v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
620 APPENDIX E. PROJECTIONE.9.2.2.2 Corollary. Unique projection via dual or normal cone.[73,4.7] (E.10.3.2, confer Theorem E.9.1.0.3) Given point x∈ R n andclosed convex cone K ⊆ R n , the following are equivalent statements:1. point Px is the unique minimum-distance projection of x on K2. Px ∈ K , x − Px ∈ −(K − Px) ∗ = −K ∗ ∩ (Px) ⊥3. Px ∈ K , 〈x − Px, Px〉 = 0, 〈x − Px, y〉 ≤ 0 ∀y ∈ K⋄E.9.2.2.3 Example. Unique projection on nonnegative orthant.(confer (1122)) From Theorem E.9.2.0.1, to project matrix H ∈ R m×n onthe self-dual orthant (2.13.5.1) of nonnegative matrices R m×n+ in isomorphicR mn , the necessary and sufficient conditions are:H ⋆ ≥ 0tr ( (H ⋆ − H) T H ⋆) = 0H ⋆ − H ≥ 0(1795)where the inequalities denote entrywise comparison. The optimal solutionH ⋆ is simply H having all its negative entries zeroed;H ⋆ ij = max{H ij , 0} , i,j∈{1... m} × {1... n} (1796)Now suppose the nonnegative orthant is translated by T ∈ R m×n ; id est,consider R m×n+ + T . Then projection on the translated orthant is [73,4.8]H ⋆ ij = max{H ij , T ij } (1797)E.9.2.2.4 Example. Unique projection on truncated convex cone.Consider the problem of projecting a point x on a closed convex cone thatis artificially bounded; really, a bounded convex polyhedron having a vertexat the origin:minimize ‖x − Ay‖ 2y∈R Nsubject to y ≽ 0(1798)‖y‖ ∞ ≤ 1
E.9. PROJECTION ON CONVEX SET 621where the convex cone has vertex-description (2.12.2.0.1), for A∈ R n×NK = {Ay | y ≽ 0} (1799)and where ‖y‖ ∞ ≤ 1 is the artificial bound. This is a convex optimizationproblem having no known closed-form solution, in general. It arises, forexample, in the fitting of hearing aids designed around a programmablegraphic equalizer (a filter bank whose only adjustable parameters are gainper band each bounded above by unity). [66] The problem is equivalent to aSchur-form semidefinite program (3.1.7.2)minimizey∈R N , t∈Rsubject tot[tI x − Ay(x − Ay) T t]≽ 0(1800)0 ≼ y ≼ 1E.9.3nonexpansivityE.9.3.0.1 Theorem. Nonexpansivity. [124,2] [73,5.3]When C ⊂ R n is an arbitrary closed convex set, projector P projecting on Cis nonexpansive in the sense: for any vectors x,y ∈ R n‖Px − Py‖ ≤ ‖x − y‖ (1801)with equality when x −Px = y −Py . E.16⋄Proof. [40]‖x − y‖ 2 = ‖Px − Py‖ 2 + ‖(I − P )x − (I − P )y‖ 2+ 2〈x − Px, Px − Py〉 + 2〈y − Py , Py − Px〉(1802)Nonnegativity of the last two terms follows directly from the uniqueminimum-distance projection theorem (E.9.1.0.2).E.16 This condition for equality corrects an error in [56] (where the norm is applied to eachside of the condition given here) easily revealed by counter-example.
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E.9. PROJECTION ON CONVEX SET 621where the convex cone has vertex-description (2.12.2.0.1), for A∈ R n×NK = {Ay | y ≽ 0} (1799)and where ‖y‖ ∞ ≤ 1 is the artificial bound. This is a convex optimizationproblem having no known closed-form solution, in general. It arises, forexample, in the fitting of hearing aids designed around a programmablegraphic equalizer (a filter bank whose only adjustable parameters are gainper band each bounded above by unity). [66] The problem is equivalent to aSchur-form semidefinite program (3.1.7.2)minimizey∈R N , t∈Rsubject tot[tI x − Ay(x − Ay) T t]≽ 0(1800)0 ≼ y ≼ 1E.9.3nonexpansivityE.9.3.0.1 Theorem. Nonexpansivity. [124,2] [73,5.3]When C ⊂ R n is an arbitrary closed convex set, projector P projecting on Cis nonexpansive in the sense: for any vectors x,y ∈ R n‖Px − Py‖ ≤ ‖x − y‖ (1801)with equality when x −Px = y −Py . E.16⋄Proof. [40]‖x − y‖ 2 = ‖Px − Py‖ 2 + ‖(I − P )x − (I − P )y‖ 2+ 2〈x − Px, Px − Py〉 + 2〈y − Py , Py − Px〉(1802)Nonnegativity of the last two terms follows directly from the uniqueminimum-distance projection theorem (E.9.1.0.2).E.16 This condition for equality corrects an error in [56] (where the norm is applied to eachside of the condition given here) easily revealed by counter-example.
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