v2010.10.26 - Convex Optimization

352 CHAPTER 4. SEMIDEFINITE PROGRAMMINGwhere ε is a relatively small positive constant. This sequence is iterated untila direction vector y is found that makes |x ⋆ | T y ⋆ vanish. The term 〈t , ε1〉in (776) is necessary to determine absolute value |x ⋆ | = t ⋆ (3.2) becausevector y can have zero-valued entries. By initializing y to (1−ε)1, thefirst iteration of problem (776) is a 1-norm problem (514); id est,minimize 〈t , 1〉x∈R n , t∈R nsubject to Ax = b−t ≼ x ≼ t≡minimizex∈R n ‖x‖ 1subject to Ax = b(518)Subsequent iterations of problem (776) engaging cardinality term 〈t , y〉 canbe interpreted as corrections to this 1-norm problem leading to a 0-normsolution; vector y can be interpreted as a direction of search.4.5.2.1 local convergenceAs before (4.5.1.3), convex iteration (776) (525) always converges to a locallyoptimal solution; a fixed point of possibly infeasible cardinality.4.5.2.2 simple variations on a signed variableSeveral useful equivalents to linear programs (776) (525) are easily devised,but their geometrical interpretation is not as apparent: e.g., equivalent inthe limit ε→0 +minimize 〈t , y〉x∈R n , t∈R nsubject to Ax = b(777)−t ≼ x ≼ tminimizey∈R n 〈|x ⋆ |, y〉subject to 0 ≼ y ≼ 1y T 1 = n − k(525)We get another equivalent to linear programs (776) (525), in the limit, byinterpreting problem (518) as infimum to a vertex-description of the 1-normball (Figure 70, Example 3.2.0.0.1, confer (517)):minimizex∈R n ‖x‖ 1subject to Ax = b≡minimizea∈R 2n 〈a , y〉subject to [A −A ]a = ba ≽ 0(778)