v2010.10.26 - Convex Optimization

254 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSSimilarly, the matrix-valued affine function of real variable x , for anyparticular matrix A∈ R M×N ,describes a line in R M×N in direction Aand describes a line in R×R M×N{[h(x) : R→R M×N = Ax + B (601){Ax + B | x∈ R} ⊆ R M×N (602)xAx + B]}| x∈ R ⊂ R×R M×N (603)whose slope with respect to x is A .3.6.1 monotonic functionA real function of real argument is called monotonic when it is exclusivelynonincreasing or nondecreasing over the whole of its domain. A realdifferentiable function of real argument is monotonic when its first derivative(not necessarily continuous) maintains sign over the function domain.3.6.1.0.1 Definition. Monotonicity.In pointed closed convex cone K , multidimensional function f(X) isnondecreasing monotonic whennonincreasing monotonic when∀X,Y ∈ dom f .Y ≽ K X ⇒ f(Y ) ≽ f(X)Y ≽ K X ⇒ f(Y ) ≼ f(X)△(604)For monotonicity of vector-valued functions, f compared with respect tothe nonnegative orthant, it is necessary and sufficient for each entry f i to bemonotonic in the same sense.Any affine function is monotonic. In K = S M + , for example, tr(Z T X) is anondecreasing monotonic function of matrix X ∈ S M when constant matrixZ is positive semidefinite; which follows from a result (378) of Fejér.A convex function can be characterized by another kind of nondecreasingmonotonicity of its gradient: