A primal (all-integer) integer programming algorithm
Year: 1965
Type: article
Abstract: The algorithm is most closely related to three existing procedures: the simplex method of G. B. Dantzig for linear programming problems, the Gomory all-integer integer programming algorithm, and the direct algorithm for integer programming of Ben-Israel and Charnes. The algorithm is similar to the Gomory all- integer algorithm in these respects: (i) it is an all-integer algorithm ; (ii) it uses t he same cut generation procedure; (iii) it uses the cut row as the pivot row; and (iv) the pivot coefficient always has unit value. While the dual method provides the vehicle for moving from tableau to tableau in the Gomory all-integer algorithm, the simplex method has the analagous role in the primal algorithm. Thus in a general sense this algorithm is a primal analog to the (dual) Gomory all-integer algorithm. The direct algorithm of Ben-Israel and Charnes also has the above similarities to the Gomory all-integer algorithm, but has one significant difference: an itera tion or cycle of the direct algorithm must frequently include the solution of an "auxiliary problem" (which is itself an integer programming problem) or a determination that no solution to the "auxiliary probl em" exists. In contrast, the cycles of the primal algorithm include only the adjoining of a Gomory cut and the execution of the change of basis procedure of the simplex method. The procedure of the algorithm and the proof of finiteness are founded on a classification of cycles of the algorithm and on two theorems. Two types of procedural restrictions are imposed as a basis for proving finiteness: (a) selection of the incoming variable is subjected to regulation (beyond that required by the simplex method), and the rules applied are a function of the type of cycle being executed; (b) selection of the row used as the source of the data for the Gomory cut is restricted (in addition to the restriction implied by (ii), (iii), and (iv) above) in certain cycles of the algorithm.
Author: Richard D. Young
Institution: Rice University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2329417820