v2009.01.01 - Convex Optimization

296 CHAPTER 4. SEMIDEFINITE PROGRAMMING
The set {u∈ R n | cardu = n−k , u i ∈ {0, 1}} comprises the extreme points
of set (684) which is a nonnegative hypercube slice. An optimal solution y
to (467), that is an extreme point of its feasible set, is known in closed
form: it has 1 in each entry corresponding to the n−k smallest entries of x ⋆
and has 0 elsewhere. By Proposition 7.1.3.0.3, the polar of that particular
direction −y can be interpreted 4.30 as pointing toward the set of all vectors
having the same ordering as x ⋆ and cardinality-k . When y = 1, as in 1-norm
minimization for example, then polar direction −y points directly at the
origin (the cardinality-0 vector).
4.5.1.2 convergence
Convex iteration (143) (467) always converges to a locally optimal solution
by virtue of a monotonically nonincreasing real objective sequence. [222,1.2]
[37,1.1] There can be no proof of global optimality, defined by (683).
Constraining cardinality, solution to problem (682), can often be achieved
but simple examples can be contrived that cause an iteration to stall at a
solution of undesirable cardinality. The direction vector is then manipulated
to steer out of local minima:
4.5.1.2.1 Example. Sparsest solution to Ax = b. [61] [100]
Problem (616) has sparsest solution not easily recoverable by least 1-norm;
id est, not by compressed sensing because of proximity to a theoretical lower
bound on number of measurements m depicted in Figure 80: for A∈ R m×n
Given data from Example 4.2.3.1.1, for m=3, n=6, k=1
⎡
⎤ ⎡ ⎤
−1 1 8 1 1 0
1
⎢
1 1 1
A = ⎣ −3 2 8 − 1 ⎥ ⎢
2 3 2 3 ⎦ , b = ⎣
−9 4 8
1
4
1
9
1
4 − 1 9
1
2
1
4
⎥
⎦ (616)
the sparsest solution to classical linear equation Ax = b is x = e 4 ∈ R 6
(confer (629)).
Although the sparsest solution is recoverable by inspection, we discern
it instead by convex iteration; namely, by iterating problem sequence
4.30 Convex iteration (143) (467) is not a projection method because there is no
thresholding or discard of variable-vector x entries.