×

A weighted analytic center for linear matrix inequalities. (English) Zbl 1017.90113

Summary: Let \(\mathcal R\) be the convex subset of \(\mathbb{R}^n\) defined by \(q\) simultaneous linear matrix inequalities (LMI) \(A_0^{(j)}+\sum^n_{i=1}x_iA_i^{(j)}\succ 0\), \(j=1,2,\dots, q\). Given a strictly positive vector \(\omega=(\omega_1,\omega_2,\cdots,\omega_q)\) the weighted analytic center \(x_{ac}(\omega)\) is the minimizer \(\text{argmin}(\phi_\omega(x))\) of the strictly convex function over \(\phi_\omega(x)=\sum^q_{j=1}\omega_j\log \det[A^{(j)}(x)]^{-1}\) 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 \(\mathcal W=\{x_{ac}(\omega)\mid \omega> 0\}\subseteq\mathcal R\) 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 \(\mathcal W\) nd its boundary are described by real algebraic varieties, and provide slices of a non-trivial real algebraic variety to show that \(\mathcal W\) is not 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.

MSC:

90C32 Fractional programming
90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
PDFBibTeX XMLCite
Full Text: EuDML