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Abstract
In this paper, we consider a subclass of linear semi-infinite programming problems whose constraint functions are polynomials in parameters and index sets are polyhedra. Based on Handelman’s representation of positive polynomials on a polyhedron, we propose two hierarchies of LP relaxations of the considered problem which respectively provide two sequences of upper and lower bounds of the optimum. These bounds converge to the optimum under some mild assumptions. Sparsity in the LP relaxations is explored for saving computational time and avoiding numerical ill behaviors.
Acknowledgements
The authors would like to acknowledge many helpful comments and suggestions from the referees.
Notes
No potential conflict of interest was reported by the authors.
Additional information
Funding
Feng Guo is supported by the Fundamental Research Funds for the Central Universities, the Chinese National Natural Science Foundation under [grant number 11401074], [grant number 11571350]. This work is supported by Fields Institute July–December 2015 Thematic Program on Computer Algebra in Toronto, Canada.