v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 211
3.1.6.0.2 Example. Linear objective.
Consider minimization of a real affine function f(z)= a T z + b over the
convex feasible set C in its domain R 2 illustrated in Figure 63. Since
vector b is fixed, the problem posed is the same as the convex optimization
minimize a T z
z
subject to z ∈ C
(490)
whose objective of minimization is a real linear function. Were convex set C
polyhedral (2.12), then this problem would be called a linear program. Were
convex set C an intersection with a positive semidefinite cone, then this
problem would be called a semidefinite program.
There are two distinct ways to visualize this problem: one in the objective
function’s domain R 2 ,[ the]
other including the ambient space of the objective
R
2
function’s range as in . Both visualizations are illustrated in Figure 63.
R
Visualization in the function domain is easier because of lower dimension and
because
level sets (492) of any affine function are affine. (2.1.9)
In this circumstance, the level sets are parallel hyperplanes with respect
to R 2 . One solves optimization problem (490) graphically by finding that
hyperplane intersecting feasible set C furthest right (in the direction of
negative gradient −a (3.1.8)).
When a differentiable convex objective function f is nonlinear, the
negative gradient −∇f is a viable search direction (replacing −a in (490)).
(2.13.10.1, Figure 59) [128] Then the nonlinear objective function can be
replaced with a dynamic linear objective; linear as in (490).
3.1.6.0.3 Example. Support function. [173,C.2.1]
For arbitrary set Y ⊆ R n , its support function σ Y (a) : R n → R is defined
σ Y (a) = ∆ supa T z (491)
z∈Y
whose range contains ±∞ [215, p.135] [173,C.2.3.1]. For each z ∈ Y ,
a T z is a linear function of vector a . Because σ Y (a) is the pointwise
supremum of linear functions, it is convex in a . (Figure 64) Application
of the support function is illustrated in Figure 25(a) for one particular
normal a .