v2006.03.09 - Convex Optimization

84 CHAPTER 2. CONVEX GEOMETRYMany optimization problems of interest and some older methods ofsolution require nonnegative variables. The method illustrated below splitsa variable into its nonnegative and negative parts; x = x + − x − (extensibleto vectors). Under what conditions on vector a and scalar b is an optimalsolution x ⋆ negative infinity?minimizex + ∈ R , x − ∈ Rx + − x −subject to x − ≥ 0x + ≥ 0[ ]a T x+= bx −Minimization of the objective entails maximization of x − .(109)2.4.2.7 PRINCIPLE 3: Separating hyperplaneThe third most fundamental principle of convex geometry again follows fromthe geometric Hahn-Banach theorem [154,5.12] [14,1] [70,I.1.2] thatguarantees existence of a hyperplane separating two nonempty convex sets inR n whose relative interiors are nonintersecting. Separation intuitively meanseach set belongs to a halfspace on an opposing side of the hyperplane. Thereare two cases of interest:1) If the two sets intersect only at their relative boundaries (2.6.1.3), 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. [38,2.5.1]2) A strictly separating hyperplane ∂H intersects the closure of neitherset; its existence is guaranteed when the intersection of the closures isempty and at least one set is bounded. [123,A.4.1]