v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1692.13.9.4 Subspace M = aff KAssume now a subspace M that is the affine hull of cone K : Consider againa pointed polyhedral cone K denoted by its extreme directions arrangedcolumnar in matrix X such thatrank(X ∈ R n×N ) = N ∆ = dim aff K ≤ n (353)We want expressions for the convex cone and its dual in subspace M=aff K :Cone Table A K K ∗ ∩ aff Kvertex-description X X †Thalfspace-description X † , X ⊥T X T , X ⊥TWhen dim aff K = n , this table reduces to Cone Table S. These descriptionsfacilitate work in a proper subspace. The subspace of symmetric matricesS N , for example, often serves as ambient space. 2.592.13.9.4.1 Example. Monotone nonnegative cone.[46, exer.2.33] [262,2] Simplicial cone (2.12.3.1.1) K M+ is the cone of allnonnegative vectors having their entries sorted in nonincreasing order:K M+ ∆ = {x | x 1 ≥ x 2 ≥ · · · ≥ x n ≥ 0} ⊆ R n += {x | (e i − e i+1 ) T x ≥ 0, i = 1... n−1, e T nx ≥ 0}= {x | X † x ≽ 0}(370)a halfspace-description where e i is the i th standard basis vector, and whereX †T ∆ = [ e 1 −e 2 e 2 −e 3 · · · e n ] ∈ R n×n (371)(With X † in hand, we might concisely scribe the remaining vertex andhalfspace-descriptions from the tables for K M+ and its dual. Instead we usedual generalized inequalities in their derivation.) For any vectors x and y ,simple algebra demandsn∑x T y = x i y i = (x 1 − x 2 )y 1 + (x 2 − x 3 )(y 1 + y 2 ) + (x 3 − x 4 )(y 1 + y 2 + y 3 ) + · · ·i=1+ (x n−1 − x n )(y 1 + · · · + y n−1 ) + x n (y 1 + · · · + y n ) (372)2.59 The dual cone of positive semidefinite matrices S N∗+ = S N + remains in S N by convention,whereas the ordinary dual cone would venture into R N×N .