LMI Studygroup Questions - Session 1 (2/7/94)

1. (i) By using Schur complements, or otherwise, write the quadratic matix inequality

Y′Y ≤ γ^{2}U′U

as an LMI in the variable matrix Y.

(ii) Prove your result in part (i). (HINT: Try to remember one of Sanj’s two proofs of Schur complements.)

2. Let E be a “hyperellipsoid” in IR^{n} defined by

E = {x : x′P x + 2q′x + r ≤ 0, P = P′ > 0} .

Let x_{0} be a point not in E (i.e. x_{0} satisfies the inequality x′_{0}P x_{0} + 2q′x_{0} + r > 0.) You are required to calculate the equation of a cutting plane which has x_{0} on one side, and the entire ellipsoid on the other. To make the “best cut possible” calculate the equation of the cutting plane that touches the hyperellipsoid at the point where the line joining x_{0} and the ellipsoid’s centre, intersects it (draw a picture!) Proceed as follows:

(i) Find the centre, c, of E where c is the value of x such that x′P x + 2q′x + r is minimised.

(ii) Let a point on the line through x_{0} and c be given by

x = c + α(x_{0} − c).  (1)

Calculate the value of α for which x lies on the boundary of E.

(iii) Let α̂ be the value of α calculated in (ii) and x̂ be the corresponding point from equation 1. Calculate the normal, ∇f(x̂) to E at x̂ and hence show that the required hyperplane satisfies

∇f(x̂)(x − x̂) = 0.

3. Let E_{i} be ellipsoids as defined in the previous question. Use the S-procedure to form an LMI that finds an ellipsoid, E_{0}, that

(i) lies inside the intersection of n E_{i}’s.

(ii) encloses the intersection of n E_{i}’s.

Rob Davis - 27/7/94