Skip Navigation


IMA Journal of Mathematical Control and Information Advance Access originally published online on October 30, 2009
IMA Journal of Mathematical Control and Information 2009 26(4):417-450; doi:10.1093/imamci/dnp023
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
26/4/417    most recent
dnp023v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Yildiran, U.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Convex hull of two quadratic constraints is an LMI set

Ugur Yildiran{dagger}

Department of Systems Engineering, Yeditepe University, Istanbul, Turkey and Department of Electrical and Electronics Engineering, Bogaziçi University, Istanbul, Turkey

{dagger} Email: uyildiran{at}yeditepe.edu.tr, yildiranu{at}hotmail.com

Received on March 26, 2008; Revision received September 15, 2009. Accepted on September 16, 2009

In this work, we are interested in the convex hull of the region determined by two quadratic polynomial constraints. The main result is that if this region is not empty, the convex hull is either Rn or the feasible set of another pair of quadratic constraints which are, in fact, positive linear combinations of the original ones. Based on this result, a losslessness condition is also derived for the classical semidefinite programming relaxation. The characterization of the convex hull we found does not have to be composed of concave quadratic constraints. However, we propose an algorithm to convert them into linear matrix inequalities (LMIs) and explain how the LMI characterization can be employed to solve a certain class of non-convex optimization problems. It is shown that this approach may perform better than the available relaxation methods for the optimization problem considered. Lastly, we show how the results developed can be applied to a certain class of control problems.

Keywords: S-lemma; SDP relaxation; LMI representation; quadratic programming; convex hull; matrix pencil.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.