v2010.10.26 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 3350R 3 +∂H(confer Figure 84)1∂H = {x | 〈x, 1〉 = κ}Figure 99: 1-norm heuristic for cardinality minimization can be interpreted asminimization of a hyperplane, ∂H with normal 1, over nonnegative orthantdrawn here in R 3 . Polar of direction vector y = 1 points toward origin.at most k . Optimal direction vector y ⋆ is defined as any nonnegative vectorfor which(530)find x ∈ R nsubject to Ax = bx ≽ 0‖x‖ 0 ≤ k≡minimizex∈R n 〈x , y ⋆ 〉subject to Ax = bx ≽ 0(156)Existence of such a y ⋆ , whose nonzero entries are complementary to thoseof x ⋆ , is obvious assuming existence of a cardinality-k solution x ⋆ .4.5.1.2 direction vector interpretation(confer4.4.1.1) Vector y may be interpreted as a negative search direction;it points opposite to direction of movement of hyperplane {x | 〈x , y〉= τ}in a minimization of real linear function 〈x , y〉 over the feasible set inlinear program (156). (p.75) Direction vector y is not unique. The feasibleset of direction vectors in (525) is the convex hull of all cardinality-(n−k)one-vectors; videlicet,