v2010.10.26 - Convex Optimization

742 APPENDIX E. PROJECTIONwhere 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} (2032)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 [109,4.8]H ⋆ ij = max{H ij , T ij } (2033)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(2034)‖y‖ ∞ ≤ 1where the convex cone has vertex-description (2.12.2.0.1), for A∈ R n×NK = {Ay | y ≽ 0} (2035)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). [96] The problem is equivalent to aSchur-form semidefinite program (3.5.2)minimizey∈R N , t∈Rsubject tot[tI x − Ay(x − Ay) T t]≽ 0(2036)0 ≼ y ≼ 1