Convex Optimization · Problem 11.15

Interior-point methods — Boyd & Vandenberghe

Mathematics — Optimization · generalized-logarithm-properties

Problem

Let ψ:Rq→R\psi : \mathbf{R}^q \to \mathbf{R} be a generalized logarithm for the proper cone K⊆RqK \subseteq \mathbf{R}^q, with degree θ>0\theta > 0. Show that if y≻K0y \succ_K 0, then ∇ψ(y)≻K∗0,\nabla \psi(y) \succ_{K^*} 0, i.e., ψ\psi is KK-increasing.

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