JIPAM logo: Home Link
Home Editors Submissions Reviews Volumes RGMIA About Us

  Volume 2, Issue 3, Article 29
A Weighted Analytic Center for Linear Matrix Inequalities

    Authors: Irwin S. Pressman, Shafiu Jibrin,  
    Keywords: Weighted analytic center, Semidefinite Programming, Linear Matrix Inequalities, Convexity, Real Algebraic Variety.  
    Date Received: 21/03/01  
    Date Accepted: 21/03/01  
    Subject Codes:


    Editors: Jonathan Borwein,  

Let ${mathcal R}$ be the convex subset of $ {{rlap{rm I}hskip0.11emhbox{rm R}^{n}}}$ defined by $q$ simultaneous linear matrix inequalities (LMI)$A_{0}^{(j)}+sum_{i=1}^{n}x_{i}A_{i}^{(j)}succ 0, j=1,2,dots,q$. Given a strictly positive vector$boldsymbol{omega}=(omega_{1},omega_{2},cdots,omega_{q})$, the weighted analytic center $x_{ac}(boldsymbol{omega})$ is the minimizerargmin $(phi_{omega}(x))$ of the strictly convex function $phi_{omega}(x)=sum_{j=1}^{q}omega_{j}logdet[A^{(j)}(x)]^{-1$over ${mathcal R}$. We give a necessary and sufficient condition for a point of${mathcal R}$ to be a weighted analytic center. We study the argmin function in this instance and show that it is a continuously differentiable open function.

In the special case of linear constraints, all interior points are weighted analytic centers. We show that the region ${mathcal W} = left{x_{ac}(boldsymbol{omega})mid boldsymbol{omega>0 right}subseteq {mathcal R}$ of weighted analytic centers for LMI's is not convex and does not generally equal ${mathcal R}$. These results imply that the techniques in linear programming of following paths of analytic centers may require special consideration when extended to semidefinite programming. We show that the region ${mathcal W}$ and its boundary are described by real algebraic varieties, and provide slices of a non-trivial real algebraic variety to show that ${mathcal W}$ isn't convex. Stiemke's Theorem of the alternative provides a practical test of whether a point is in ${mathcal W}$. Weighted analytic centers are used to improve the location of standing points for the Stand and Hit method of identifying necessary LMI constraints in semidefinite programming.

  Download Screen PDF
  Download Print PDF
  Send this article to a friend
  Print this page

      search [advanced search] copyright 2003 terms and conditions login