v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 225
Similarly, the matrix-valued affine function of real variable x , for any
particular matrix A∈ R M×N ,
describes a line in R M×N in direction A
and describes a line in R×R M×N
{[
x
Ax + B
]
h(x) : R→R M×N = Ax + B (538)
{Ax + B | x∈ R} ⊆ R M×N (539)
}
| x∈ R ⊂ R×R M×N (540)
whose slope with respect to x is A .
3.1.8.1 monotonic function
A real function of real argument is called monotonic when it is exclusively
nonincreasing or nondecreasing over the whole of its domain. A real
differentiable function of real argument is monotonic when its first derivative
(which is not necessarily continuous) maintains sign over the function
domain.
3.1.8.1.1 Definition. Monotonicity.
Multidimensional function f(X) is
nondecreasing monotonic when
nonincreasing monotonic when
∀X,Y ∈ dom f .
Y ≽ X ⇒ f(Y ) ≽ f(X)
Y ≽ X ⇒ f(Y ) ≼ f(X)
△
(541)
is perpendicular to
η ∆ =
[
∇f(x)
−I
]
∈ R p×M × R M×M
because
η T ([
x
Ax + b
] [ 0
−
b
])
= 0 ∀x ∈ R p
Yet η is a vector (in R p ×R M ) only when M = 1.