Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ... 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
3.1. CONVEX FUNCTION 215 Rf (b) f(X ⋆ ) f(X ⋆ ) w f(X) 2 f(X) 1 Rf (a) f f w Figure 67: (confer Figure 38) Function range is convex for a convex problem. (a) Point f(X ⋆ ) is the unique minimum element of function range Rf . (b) Point f(X ⋆ ) is a minimal element of depicted range. (Cartesian axes drawn for reference.)
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3.1. CONVEX FUNCTION 215<br />
Rf<br />
(b)<br />
f(X ⋆ )<br />
f(X ⋆ )<br />
w<br />
f(X) 2<br />
f(X) 1<br />
Rf<br />
(a)<br />
f<br />
f<br />
w<br />
Figure 67: (confer Figure 38) Function range is <strong>convex</strong> for a <strong>convex</strong> problem.<br />
(a) Point f(X ⋆ ) is the unique minimum element <strong>of</strong> function range Rf .<br />
(b) Point f(X ⋆ ) is a minimal element <strong>of</strong> depicted range.<br />
(Cartesian axes drawn for reference.)
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