Thursday, Apr 5, 2007, 11:00-12:00 pm	SOND 414
Classification of Homogeneous Cones
- Ms. Olena Shevchenko
Abstract: A cone K is called homogeneous if for any two points x and y in K there exists a non-degenerate linear automorphism A of K (i.e. AK=K) such that Ax=y. Vinberg partially classified these cones in terms of T-algebras. He proved that each homogeneous cone is isomorphic to a cone of generalized positive-definite Hermitian matrices. In the talk, we will present key results that lead to this classification. Time permitting, we will also discuss self-concordant barriers for homogeneous cones.
Thursday, Apr 12, 2007, 2:30-3:30 pm	SOND 414
On the finiteness of cone spectrum of certain linear transformations on Euclidean Jordan algebra
- Ms. Yihui Zhou
Abstract: Let $L$ be a linear transformation on a finite dimensional real Hilbert space $H$ and $K$ be a closed convex cone with dual $K^*$ in $H$. The cone spectrum of $L$ relative to $K$ is the set of all $\lambda \in R$ for which the linear complementarity problem $$x \in K,\ y=L(x)-\lambda x \in K^{*},\,\mbox{and}\,\, \langle x, y\rangle=0 $$ admits a nonzero solution $x\in H$. In the setting of a Euclidean Jordan algebra $H$ and the corresponding symmetric cone $K$, we discuss the finiteness of the cone spectrum for ${\bf Z}$-transformations and quadratic representations on $H$.
Thursday, May 3, 2007, 2:30-3:30 pm	SOND 414
Asymptotic Stability of Linear Complemenarity Systems
- Dr. Jinglai Shen
Abstract: I will talk about my recent result on stability analysis for linear complementarity systems (LCSs). Specifically, we show that asymptotic stability of an LCS is equivalent to its exponential stability. The cornerstone of the proof relies on the concepts of uniform asymptotic stability and positive homogeneity of the LCS, where the former is closely related to continuous solution dependence on initial conditions. We show the stability equivalence for both Lipschitz and non-Lipschitz LCSs. For the latter systems, the upper Lipschitz property of the LCP and differential inclusion formulation of the LCS is utilized. Finally, we will discuss Lyapunov stability of the (Lipschitz) LCSs satisfying B-singleton property from the conewise linear perspective. This perspective leads to simpler argument for the stability results.