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A finitely convergent procedure is given for maximizing a differentiable and strictly concave function subject to linear constraints. It is also assumed that the objective function attains its unconstrained maximum. No additional assumptions whatever are required. The procedure is aimed directly at constructing a solution of a certain version of the Kuhn-Tucker Conditions. Provision is made for utilizing a priori information regarding which constraints are likely to be satisfied exactly at the optimum. When applied to quadratic programming, the procedure specializes to a promising generalization of Theil and van de Panne's algorithm.