Efficient Heuristic Algorithms for Positive 0-1 Polynomial Programming Problems
We develop in this paper two types of heuristic methods for solving the positive 0-1 polynomial programming (PP) problem of finding a 0-1 vector x that maximizes cTx subject to f(x) ≤ b where c, b ≥ 0 and f is an m-vector of polynomials with non-negative coefficients. The various heuristics were fir...
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| Published in: | Management science Vol. 28; no. 7; pp. 829 - 836 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Hanover, MD., etc
Institute of Management Sciences
01.07.1982
Institute for Operations Research and the Management Sciences |
| Subjects: | |
| ISSN: | 0025-1909, 1526-5501 |
| Online Access: | Get full text |
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| Abstract | We develop in this paper two types of heuristic methods for solving the positive 0-1 polynomial programming (PP) problem of finding a 0-1 vector x that maximizes cTx subject to f(x) ≤ b where c, b ≥ 0 and f is an m-vector of polynomials with non-negative coefficients. The various heuristics were first tested on randomly generated sparse problems of up to 50 variables and 50 constraints, and their performance in terms of computational time and effectiveness was investigated. The results were very encouraging. Optimal solutions were consistently obtained by some of the heuristic methods in over 50% of the problems solved. The effectiveness was on the average better than 99% and no less than 96.5%. The computational time using these heuristics is on the average 5% of the time required to solve the PP problems to optimality. Some results for very sparse problems with up to 1000 variables and 200 constraints are also reported. |
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| AbstractList | We develop in this paper two types of heuristic methods for solving the positive 0-1 polynomial programming (PP) problem of finding a 0-1 vector x that maximizes cTx subject to f(x) ≤ b where c, b ≥ 0 and f is an m-vector of polynomials with non-negative coefficients. The various heuristics were first tested on randomly generated sparse problems of up to 50 variables and 50 constraints, and their performance in terms of computational time and effectiveness was investigated. The results were very encouraging. Optimal solutions were consistently obtained by some of the heuristic methods in over 50% of the problems solved. The effectiveness was on the average better than 99% and no less than 96.5%. The computational time using these heuristics is on the average 5% of the time required to solve the PP problems to optimality. Some results for very sparse problems with up to 1000 variables and 200 constraints are also reported. Programming - Integer Algorithms, Heuristic In this research, heuristic algorithms for the positive 0-1 polynomial programming (PP) problem were developed. Previous approaches to the PP problem have used linearization or solved the problem in its original form; however, these algorithms are computationally limited and cannot solve large size problems. The present heuristics involve primal and dual methods. The primal method begins with a feasible PP solution and improves it by gradually increasing the values of variables; the dual method begins with all variables equal to one and decreases their values until a feasible solution is obtained. The methods were initially tested on problems of up to 50 variables and constraints, with encouraging results. Average effectiveness was over 99%, with a computation time averaging only 5% of the time required with other PP methods. The methods were also applied successfully to PP problems with up to 1,000 variables and 200 constraints. |
| Author | Granot, Frieda |
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| Copyright | Copyright 1982 The Institute of Management Sciences Copyright Institute for Operations Research and the Management Sciences Jul 1982 |
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| DOI | 10.1287/mnsc.28.7.829 |
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| Snippet | We develop in this paper two types of heuristic methods for solving the positive 0-1 polynomial programming (PP) problem of finding a 0-1 vector x that... Programming - Integer Algorithms, Heuristic In this research, heuristic algorithms for the positive 0-1 polynomial programming (PP) problem were developed. Previous approaches to the PP problem have used... |
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| SubjectTerms | Algorithms Computation Computer programming Constraints Heuristic Heuristic methods Heuristics Input output Integers Linear programming Management science Methodological problems Objective functions Optimal solutions Polynomials Variables Zero |
| Title | Efficient Heuristic Algorithms for Positive 0-1 Polynomial Programming Problems |
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