v2010.10.26 - Convex Optimization

196 CHAPTER 2. CONVEX GEOMETRYThis means membership determination in subspace S requires knowledge ofdual cone only in S . For sake of completeness, for proper cone K withrespect to subspace S (confer (326))x ∈ int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ ∩ S , y ≠ 0 (425)x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ int K ∗ ∩ S (426)(By closure, we also have the conjugate relations.) Yet when S equals aff Kfor K a closed convex conex ∈ rel int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ ∩ aff K , y ≠ 0 (427)x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ rel int(K ∗ ∩ aff K) (428)2.13.9.4 Subspace S = aff KAssume now a subspace S 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 (412)We want expressions for the convex cone and its dual in subspace S =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 matrices S N ,for example, often serves as ambient space. 2.782.13.9.4.1 Exercise. Conically independent columns and rows.We suspect the number of conically independent columns (rows) of X tobe the same for X †T , where † denotes matrix pseudoinverse (E). Provewhether it holds that the columns (rows) of X are c.i. ⇔ the columns (rows)of X †T are c.i.2.78 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 .