v2010.10.26 - Convex Optimization

222 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSAny convex real function f(X) has unique minimum value over anyconvex subset of its domain. [302, p.123] Yet solution to some convexoptimization problem is, in general, not unique; e.g., given minimizationof a convex real function over some convex feasible set Cminimize f(X)Xsubject to X ∈ C(502)any optimal solution X ⋆ comes from a convex set of optimal solutionsX ⋆ ∈ {X | f(X) = inf f(Y ) } ⊆ C (503)Y ∈ CBut a strictly convex real function has a unique minimizer X ⋆ ; id est, for theoptimal solution set in (503) to be a single point, it is sufficient (Figure 68)that f(X) be a strictly convex real 3.5 function and set C convex. But strictconvexity is not necessary for minimizer uniqueness: existence of any strictlysupporting hyperplane to a convex function epigraph (p.219,3.5) at itsminimum over C is necessary and sufficient for uniqueness.Quadratic real functions x T Ax + b T x + c are convex in x iff A≽0.(3.6.4.0.1) Quadratics characterized by positive definite matrix A≻0 arestrictly convex and vice versa. The vector 2-norm square ‖x‖ 2 (Euclideannorm square) and Frobenius’ norm square ‖X‖ 2 F , for example, are strictlyconvex functions of their respective argument (each absolute norm is convexbut not strictly). Figure 68a illustrates a strictly convex real function.3.1.2.1 minimum/minimal element, dual cone characterizationf(X ⋆ ) is the minimum element of its range if and only if, for each and everyw ∈ int R M∗+ , it is the unique minimizer of w T f . (Figure 69) [61,2.6.3]If f(X ⋆ ) is a minimal element of its range, then there exists a nonzerow ∈ R M∗+ such that f(X ⋆ ) minimizes w T f . If f(X ⋆ ) minimizes w T f for somew ∈ int R M∗+ , conversely, then f(X ⋆ ) is a minimal element of its range.3.5 It is more customary to consider only a real function for the objective of a convexoptimization problem because vector- or matrix-valued functions can introduce ambiguityinto the optimal objective value. (2.7.2.2,3.1.2.1) Study of multidimensional objectivefunctions is called multicriteria- [324] or multiobjective- or vector-optimization.