v2007.09.13 - Convex Optimization

134 CHAPTER 2. CONVEX GEOMETRY2.12.4 Converting between descriptionsConversion between halfspace-descriptions (245) (246) and equivalentvertex-descriptions (75) (249) is nontrivial, in general, [13] [77,2.2] but theconversion is easy for simplices. [46,2.2] Nonetheless, we tacitly assume thetwo descriptions to be equivalent. [228,19, thm.19.1] We explore conversionsin2.13.4 and2.13.9:2.13 Dual cone & generalized inequality& biorthogonal expansionThese three concepts, dual cone, generalized inequality, and biorthogonalexpansion, are inextricably melded; meaning, it is difficult to completelydiscuss one without mentioning the others. The dual cone is critical in testsfor convergence by contemporary primal/dual methods for numerical solutionof conic problems. [296] [203,4.5] For unique minimum-distance projectionon a closed convex cone K , the negative dual cone −K ∗ plays the role thatorthogonal complement plays for subspace projection. 2.43 (E.9.2.1) Indeed,−K ∗ is the algebraic complement in R n ;K ⊞ −K ∗ = R n (257)where ⊞ denotes unique orthogonal vector sum.One way to think of a pointed closed convex cone is as a new kind ofcoordinate system whose basis is generally nonorthogonal; a conic system,very much like the familiar Cartesian system whose analogous cone isthe first quadrant or nonnegative orthant. Generalized inequality ≽ Kis a formalized means to determine membership to any pointed closedconvex cone (2.7.2.2) whereas biorthogonal expansion is, fundamentally, anexpression of coordinates in a pointed conic system. When cone K is thenonnegative orthant, then these three concepts come into alignment with theCartesian prototype; biorthogonal expansion becomes orthogonal expansion.2.43 Namely, projection on a subspace is ascertainable from its projection on the orthogonalcomplement.