Convex Optimization · Problem 6.13

Statistical estimation — Boyd & Vandenberghe

Mathematics — Optimization · interpolation-quasiconvex-functions

Problem

Interpolation with quasiconvex functions. Show that there exists a quasiconvex function

f : \mathbf{R}^k \to \mathbf{R}

, that satisfies

f(u_i) = y_i

for

i = 1, \ldots, m

, if and only if there exist

g_i \in \mathbf{R}^k

i = 1, \ldots, m

, such that

g_i^T(u_j - u_i) \leq -1 \text{ whenever } y_j < y_i, \quad i,\ j = 1, \ldots, m.