v2010.10.26 - Convex Optimization

744 APPENDIX E. PROJECTIONTo project a matrix on nonnegative orthant R m×n+ , simply clip allnegative entries to 0. Likewise, projection on nonpositive orthant R m×n−sees all positive entries clipped to 0. Projection on other orthants isequally simple with appropriate clipping.Projecting on hyperplane, halfspace, slab:E.5.0.0.8.Projection of y ∈ R n on Euclidean ball B = {x∈ R n | ‖x − a‖ ≤ c} :for y ≠ a , P B y = (y − a) + a .c‖y−a‖Clipping in excess of |1| each entry of point x∈ R n is equivalent tounique minimum-distance projection of x on a hypercube centered atthe origin. (conferE.10.3.2)Projection of x∈ R n on a (rectangular) hyperbox: [61,8.1.1]C = {y ∈ R n | l ≼ y ≼ u, l ≺ u} (2041)⎧⎨ l k , x k ≤ l kP(x) k=0...n = x k , l k ≤ x k ≤ u k (2042)⎩u k , x k ≥ u kOrthogonal projection of x on a Cartesian subspace, whose basis issome given subset of the Cartesian axes, zeroes entries correspondingto the remaining (complementary) axes.Projection of x on set of all cardinality-k vectors {y | cardy ≤k} keepsk entries of greatest magnitude and clips to 0 those remaining.Unique minimum-distance projection of H ∈ S n on positive semidefinitecone S n + in Euclidean/Frobenius sense is accomplished by eigenvaluedecomposition (diagonalization) followed by clipping all negativeeigenvalues to 0.Unique minimum-distance projection on generally nonconvex subset ofall matrices belonging to S n + having rank not exceeding ρ (2.9.2.1)is accomplished by clipping all negative eigenvalues to 0 and zeroingsmallest nonnegative eigenvalues keeping only ρ largest. (7.1.2)