v2010.10.26 - Convex Optimization

84 CHAPTER 2. CONVEX GEOMETRYinto parts; x = α − β (extensible to vectors). Under what conditions onvector a and scalar b is an optimal solution x ⋆ negative infinity?minimize α − βα∈R , β∈Rsubject to β ≥ 0α ≥ 0[ ] αa T = bβMinimization of the objective function entails maximization of β .(135)2.4.2.7 PRINCIPLE 3: Separating hyperplaneThe third most fundamental principle of convex geometry again follows fromthe geometric Hahn-Banach theorem [250,5.12] [18,1] [132,I.1.2] thatguarantees existence of a hyperplane separating two nonempty convex setsin R n whose relative interiors are nonintersecting. Separation intuitivelymeans each set belongs to a halfspace on an opposing side of the hyperplane.There are two cases of interest:1) If the two sets intersect only at their relative boundaries (2.1.7.2), thenthere exists a separating hyperplane ∂H containing the intersection butcontaining no points relatively interior to either set. If at least one ofthe two sets is open, conversely, then the existence of a separatinghyperplane implies the two sets are nonintersecting. [61,2.5.1]2) A strictly separating hyperplane ∂H intersects the closure of neither set;its existence is guaranteed when intersection of the closures is emptyand at least one set is bounded. [199,A.4.1]2.4.3 Angle between hyperspacesGiven halfspace-descriptions, dihedral angle between hyperplanes orhalfspaces is defined as the angle between their defining normals. Givennormals a and b respectively describing ∂H a and ∂H b , for example( ) 〈a , b〉(∂H a , ∂H b ) arccos radians (136)‖a‖ ‖b‖