v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 333
image U , that is known suboptimal [62], may be concisely posed
where
minimize ‖Ψ vec U‖ 1
U
subject to P vec U = f
(765)
f = (F H ⊗F H ) vec K ∈ C n2 (766)
is the known inverse subsampled Fourier data (a vectorized aliased image,
Figure 88), and where a norm of image-gradient ∇U is equivalently
expressed as norm of a linear transformation Ψ vec U .
Although this simple problem statement (765) is equivalent to a linear
program (3.1.3), its numerical solution is beyond the capability of even
the most highly regarded of contemporary commercial solvers. 4.50 Our only
recourse is to recast the problem in Lagrangian form and write customized
code to solve it: [308,IIIA] [217, p.398]
minimize 〈|Ψ vec U| , y〉 + 1λ‖P vec U −
U
2 f‖2 2 (767)
where (multiobjective optimization) parameter λ∈ R + is quite large (λ≈1E8)
so as to enforce the equality constraint in (765). We introduce a direction
vector y ∈ R 4n2
+ as part of a convex iteration (4.5.2) to overcome that known
suboptimal minimization of image-gradient cardinality: id est, there exists a
vector y ⋆ having entries yi ⋆ ∈ {0, 1} such that
minimize ‖Ψ vec U‖ 0
U
subject to P vec U = f
≡
minimize
U
〈|Ψ vec U| , y ⋆ 〉 + 1 2 λ‖P vec U − f‖2 2
(768)
Existence of such a y ⋆ , complementary to an optimal vector Ψ vec U ⋆ , is
obvious by definition of global optimality 〈|Ψ vec U ⋆ | , y ⋆ 〉= 0 (683) under
which a cardinality-c optimal objective is assumed to exist.
Because (767) is an unconstrained convex problem, a zero in the objective
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 (769)
4.50 for images as small as 128×128 pixels. Obstacle to numerical solution is not a
computer resource: e.g., execution time, memory. The obstacle is, in fact, inadequate
numerical precision. Even when all dependent equality constraints are manually removed,
the best commercial solvers fail simply because computer numerics become nonsense;
id est, numerical errors enter significant digits and the algorithm exits prematurely, loops
indefinitely, or produces an infeasible solution.