Convex Optimization · Problem 7.10

Geometric problems — Boyd & Vandenberghe

Mathematics — Optimization · nonparametric-distributions-covariance-bound

Problem

Nonparametric distributions on

\mathbf{R}^k

. We consider a random variable

x \in \mathbf{R}^k

with values in a finite set

\{\alpha_1, \ldots, \alpha_n\}

, and with distribution

p_i = \mathbf{prob}(x = \alpha_i), \quad i = 1, \ldots, n.

Show that a lower bound on the covariance of

X

S \preceq \mathbf{E}(X - \mathbf{E}\, X)(X - \mathbf{E}\, X)^T,

is a convex constraint in

p