Convex Optimization · Problem 2.29

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · cones-in-r2

Problem

Cones in

\mathbf{R}^2

. Suppose

K \subseteq \mathbf{R}^2

is a closed convex cone. ((a)) Give a simple description of

K

in terms of the polar coordinates of its elements (

x = r(\cos \phi, \sin \phi)

with

r \geq 0

). ((b)) Give a simple description of

K^*

, and draw a plot illustrating the relation between

K

and

K^*

. ((c)) When is

K

pointed? ((d)) When is

K

proper (hence, defines a generalized inequality)? Draw a plot illustrating what

x \preceq_K y

means when

K

is proper.