On the equivalence of algebraic conditions for convexity and quasiconvexity of polynomials
Name
9BF17d01.pdf
Size
204.54 KB
Format
Adobe PDF
Checksum (MD5)
55173bb1701e05928ccfb536fc5f15d2
Author(s)
Ahmadi, Amir Ali
Parrilo, Pablo A.
Date Issued
February 2011
Journal
Proceedings of the 49th IEEE Conference on Decision and Control (CDC), 2010
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Ahmadi, Amir Ali, and Pablo A. Parrilo. “On the Equivalence of Algebraic Conditions for Convexity and Quasiconvexity of Polynomials.” IEEE, 2010. 3343–3348. © Copyright 2010 IEEE
Version
Final published version
Abstract
This paper is concerned with algebraic relaxations, based on the concept of sum of squares decomposition, that give sufficient conditions for convexity of polynomials and can be checked efficiently with semidefinite programming. We propose three natural sum of squares relaxations for polynomial convexity based on respectively, the definition of convexity, the first order characterization of convexity, and the second order characterization of convexity. The main contribution of the paper is to show that all three formulations are equivalent; (though the condition based on the second order characterization leads to semidefinite programs that can be solved much more efficiently). This equivalence result serves as a direct algebraic analogue of a classical result in convex analysis. We also discuss recent related work in the control literature that introduces different sum of squares relaxations for polynomial convexity. We show that the relaxations proposed in our paper will always do at least as well the ones introduced in that work, with significantly less computational effort. Finally, we show that contrary to a claim made in the same related work, if an even degree polynomial is homogeneous, then it is quasiconvex if and only if it is convex. An example is given.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1109/CDC.2010.5717510
