v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
736 APPENDIX E. PROJECTION∂H −C(a)P C xH −H +κaP ∂H− xx(b)Figure 164: Dual interpretation of projection of point x on convex set Cin R 2 . (a) κ = (a T a) −1( a T x − σ C (a) ) (b) Minimum distance from x toC is found by maximizing distance to all hyperplanes supporting C andseparating it from x . A convex problem for any convex set, distance ofmaximization is unique.
E.9. PROJECTION ON CONVEX SET 737With reference to Figure 164, identifyingthenH + = {y ∈ R n | a T y ≥ σ C (a)} (107)‖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)|(2013)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(2014)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 (2015)E.9.1.1.1 Exercise. Dual projection technique on polyhedron.Test that projection paradigm from Figure 164 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 [199,C.2.3.1]. So problem (2014) for projection of x on K becomes‖x − P K x‖ = 1 τ maximize a T xasubject to ‖a‖ ≤ τ (2016)a ∈ K ◦The norm inequality in (2016) can be handled by Schur complement (3.5.2).Normals a to all hyperplanes supporting K belong to the polar coneK ◦ = −K ∗ by definition: (319)a ∈ K ◦ ⇔ 〈a, x〉 ≤ 0 for all x ∈ K (2017)
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- Page 699 and 700: Appendix EProjectionFor any A∈ R
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- Page 769 and 770: Appendix FNotation and a few defini
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736 APPENDIX E. PROJECTION∂H −C(a)P C xH −H +κaP ∂H− xx(b)Figure 164: Dual interpretation of projection of point x on convex set Cin R 2 . (a) κ = (a T a) −1( a T x − σ C (a) ) (b) Minimum distance from x toC is found by maximizing distance to all hyperplanes supporting C andseparating it from x . A convex problem for any convex set, distance ofmaximization is unique.
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