Based on the recently developed ABS algorithm for solving linear Diophantine equations, we present a special ABS algorithm for solving such equations which is effective in computation and storage, not requiring the co...
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Based on the recently developed ABS algorithm for solving linear Diophantine equations, we present a special ABS algorithm for solving such equations which is effective in computation and storage, not requiring the computation of the greatest common divisor. A class of equations always solvable in integers is identified. Using this result, we discuss the ILP problem with upper and lower bounds on the variables.
We describe an algorithm of the ABS class, which solves a general nonsingular linear system in n(3)/3 + 0(n(2)) multiplications without the assumption that the coefficient matrix be regular. The method can be viewed a...
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We describe an algorithm of the ABS class, which solves a general nonsingular linear system in n(3)/3 + 0(n(2)) multiplications without the assumption that the coefficient matrix be regular. The method can be viewed as a variation of the implicit LU algorithm of the ABS class, whose associated factorization contains a factor which is not triangular (but can be reduced to triangular form after suitable row permutations). We describe properties of the method, including in particular an efficient way of updating the Abaffan matrix after column interchanges. Such a problem arises in the application to the simplex algorithm, where the implicit lx algorithm provides a faster technique than the standard LU factorization for the pivoting operation if the number of equality constraints m is greater than n/2.
In this paper, we discuss the application of the ABS algorithm to the simplex method, the dual simplex method, the linear complementary problem. We consider the ABS formulation of the stopping criterion, the search di...
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In this paper, we discuss the application of the ABS algorithm to the simplex method, the dual simplex method, the linear complementary problem. We consider the ABS formulation of the stopping criterion, the search direction, the minimal rule to determine the vectors entering and leaving the basis matrix and the updating of the Abaffian matrix after a basis vector exchange. The Lemke algorithm for the LCP problem is reformulated in terms of the ABS procedure.
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