Chapter 3 Geometry of convex functions - Meboo Publishing ...

214 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
Any convex real function f(X) has unique minimum value over any
convex subset of its domain. [296, p.123] Yet solution to some convex
optimization problem is, in general, not unique; e.g., given minimization
of a convex real function over some convex feasible set C
minimize f(X)
X
subject to X ∈ C
any optimal solution X ⋆ comes from a convex set of optimal solutions
(480)
X ⋆ ∈ {X | f(X) = inf f(Y ) } ⊆ C (481)
Y ∈ C
But a strictly convex real function has a unique minimizer X ⋆ ; id est, for the
optimal solution set in (481) to be a single point, it is sufficient (Figure 66)
that f(X) be a strictly convex real 3.4 function and set C convex. [315]
Quadratic real functions x T Ax + b T x + c are convex in x iff A≽0.
Quadratics characterized by positive definite matrix A≻0 are strictly convex.
The vector 2-norm square ‖x‖ 2 (Euclidean norm square) and Frobenius’
norm square ‖X‖ 2 F , for example, are strictly convex functions of their
respective argument (each absolute norm is convex but not strictly convex).
Figure 66a illustrates a strictly convex real function.
3.1.2.1 minimum/minimal element, dual cone characterization
f(X ⋆ ) is the minimum element of its range if and only if, for each and every
w ∈ int R M∗
+ , it is the unique minimizer of w T f . (Figure 67) [59,2.6.3]
If f(X ⋆ ) is a minimal element of its range, then there exists a nonzero
w ∈ R M∗
+ such that f(X ⋆ ) minimizes w T f . If f(X ⋆ ) minimizes w T f for some
w ∈ int R M∗
+ , conversely, then f(X ⋆ ) is a minimal element of its range.
3.1.2.1.1 Exercise. Cone of convex functions.
Prove that relation (476) implies: the set of all vector-valued convex functions
in R M is a convex cone. So, the trivial function f = 0 is convex. Indeed, any
nonnegatively weighted sum of (strictly) convex functions remains (strictly)
convex. 3.5 Interior to the cone are the strictly convex functions.
3.4 It is more customary to consider only a real function for the objective of a convex
optimization problem because vector- or matrix-valued functions can introduce ambiguity
into the optimal objective value. (2.7.2.2,3.1.2.1) Study of multidimensional objective
functions is called multicriteria- [318] or multiobjective- or vector-optimization.
3.5 Strict case excludes cone’s point at origin. By these definitions (476) (479), positively
weighted sums mixing convex and strictly convex real functions are not strictly convex