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