PIPZ1: A PARTIAL ENUMERATION ALGORITHM FOR PURE 0-1 POLYNOMIAL INTEGER PROGRAMMING PROBLEMS
Abstract
A partial enumeration algorithm, PIPZ1, for the 0-1 polynomial integer programming problem is developed. A problem to be solved is transformed to an equivalent problem with nonnegative coefficients only. In the worst case, the number of decision variables is doubled and n multiple choice constraints are added, where n is the number of decision variables in the original problem. As a result of this transformation, optimality and feasibility tests are performed more efficiently. Though the transformation may increase the size of the problem, the transformed problem is no harder to solve (in the complexity theoretic sense) because at most 2('n), not 2('2n), solutions will be examined. Furthermore, the multiple choice constraints added to the problem are enforced implicitly. Hence the transformation does not further complicate the enumeration procedure. A second unique feature of PIPZ1 is the option of solving imbedded linear programs periodically in order to construct linear surrogate constraints. The surrogate constraints constructed are similar to those devised by Geoffrion {1969} to accelerate the solution of linear integer programs. The imbedded linear program is the dual problem of the continuous linear relaxation of the nonlinear integer program associated with the current partial solution. Computational results on over 90 problems indicate that the PIPZ1 transformation can be effective in rendering a transformed problem which can be solved in less time than the original problem due to the monotonicity property of the objective and constraint functions. The surrogate constraint option is shown to be a valuable enhancement to the PIPZ1 algorithm. All of the test problems were also solved with a computerized version of an implicit enumeration algorithm developed by Hansen {1972}. The PIPZ1 results compare most favorably to those of Hansen's algorithm when the number of variables added to a problem during the transformation step is small.
Degree
Ph.D.
Subject Area
Industrial engineering
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