LMI sets (SDP-representable sets, spectrahedra): definition, basic properties and 2D examples

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 17:42

Summary:

This video defines an ‘LMI set’, also known as ‘SDP-representable’ set, as the set of feasible values of a linear matrix inequality $LMI(x)\succcurlyeq 0$, where $\succcurlyeq 0$ means being positive semidefinite. The name of ‘spectrahedron’ (plural spectrahedra) is, too, used to denote these sets of feasible values of a semidefinite program.

Convexity of the cone of positive semidefinite matrices implies that LMi sets are convex sets. As positive-semidefiniteness of a matrix is equivalent to all principal minors being non-negative, then, LMI sets are semialgebraic ones with polynomial boundary.

Examples of LMI sets are given, such as circles, ellipses, polyhedra, cones and a cubic egg-shaped set. All examples are in 2D, for ease of visualization, albeit of course LMI sets can be defined in any dimension.

The next video [lmisets2EN] will describe examples of the so-called ‘lifted’ LMI sets, projections on a reduced dimension of LMI sets in a higher-dimensional space.

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