; Department of Information Sciences; Tokyo Institute of Technology; Oh-okayama, Meguro-ku; 8
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; Tokyo 152, Japan
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. In this short paper, we prove the finiteness of the criss-cross method by showing a certain binary number of bounded digits associated with each iteration increases monotonically. This new proof immediately suggests the possibility of relaxing the pivoting selection in the criss-cross method without sacrificing the finiteness. Key Words: linear programming. simplex method, finite pivoting rules. 1 The Criss-Cross Method Let A be an m2 n matrix. Let E be the index set of columns of the matrix A; and f; g be two distinct members of E: Here we consider the standard form linear program: (P ) maximize x f (1.1) subject to A x = 0; (1.2) x g = 1; (1.3) x j 0; 8 j 2 E 0 ff; gg: (1.4) A vector x is said to be feasible if it satisfies the constraints (1.2), (1.3), and (1.4). If a linear program has a feasible solution, then it is called feasible, otherwise it is called infeasible. For any linear program, we will refer to following three situations as characters: 3 Supported by Grant...