v2007.09.13 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 71There is no geometric difference 2.19 between supporting hyperplane ∂H +or ∂H − or ∂H and an ordinary hyperplane ∂H coincident with them.2.4.2.6.2 Example. Minimization on the unit cube.Consider minimization of a linear function on a hypercube, given vector cminimize c T xxsubject to −1 ≼ x ≼ 1(111)This convex optimization problem is called a linear program because theobjective of minimization is linear and the constraints describe a polyhedron(intersection of a finite number of halfspaces and hyperplanes). Applyinggraphical concepts from Figure 17, Figure 19, and Figure 20, an optimalsolution can be shown to be x ⋆ = − sgn(c) but is not necessarily unique.Because a solution always exists at a hypercube vertex (2.6.1.0.1) regardlessof the value of nonzero vector c [64], mathematicians see this geometryas a means to relax a discrete problem (whose desired solution is integer,confer Example 4.2.3.0.2).2.4.2.6.3 Exercise. Unbounded below.Suppose instead we minimize over the unit hypersphere in Example 2.4.2.6.2;‖x‖ ≤ 1. What is an expression for optimal solution now? Is that programstill linear?Now suppose we instead minimize absolute value in (111). Are thefollowing programs equivalent for some arbitrary real convex set C ?(confer (433))minimize |x|x∈Rsubject to −1 ≤ x ≤ 1x ∈ C≡minimize x + + x −x + , x −subject to 1 ≥ x − ≥ 01 ≥ x + ≥ 0x + − x − ∈ C(112)Many 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 − (extensible2.19 If vector-normal polarity is unimportant, we may instead signify a supportinghyperplane by ∂H .