v2007.09.13 - Convex Optimization

616 APPENDIX E. PROJECTIONWith reference to Figure 119, identifyingthenH + = {y ∈ R n | a T y ≥ σ C (a)} (87)‖x − P C x‖ = sup ‖x − P ∂H− x‖ = sup ‖a(a T a) −1 (a T x − σ C (a))‖∂H − | x∈H + a | x∈H += supa | x∈H +1‖a‖ |aT x − σ C (a)|(1776)which can be expressed as a convex optimization, for arbitrary positiveconstant τ‖x − P C x‖ = 1 τ maximize a T x − σ C (a)a(1777)subject to ‖a‖ ≤ τThe unique minimum-distance projection on convex set C is thereforewhere optimally ‖a ⋆ ‖= τ .P C x = x − a ⋆( a ⋆T x − σ C (a ⋆ ) ) 1τ 2 (1778)E.9.1.1.1 Exercise. Dual projection technique on polyhedron.Test that projection paradigm from Figure 119 on any convex polyhedralset.E.9.1.2Dual interpretation of projection on coneIn the circumstance set C is a closed convex cone K and there exists ahyperplane separating given point x from K , then optimal σ K (a ⋆ ) takesvalue 0 [147,C.2.3.1]. So problem (1777) for projection of x on K becomes‖x − P K x‖ = 1 τ maximize a T xasubject to ‖a‖ ≤ τ (1779)a ∈ K ◦The norm inequality in (1779) can be handled by Schur complement(3.1.7.2). Normals a to all hyperplanes supporting K belong to the polarcone K ◦ = −K ∗ by definition: (275)a ∈ K ◦ ⇔ 〈a, x〉 ≤ 0 for all x ∈ K (1780)