Convex Optimization · Problem 2.31

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · properties-of-dual-cones

Problem

Properties of dual cones. Let

K^*

be the dual cone of a convex cone

K

, as defined in (2.19). Prove the following. ((a))

K^*

is indeed a convex cone. ((b))

K_1 \subseteq K_2

implies

K_2^* \subseteq K_1^*

. ((c))

K^*

is closed. ((d)) The interior of

K^*

is given by

\mathbf{int}\, K^* = \{y \mid y^T x > 0 \text{ for all } x \in \mathbf{cl}\, K\}

. ((e)) If

K

has nonempty interior then

K^*

is pointed. ((f))

K^{**}

is the closure of

K

. (Hence if

K

is closed,

K^{**} = K

.) ((g)) If the closure of

K

is pointed then

K^*

has nonempty interior.