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An algorithm for the optimal solution of consistent and inconsistent linear inequalities is presented, where the optimality criterion is the maximization of the number of satisfied constraints. The algorithm is developed as a nonenumerative search procedure based on two new theorems established in this paper. It is shown that the number of iterative steps before termination is strictly less than that required by an exhaustive search. Experimental results with various types of data establish the computational tractability of the procedure under nontrivial conditions.