LMI sets (4): scaling, perspective cones, set interpolation, convex hull

Antonio Sala, UPV

Difficulty: ***** ,       Relevance: ,      Duration: 20:15

Summary:

This video discusses some transformations and operations with LMI sets (SDP-representable sets), introduced in videos [lmisets1EN] and [lmisets2EN].

The basic ideas are the following two ones:

1. Even if we name LMIs as ‘Linear’ matrix inequalities, they are actually ’ affine’, with a constant term $LMI(0)$ plus a truly linear one $LM{I}_L(x)$ so $LM{I}_L(ax+by)=a\cdot LMI(x)+b\cdot LMI(y)$. The linear component is, henceforth, $LM{I}_L(x)=LMI(x)-LMI(0)$.

2. if $x$ is in the set such that $LMI(x)\succcurlyeq 0$, the transformation $z=Tx+q$ converts it to an LMI set in variable $z$ given by $LMI(T^{-1}(z-q))\succcurlyeq 0$.

From these two ideas, in this video we discuss:

• Scaling an LMI set

• Forming the ‘perspective cone’ in one more dimension, so the sections are the LMI set, whose projection is the union of all scalings in an interval

• Changing the apex (scaling point)

• Convex hull of LMI sets

• Convex interpolation of LMI sets (direct sum of $\lambda$ and $(1-\lambda )$ scaled versions).

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.