Applying Gröbner Basis Method to Multiparametric Polynomial Nonlinear Programming
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In this paper, we present first a new algorithm based on Gröbner basis to analyze and/or solve a convex polynomial nonlinear programming problem that is a convex nonlinear programming in which the objective and constraints are algebraic polynomials. The main efficiency of our algorithm is that there is no need to compute feasible points to find the optimum value. Next, we generalize our results to analyze and/or solvemultiparametric convex polynomial nonlinear programming problems. The mainproperty of this method is that it preserves the parametric scheme of theproblem until the end of the algorithm. Even the output of our algorithm depends onparameters and so one can present the optimum value and optimizer points as afunction on parameters. To show the ability of this algorithm, we will state two applied examples: the problem of minimizing the expense of forage in a chicken farm with unpredictable price of forage and the number of chickens, and an optimal control problem in model predictive control. The presented algorithms in this paper are all implemented in the Maple software.
KeywordsPolynomial nonlinear programming Multiparametric polynomial nonlinear programming Eigenvalue system Gröbner basis Comprehensive Gröbner system
Mathematics Subject Classification13P25 13P10
The authors are thankful to Dr. Amir Hashemi for his useful comments onpreparing this paper. They would also like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.
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