Convex Optimization · Problem 9.14

Equality constrained minimization — Boyd & Vandenberghe

Mathematics — Optimization · composition-with-logarithm-self-concordance

Problem

Composition with logarithm. Let

g : \mathbf{R} \to \mathbf{R}

be a convex function with

\mathbf{dom}\, g = \mathbf{R}_{++}

, and

|g'''(x)| \leq 3 \frac{g''(x)}{x}

for all

x

. Prove that

f(x) = -\log(-g(x)) - \log x

is self-concordant on

\{x \mid x > 0,\, g(x) < 0\}

. Hint. Use the inequality

\frac{3}{2} r p^2 + q^3 + \frac{3}{2} p^2 q + r^3 \leq 1

which holds for

p, q, r \in \mathbf{R}_+

with

p^2 + q^2 + r^2 = 1