v2006.03.09 - Convex Optimization

6.2. FRAMEWORK 369When equality is attained in (870)rankX ⋆ + rankS ⋆ = n (872)there are no coinciding main-diagonal zeros in Λ X ⋆Λ S ⋆ , and so we have whatis called strict complementarity. 6.10 Logically it follows that a necessary andsufficient condition for strict complementarity of an optimal primal and dualsolution isX ⋆ + S ⋆ ≻ 0 (873)The beauty of Corollary 6.2.3.0.1 is its conjugacy; id est, one can solveeither the primal or dual problem and then find a solution to the other via theoptimality conditions. When a dual optimal solution is known, for example,a primal optimal solution belongs to the hyperplane {X | 〈S ⋆ , X〉=0}.6.2.3.0.2 Example. Minimum cardinality Boolean. [52] [24,4.3.4] [223]Consider finding a minimum cardinality Boolean solution x to the classiclinear algebra problem Ax = b given noiseless data A∈ R m×n and b∈ R m ;minimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... n(874)where ‖x‖ 0 denotes cardinality of vector x (a.k.a, 0-norm; not a convexfunction).A minimum cardinality solution answers the question: “Which fewestlinear combination of columns in A constructs vector b ?” Cardinalityproblems have extraordinarily wide appeal, arising in many fields of scienceand across many disciplines. [203] [132] [98] [99] Yet designing an efficientalgorithm to optimize cardinality has proved difficult. In this example, wealso constrain the variable to be Boolean. The Boolean constraint forcesan identical solution were the norm in problem (874) instead the 1-norm or2-norm; id est, the two problems6.10 distinct from maximal complementarity (6.1.1).