v2010.10.26 - Convex Optimization

294 CHAPTER 4. SEMIDEFINITE PROGRAMMINGWe can reformulate this minimal cardinality Boolean problem (680) as asemidefinite program: First transform the variablex (ˆx + 1) 1 2(688)so ˆx i ∈ {−1, 1} ; equivalently,minimize ‖(ˆx + 1) 1‖ ˆx2 0subject to A(ˆx + 1) 1 = b 2δ(ˆxˆx T ) = 1(689)where δ is the main-diagonal linear operator (A.1). By assigning (B.1)[ ˆxG =1] [ ˆx T 1] =[ X ˆxˆx T 1][ ˆxˆxT ˆxˆx T 1]∈ S n+1 (690)problem (689) becomes equivalent to: (Theorem A.3.1.0.7)minimize 1 TˆxX∈ S n , ˆx∈R nsubject to A(ˆx + 1) 1[ = b 2] X ˆxG = (≽ 0)ˆx T 1δ(X) = 1rankG = 1(691)where solution is confined to rank-1 vertices of the elliptope in S n+1(5.9.1.0.1) by the rank constraint, the positive semidefiniteness, and theequality constraints δ(X)=1. The rank constraint makes this problemnonconvex; by removing it 4.15 we get the semidefinite program4.15 Relaxed problem (692) can also be derived via Lagrange duality; it is a dual of adual program [sic] to (691). [304] [61,5, exer.5.39] [377,IV] [150,11.3.4] The relaxedproblem must therefore be convex having a larger feasible set; its optimal objective valuerepresents a generally loose lower bound (1683) on the optimal objective of problem (691).