Convex Optimization · Problem 9.13

Equality constrained minimization — Boyd & Vandenberghe

Mathematics — Optimization · self-concordance-inverse-barrier

Problem

Self-concordance and the inverse barrier. ((a)) Show that

f(x) = 1/x

with domain

(0, 8/9)

is self-concordant. ((b)) Show that the function

f(x) = \alpha \sum_{i=1}^m \frac{1}{b_i - a_i^T x}

with

\mathbf{dom}\, f = \{x \in \mathbf{R}^n \mid a_i^T x < b_i,\, i = 1, \ldots, m\}

, is self-concordant if

\mathbf{dom}\, f

is bounded and

\alpha > (9/8) \max_{i=1,\ldots,m} \sup_{x \in \mathbf{dom}\, f} (b_i - a_i^T x).