Convex Optimization 2015.12.02

(\(f\) is log-convex \(\iff\) \(\log f\) is convex) \(\implies\) \(e^{\log f}\) convex \(\implies\) \(f\) is convex

(\(f\) is log-concave \(\iff\) \(\log f\) is concave)

  • Example: \(f(x) = a^T x + b\)

  • Example: \(f(X) = \det (X), f : S_{++}^n \to \mathbb{R}_+\)

is log-concave. Actually, already proved \(\log(\det(X))\) is a convex function.

  • Proof: \(g(t) = \log \det(X + tZ)\) for fiexed \(X, Z \in S_{++}^n\)

\(g'(t) = \frac{1}{\det(X + tZ)} \cdot \frac{d}{dt} \det (X + tZ)\)

\[\begin{align} g(t) & = \log (\det(X + tZ)) \\ & = \log (\det (X^{1/2} (I + tX^{-1/2} Z X^{-1/2}) X^{1/2})) \\ & = \end{align}\]