Convex Optimization · Problem 2.17

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · image-of-polyhedral-sets-under-perspective

Problem

Image of polyhedral sets under perspective function. In this problem we study the image of hyperplanes, halfspaces, and polyhedra under the perspective function

P(x, t) = x/t

, with

\mathbf{dom}\, P = \mathbf{R}^n \times \mathbf{R}_{++}

. For each of the following sets

C

, give a simple description of

P(C) = \{v/t \mid (v, t) \in C,\, t > 0\}

. ((a)) The polyhedron

C = \mathbf{conv}\{(v_1, t_1), \ldots, (v_K, t_K)\}

where

v_i \in \mathbf{R}^n

and

t_i > 0

. ((b)) The hyperplane

C = \{(v, t) \mid f^T v + g t = h\}

(with

f

and

g

not both zero). ((c)) The halfspace

C = \{(v, t) \mid f^T v + g t \leq h\}

(with

f

and

g

not both zero). ((d)) The polyhedron

C = \{(v, t) \mid Fv + gt \preceq h\}

Related theory

Image To Image Translation

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