v2009.01.01 - Convex Optimization

140 CHAPTER 2. CONVEX GEOMETRY
2.12.4 Converting between descriptions
Conversion between halfspace-descriptions (258) (259) and equivalent
vertex-descriptions (78) (262) is nontrivial, in general, [14] [96,2.2] but the
conversion is easy for simplices. [53,2.2] Nonetheless, we tacitly assume the
two descriptions to be equivalent. [266,19, thm.19.1] We explore conversions
in2.13.4,2.13.9, and2.13.11:
2.13 Dual cone & generalized inequality
& biorthogonal expansion
These three concepts, dual cone, generalized inequality, and biorthogonal
expansion, are inextricably melded; meaning, it is difficult to completely
discuss one without mentioning the others. The dual cone is critical in tests
for convergence by contemporary primal/dual methods for numerical solution
of conic problems. [342] [239,4.5] For unique minimum-distance projection
on a closed convex cone K , the negative dual cone −K ∗ plays the role that
orthogonal complement plays for subspace projection. 2.50 (E.9.2.1) Indeed,
−K ∗ is the algebraic complement in R n ;
K ⊞ −K ∗ = R n (269)
where ⊞ denotes unique orthogonal vector sum.
One way to think of a pointed closed convex cone is as a new kind of
coordinate system whose basis is generally nonorthogonal; a conic system,
very much like the familiar Cartesian system whose analogous cone is the
first quadrant (the nonnegative orthant). Generalized inequality ≽ K is a
formalized means to determine membership to any pointed closed convex
cone K (2.7.2.2) whereas biorthogonal expansion is, fundamentally, an
expression of coordinates in a pointed conic system whose axes are linearly
independent but not necessarily orthogonal. When cone K is the nonnegative
orthant, then these three concepts come into alignment with the Cartesian
prototype: biorthogonal expansion becomes orthogonal expansion, the dual
cone becomes identical to the orthant, and generalized inequality obeys a
total order entrywise.
2.50 Namely, projection on a subspace is ascertainable from its projection on the orthogonal
complement.