## Polynomial Programming Using Groebner Bases (1994)

### BibTeX

@MISC{Chang94polynomialprogramming,

author = {Yao-jen Chang and Benjamin W. Wah},

title = {Polynomial Programming Using Groebner Bases},

year = {1994}

}

### OpenURL

### Abstract

Finding the global optimal solution for a general nonlinear program is a difficult task except for very small problems. In this paper we identify a class of nonlinear programming problems called polynomial programming problems (PP). A polynomial program is an optimization problem with a scalar polynomial objective function and a set of polynomial constraints. By using Groebner Bases, we can determine the global minimum of a polynomial program in a reasonable amount of time and memory. 1 Introduction In almost every engineering discipline, the need for optimizing a particular application at hand arises constantly. In engineering applications, global solutions mean the most efficient use of resources under practical constraints. Optimization techniques have been a fruitful domain in engineering research. For linear and quadratic programming problems, efficient algorithms have been developed to guarantee global solutions with convexity. For general nonlinear programs, various optimizatio...

### Citations

124 |
Solving polynomial systems using continuation for engineering and scientific problems
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- 1987
(Show Context)
Citation Context ...athematical form becomes robust enough to tolerate numerical errors. 2 Modeling Power of Polynomial Programming Polynomial programs have a wide spectrum of applications. Many examples can be found in =-=[1, 2]-=-. A typical chemical equilibrium system can be modeled by ten to twenty equations in ten to twenty unknowns. In robotics, a six-joint robot can involve a polynomial program with eighteen equations in ... |

105 |
An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal (in German
- Buchberger
- 1965
(Show Context)
Citation Context ...ys exist, are canonical and finite, and are able to derive inconsistencies if the original set of equations are inconsistent. The algorithm for constructing Groebner bases is due to Buchberger (1965) =-=[5]-=-. We do not intend to state it in a rigorous and formal fashion. Instead, we give the following pseudo code. Groebner Basis Construction Algorithm Given P: a polynomial system consisting of p 1 ; p 2 ... |

54 |
GrÃ¶bner Bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations
- Lazard
- 1983
(Show Context)
Citation Context ...epeated. This is the essence of the algorithm used to construct a Groebner basis. Principal terms are defined with respect to an ordering that is lexicographic or inverse lexicographic. Lazard (1983) =-=[6]-=- has shown that lexicographic orderings lead to better bounds on the degree of the Groebner basis obtained. We see in this algorithm that s i;j is defined such that principal terms p i and p j are can... |

29 |
On the stationary values of a second-degree polynomial on the unit sphere
- Forsythe, Golub
- 1965
(Show Context)
Citation Context ...an n-vector b, and a scalar ff, the following polynomial program minimize x T Ax + b T x subject to : jjxjj 2 2 = n X i=1 x 2 i = ff 2 is the Trust-Region Problem studied by Forsythe and Golub (1965) =-=[9]-=-, Spjoetvoll (1972) [10], and Lyle and Szularz (1994) [11]. It is equivalent to minimizing a second-degree polynomial equation in the unit sphere. Historically, eigensystems are analyzed to solve this... |

18 |
Algebraic Computing with REDUCE
- MacCallum, Wright
- 1991
(Show Context)
Citation Context ...omial system is inconsistent if and only if its Groebner basis contains a non-zero constant. 4. Groebner bases are finite. These results have been stated more precisely by MacCallum and Wright (1991) =-=[4]-=- and Buchberger (1983) [3]. These results show that Groebner bases always exist, are canonical and finite, and are able to derive inconsistencies if the original set of equations are inconsistent. The... |

15 |
Linear Programs and Related Problems
- Nering, Tucker
- 1993
(Show Context)
Citation Context ...nstrate the ability of the optimization algorithm presented in the last section to find global optima for general polynomial programs. The first two problems are adapted from Nering and Tucker (1993) =-=[7]-=- and Barbeau (1989) [8], respectively. The third problem is a famous problem in control theory, namely, the trust-region problem. In the last problem, we show how a general nonlinear program can be ap... |

5 |
Local minima of the trust region problem
- Lyle, Szularz
- 1994
(Show Context)
Citation Context ...program minimize x T Ax + b T x subject to : jjxjj 2 2 = n X i=1 x 2 i = ff 2 is the Trust-Region Problem studied by Forsythe and Golub (1965) [9], Spjoetvoll (1972) [10], and Lyle and Szularz (1994) =-=[11]-=-. It is equivalent to minimizing a second-degree polynomial equation in the unit sphere. Historically, eigensystems are analyzed to solve this problem numerically. Since this problem can be convenient... |

2 |
Pardalos, "A collection of test problems for constrained global optimization algorithms
- Floudas, M
- 1990
(Show Context)
Citation Context ...athematical form becomes robust enough to tolerate numerical errors. 2 Modeling Power of Polynomial Programming Polynomial programs have a wide spectrum of applications. Many examples can be found in =-=[1, 2]-=-. A typical chemical equilibrium system can be modeled by ten to twenty equations in ten to twenty unknowns. In robotics, a six-joint robot can involve a polynomial program with eighteen equations in ... |

2 |
Committee on Information Systems Trustworthiness
- Buchberger
- 1983
(Show Context)
Citation Context ...out much difficulty. We know that Gaussian elimination serves this purpose for a linear system. Buchberger carried this procedure one step further to extend Gaussian elimination to polynomial systems =-=[3]-=-. Groebner Basis. A Groebner basis for a polynomial system is a canonical form that represents the original system. It can be shown that a Groebner basis contains the same information as the original ... |

1 |
A note on a theorem of forsythe and golub
- Spjoetvoll
- 1972
(Show Context)
Citation Context ...alar ff, the following polynomial program minimize x T Ax + b T x subject to : jjxjj 2 2 = n X i=1 x 2 i = ff 2 is the Trust-Region Problem studied by Forsythe and Golub (1965) [9], Spjoetvoll (1972) =-=[10]-=-, and Lyle and Szularz (1994) [11]. It is equivalent to minimizing a second-degree polynomial equation in the unit sphere. Historically, eigensystems are analyzed to solve this problem numerically. Si... |