### Geometric properties for level sets of quadratic functions and the theorem of S-lemma with equality

** Time:** 10:00 to 11:30 Ngày 09/01/2019

**Venue/Location: ** C2-714, VIASM

**Speaker:**
Prof. Ruey-Lin Sheu

**Content:**

This talk is motivated by a recent work of Xia, Wangand, Sheu where they studied the theorem of S-lemma withequality. Instead of a hard lengthy analysis, we provide an elegantand neat geometric characterization for the S-lemma with equality.We show that, for any pair of quadratic functions $(f,g)$ with $g$satisfying the two-side Slater condition, the theorem fails if andonly if the sublevel set $\{x\in{R}^n|f(x)<0\}$ consists of twoconvex connected components; the hypersurface $\{x\in{R}^n|g(x)=0\}$is an affine subspace; and the affine subspace $\{g=0\}$ separatesthe two convex branches of $\{f<0\}.$ It can be also shown that,under the two-side slater condition, the theorem holds if and onlyif the sublevel set $\{f<0\}$ is connected or the hypersurface$\{x\in{R}^n|g(x)=0\}$ is not an affine subspace. Variousapplications of the related S-lemma and the separation theorem arealso discussed.