v2006.03.09 - Convex Optimization

2.10. CONIC INDEPENDENCE (C.I.) 1312.10.0.0.1 Table: Maximum number c.i. directionsn supk (pointed) supk (not pointed)0 0 01 1 22 2 43.∞.∞.Assuming veracity of this table, there is an apparent vastness between twoand three dimensions. The finite numbers of conically independent directionsindicate:Convex cones in dimensions 0, 1, and 2 must be polyhedral. (2.12.1)Conic independence is certainly one convex idea that cannot be completelyexplained by a two-dimensional picture. [17, p.vii]From this table it is also evident that dimension of Euclidean space cannotexceed the number of conically independent directions possible;n ≤ supkWe suspect the number of conically independent columns (rows) of X tobe the same for X †T .The columns (rows) of X are c.i. ⇔ the columns (rows) of X †T are c.i.Proof. Pending.2.10.1 Preservation of conic independenceIndependence in the linear (2.1.2.1), affine (2.4.2.4), and conic senses canbe preserved under linear transformation. Suppose a matrix X ∈ R n×N (228)holds a conically independent set columnar. Consider the transformationT(X) : R n×N → R n×N ∆ = XY (230)where the given matrix Y = ∆ [y 1 y 2 · · · y N ]∈ R N×N is represented by linearoperator T . Conic independence of {Xy i ∈ R n , i=1... N} demands, bydefinition (227),Xy i ζ i + · · · + Xy j ζ j − Xy l ζ l = 0, i≠ · · · ≠j ≠l = 1... N (231)have no nontrivial solution ζ ∈ R N + . That is ensured by conic independenceof {y i ∈ R N } and by R(Y )∩ N(X) = 0 ; seen by factoring X .