v2007.09.13 - Convex Optimization

4.2. FRAMEWORK 243where δ is the main-diagonal linear operator (A.1). By assigning (B.1)[ ] [ ] [ ]ˆx [ ˆxG =T 1] X ˆx ∆ ˆxˆxTˆx=1ˆx T =1 ˆx T ∈ S n+1 (586)1problem (585) becomes equivalent to: (Theorem A.3.1.0.7)minimize 1 T ˆxX∈ S n , ˆx∈R nsubject to A(ˆx + 1) 1[ = b 2] X ˆxG =ˆx T 1δ(X) = 1(G ≽ 0)rankG = 1(587)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 4.12 we get the semidefinite programminimize 1 T ˆxX∈ S n , ˆx∈R nsubject to A(ˆx + 1) 1[ = b 2] X ˆxG =ˆx T 1δ(X) = 1≽ 0(588)whose optimal solution x ⋆ (584) is identical to that of minimum cardinalityBoolean problem (576) if and only if rankG ⋆ =1. Hope 4.13 of acquiring arank-1 solution is not ill-founded because 2 n elliptope vertices have rank 1,and we are minimizing an affine function on a subset of the elliptope(Figure 86) containing rank-1 vertices; id est, by assumption that thefeasible set of minimum cardinality Boolean problem (576) is nonempty,4.12 Relaxed problem (588) can also be derived via Lagrange duality; it is a dual of adual program [sic] to (587). [225] [46,5, exer.5.39] [282,IV] [100,11.3.4] The relaxedproblem must therefore be convex having a larger feasible set; its optimal objective valuerepresents a generally loose lower bound (1457) on the optimal objective of problem (587).4.13 A more deterministic approach to constraining rank and cardinality is developed in4.4.3.0.8.