Lifted LMI sets, 2D examples (Matlab, YALMIP)

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 13:56

Summary:

This video presents examples of ‘lifted’ LMI sets (SDP-representable sets), i.e., projections of higher-dimensional LMI sets, involving additional ‘dummy’ decision variables. This video is a continuation of video [lmisets1EN], which introduced the concept of LMI set, its convexity and polynomial boundaries, and posed some non-lifted examples. The first three minutes of the present video review such concepts.

Then, several examples of lifted sets are posed:

• a square with rounded edges, a 4D LMI set projected in 2D,

• an ellipse defined by its foci, also a 4D to 2D lifted set.

• a 3-ellipse (guitar pick), generalising from 2 to 3 the number of foci, so it is represented as a 2D projection of a 5D LMI set,‘

• a generic, arbitrarily invented $4\times 4$ LMI in 5 variables, giving a convex shape with 4th-degree polynomial boundary (well, an extra linear constraint is added so degree might be considered to be 5) in the higher dimension.

Video [lmisets3EN] will discuss the details on plotting code, plus geometric and optimization problems that may be solved with these sets.

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