v2010.10.26 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 333minimizeW ∈ S N f(Z ⋆ ) + w〈Z ⋆ , W 〉subject to 0 ≼ W ≼ ItrW = N − n(757)whose purpose is to constrain rank, affine dimension, or cardinality:The abstraction, that is Figure 98, is a synopsis; a broad generalizationof accumulated empirical evidence: There exists a critical (smallest) weightw c • for which a minimal-rank constraint is met. Graphical discontinuitycan subsequently exist when there is a range of greater w providing requiredrank k but not necessarily increasing a minimization objective function f ;e.g., Example 4.6.0.0.2. Positive scalar w is well chosen by cut-and-try.4.5 Constraining cardinalityThe convex iteration technique for constraining rank can be applied tocardinality problems. There are parallels in its development analogous tohow prototypical semidefinite program (649) resembles linear program (302)on page 275 [382]:4.5.1 nonnegative variableOur goal has been to reliably constrain rank in a semidefinite program. Thereis a direct analogy to linear programming that is simpler to present but,perhaps, more difficult to solve. In Optimization, that analogy is known asthe cardinality problem.Consider a feasibility problem Ax = b , but with an upper bound k oncardinality ‖x‖ 0 of a nonnegative solution x : for A∈ R m×n and vectorb∈R(A)find x ∈ R nsubject to Ax = bx ≽ 0‖x‖ 0 ≤ k(530)where ‖x‖ 0 ≤ k means 4.31 vector x has at most k nonzero entries; sucha vector is presumed existent in the feasible set. Nonnegativity constraint4.31 Although it is a metric (5.2), cardinality ‖x‖ 0 cannot be a norm (3.2) because it isnot positively homogeneous.