Problem: Linear Matrix Inequality Solution

Problem:

Prove that set of points \(x\) satisfying Linear Matrix Inequality is convex: \(x_1 A_1 + \dots + x_n A_n \preceq B\), with \(A_i , B \in \mathbb{S}^m\)

Proof:

Consider the set: \(C = \{x\| A(x) \preceq B\}\). If \(x\) belongs to this set, then \(B-A(x)\) is Postive Semi Definite matrix. In fact all such matrices form a cone which is convex. \(C\) is the inverse image of this cone under the affine function \(f: \mathbb{R}^n \longrightarrow \mathbb{S}^m, f(x) = B- A(x)\).

Other relevant problem: proof for convexity of ellipsoid:

Prove that ellipsoid is covex: \(\zeta = \{x \| (x-x_c)^T P^{-1} (x-x_c) \leq 1\}\).

Proof:

Ellipsoid is the invese image of unit ball under the affine mapping \(f(x) = P^{-1/2}(x-x_c)\).