. The PP algorithm may be implemented by the simplified Fenchel dual.  The results permit us to specify linear sets and data such that the empiricallikelihood-maximizing distribution exists and is the same as the multinomiallikelihood-maximizing distribution. The multinomial likelihood ratio reaches,in general, a different conclusion than the empirical likelihood ratio.  Implications for minimum discrimination information, compositional dataanalysis, Lindsay geometry, bootstrap with auxiliary information, and Lagrangemultiplier tests are discussed.

Multinomial and empirical likelihood under convex constraints: directions of recession, Fenchel duality, perturbations