v2010.10.26 - Convex Optimization

334 CHAPTER 4. SEMIDEFINITE PROGRAMMINGx ≽ 0 is analogous to positive semidefiniteness; the notation means vector xbelongs to the nonnegative orthant R n + . Cardinality is quasiconcave on R n +just as rank is quasiconcave on S n + . [61,3.4.2]4.5.1.1 direction vectorWe propose that cardinality-constrained feasibility problem (530) can beequivalently expressed as a sequence of convex problems: for 0≤k ≤n−1minimizex∈R n 〈x , y〉subject to Ax = bx ≽ 0(156)n∑π(x ⋆ ) i =i=k+1minimizey∈R n 〈x ⋆ , y〉subject to 0 ≼ y ≼ 1y T 1 = n − k(525)where π is the (nonincreasing) presorting function. This sequence is iterateduntil x ⋆T y ⋆ vanishes; id est, until desired cardinality is achieved. But thisglobal convergence cannot always be guaranteed. 4.32Problem (525) is analogous to the rank constraint problem; (p.308)N∑λ(G ⋆ ) ii=k+1= minimizeW ∈ S N 〈G ⋆ , W 〉subject to 0 ≼ W ≼ ItrW = N − k(1700a)Linear program (525) sums the n−k smallest entries from vector x . Incontext of problem (530), we want n−k entries of x to sum to zero; id est,we want a globally optimal objective x ⋆T y ⋆ to vanish: more generally,n∑π(|x ⋆ |) i = 〈|x ⋆ |, y ⋆ 〉 0 (758)i=k+1defines global convergence for the iteration. Then n−k entries of x ⋆ arethemselves zero whenever their absolute sum is, and cardinality of x ⋆ ∈ R n is4.32 When it succeeds, a sequence may be regarded as a homotopy to minimal 0-norm.