Global optimization of the difference of affine increasing and positively homogeneous functions

Abstract

The theory of increasing and positively homogeneous (IPH) functions defined on a real topological vector space X has well been developed. In this paper, we first give various characterizations for maximal elements of the support set of this class of functions. As an application, we present various characterizations for maximal elements of the support set of affine IPH functions. Finally, we investigate necessary and sufficient conditions for the global minimum of the difference of two strictly affine IPH functions.

Keywords:

• Monotonic analysis
• increasing and positively homogeneous function
• affine IPH function
• support set
• maximal element
• global minimum
• abstract convexity

AMS Subject Classifications:

• 90C46
• 26B25
• 26A48

Acknowledgements

The authors are very grateful to the anonymous referee and the Associate Editor for their useful suggestions on an earlier version of this paper. The comments of the referee and the Associate Editor were very fruitful and these comments have enabled the authors to improve the paper significantly.

Notes

No potential conflict of interest was reported by the authors.