v2010.10.26 - Convex Optimization

380 CHAPTER 4. SEMIDEFINITE PROGRAMMINGFigure 110: Neighboring-pixel stencil [356] for image-gradient estimation onCartesian grid. Implementation selects adaptively from darkest four • aboutcentral. Continuous image-gradient from two pixels holds only in a limit. Fordiscrete differences, better practical estimates are obtained when centered.where multiobjective parameter λ∈ R + is quite large (λ≈1E8) so as to enforcethe equality constraint: P vec U −f = 0 ⇔ ‖P vec U −f‖ 2 2=0 (A.7.1). Weintroduce a direction vector y ∈ R 4n2+ as part of a convex iteration (4.5.2)to overcome that known suboptimal minimization of discrete image-gradientcardinality: id est, there exists a vector y ⋆ with entries yi ⋆ ∈ {0, 1} such thatminimize ‖Ψ vec U‖ 0Usubject to P vec U = f≡minimizeU〈|Ψ vec U| , y ⋆ 〉 + 1 2 λ‖P vec U − f‖2 2(857)Existence of such a y ⋆ , complementary to an optimal vector Ψ vec U ⋆ , isobvious by definition of global optimality 〈|Ψ vec U ⋆ | , y ⋆ 〉= 0 (758) underwhich a cardinality-c optimal objective ‖Ψ vec U ⋆ ‖ 0 is assumed to exist.Because (856b) is an unconstrained convex problem, a zero in theobjective function’s gradient is necessary and sufficient for optimality(2.13.3); id est, (D.2.1)Ψ T δ(y) sgn(Ψ vec U) + λP H (P vec U − f) = 0 (858)Because of P idempotence and Hermitian symmetry and sgn(x)= x/|x| ,