Convex Optimization · Problem 11.17

Interior-point methods — Boyd & Vandenberghe

Mathematics — Optimization · dual-generalized-logarithm

Problem

Let ψ\psi be a generalized logarithm for the proper cone KK, with degree θ\theta. Define the dual logarithm ψ‾(v)=−(−ψ)∗(−v)=inf⁡u(vTu−ψ(u)),dom ψ‾=int K∗.\overline{\psi}(v) = -(-\psi)^*(-v) = \inf_u \left( v^T u - \psi(u) \right), \quad \mathbf{dom}\,\overline{\psi} = \mathbf{int}\,K^*. Show that ψ‾\overline{\psi} is a generalized logarithm for the dual cone K∗K^*, with the same degree θ\theta.

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