v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 217N∑minimize Γ ∗TΓ ∗ i (x ⋆ − t ⋆ iΓ i )i ∈Rn i=1subject to Γ ∗Ti Γ i = 1 ,Γ ∗ i ∈ K ∗ ,i=1... Ni=1... N(494)where dual extreme directions Γ ∗ i are initialized arbitrarily and ultimatelyascertained by the alternation. Convex problems (493) and (494) areiterated until convergence which is guaranteed by virtue of a monotonicallynonincreasing real sequence of objective values. Convergence can be fast.The mapping t ⋆ (x) is uniquely inverted when the necessarily nonnegativeobjective vanishes; id est, when Γ ∗Ti (x ⋆ − t ⋆ iΓ i )=0 ∀i. Here, a zeroobjective can occur only at the true solution x . But this global optimalitycondition cannot be guaranteed by the alternation because the commonobjective function, when regarded in both primal x and dual Γ ∗ i variablessimultaneously, is generally neither quasiconvex or monotonic. (3.8.0.0.3)Conversely, a nonzero objective at convergence is a certificate thatinversion was not performed properly. A nonzero objective indicates thatthe global minimum of a multimodal objective function could not be foundby this alternation. That is a flaw in this particular iterative algorithm forinversion; not in theory. 2.89 A numerical remedy is to reinitialize the Γ ∗ i todifferent values.2.89 The Proof 2.13.12.0.2, that suitable dual extreme directions {Γj ∗ } always exist, meansthat a global optimization algorithm would always find the zero objective of alternation(493) (494); hence, the unique inversion x. But such an algorithm can be combinatorial.