v2010.10.26 - Convex Optimization

70 CHAPTER 2. CONVEX GEOMETRYconvex combination of at most (n−1) 2 +1 permutation matrices. [202,8.7][55, thm.1.2.5] This polyhedron, then, can be a device for relaxing an integer,combinatorial, or Boolean optimization problem. 2.16 [66] [283,3.1] 2.3.2.0.5 Example. Convex hull of orthonormal matrices. [27,1.2]Consider rank-k matrices U ∈ R n×k such that U T U = I . These are theorthonormal matrices; a closed bounded submanifold, of all orthogonalmatrices, having dimension nk − 1 k(k + 1) [52]. Their convex hull is2expressed, for 1 ≤ k ≤ nconv{U ∈ R n×k | U T U = I} = {X ∈ R n×k | ‖X‖ 2 ≤ 1}= {X ∈ R n×k | ‖X T a‖ ≤ ‖a‖ ∀a∈ R n }(101)When k=n, matrices U are orthogonal and the convex hull is called thespectral norm ball which is the set of all contractions. [203, p.158] [329, p.313]The orthogonal matrices then constitute the extreme points (2.6.0.0.1) ofthis hull.By Schur complement (A.4), the spectral norm ‖X‖ 2 constraining largestsingular value σ(X) 1 can be expressed as a semidefinite constraint[ ] I X‖X‖ 2 ≤ 1 ⇔X T ≽ 0 (102)Ibecause of equivalence X T X ≼ I ⇔ σ(X) ≼ 1 with singular values. (1566)(1450) (1451) 2.3.3 Conic hullIn terms of a finite-length point list (or set) arranged columnar in X ∈ R n×N(76), its conic hull is expressedK cone {x l , l=1... N} = cone X = {Xa | a ≽ 0} ⊆ R n (103)id est, every nonnegative combination of points from the list. Conic hull ofany finite-length list forms a polyhedral cone [199,A.4.3] (2.12.1.0.1; e.g.,Figure 50a); the smallest closed convex cone (2.7.2) that contains the list.2.16 Relaxation replaces an objective function with its convex envelope or expands a feasibleset to one that is convex. Dantzig first showed in 1951 that, by this device, the so-calledassignment problem can be formulated as a linear program. [316] [26,II.5]