v2009.01.01 - Convex Optimization

76 CHAPTER 2. CONVEX GEOMETRY
A set of nonempty interior that has a supporting hyperplane at every
point on its boundary, conversely, is convex. A convex set C ⊂ R n , for
example, can be expressed as the intersection of all halfspaces partially
bounded by hyperplanes supporting it; videlicet, [215, p.135]
C = ⋂
a∈R n {
y | a T y ≤ σ C (a) } (122)
by the halfspaces theorem (2.4.1.1.1).
There is no geometric difference between supporting hyperplane ∂H + or
∂H − or ∂H and 2.22 an ordinary hyperplane ∂H coincident with them.
2.4.2.6.2 Example. Minimization on a hypercube.
Consider minimization of a linear function on a hypercube, given vector c
minimize c T x
x
subject to −1 ≼ x ≼ 1
(123)
This convex optimization problem is called a linear program 2.23 because the
objective of minimization is linear and the constraints describe a polyhedron
(intersection of a finite number of halfspaces and hyperplanes). Applying
graphical concepts from Figure 22, Figure 24, and Figure 25, an optimal
solution can be shown to be x ⋆ = − sgn(c) but is not necessarily unique.
Because an optimal solution always exists at a hypercube vertex (2.6.1.0.1)
regardless of the value of nonzero vector c [82], mathematicians see this
geometry as a means to relax a discrete problem (whose desired solution is
integer or combinatorial, confer Example 4.2.3.1.1). [206,3.1] [207]
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 program
still linear?
Now suppose we instead minimize absolute value in (123). Are the
following programs equivalent for some arbitrary real convex set C ?
2.22 If vector-normal polarity is unimportant, we may instead signify a supporting
hyperplane by ∂H .
2.23 The term program has its roots in economics. It was originally meant with regard to
a plan or to efficient organization of some industrial process. [82,2]