A Counterexample to De Pierros conjecture 19 February 2018 - - PowerPoint PPT Presentation

A Counterexample to De Pierro’s conjecture

19 February 2018

Variational Analysis Down Under (VADU2018) in honour of Prof. Asen Dontchev’s 70th Birthday

Vera Roshchina RMIT University and Federation University Australia vera.roshchina@gmail.com

Joint work with Roberto Cominetti (Universidad Adolfo Ib´ a˜ nez, Chile) and Andrew Williamson (RMIT University).

Alternating projections

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Infeasible problem

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Convergence of alternating projections

For two closed convex sets C1, C2 in a Hilbert space H the method of alternating projections converges weakly to a fixed two-point cycle that solves the minimal distance problem min

x1∈C1 x2∈C2

x1 − x2 (1) if and only if such problem has a solution (e.g. one of the sets is bounded). 3/18

What about more than two sets?

C1 C2 C3

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Convergence of cyclic projections

Under mild assumptions (e.g. one of the sets is bounded) cyclic projections converge weakly either to a feasible point in the in- tersection of all sets, or to a fixed cycle if the intersection is

• empty. We have explicitly for C1, C2, . . . , Cm closed convex sets

in a Hilbert space H ukm+1 = ΠC1(ukm), ukm+2 = ΠC2(ukm+1), . . . . . . ukm+m = ΠCm(ukm+m−1), where by ΠC(u) we denote the projection of a point u ∈ H onto a closed convex set C, ΠC(u) = arg min

x∈C

x − u. 5/18

There is no variational characterisation

C1 C2 C3

It was shown by Baillon, Combettes and Cominetti in 2012 that for m ≥ 3 there is no function Φ : Hm → R such that for any collection of compact convex sets C1, C2, . . . , Cm ∈ H such limit cycles are precisely the solutions to the minimisation problem min

xi∈Ci

Φ(x1, x2, . . . , xm). 6/18

Under-relaxed projections

An under-relaxed version of the cyclic projection algorithm was suggested by De Pierro in 2001. Let ε ∈ (0, 1]. Given a finite ordered family of compact convex sets C1, C2, . . . , Cm ⊆ H and an initial point u0 ∈ H, we let ukm+1 = ukm + ε(ΠC1(ukm) − ukm), ukm+2 = ukm+1 + ε(ΠC2(ukm+1) − ukm+1), (2) . . . . . . ukm+m = ukm+m−1 + ε(ΠCm(ukm+m−1) − ukm+m−1), 7/18

Under-relaxed projections

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Limit A

Limits of tuples of under-relaxed projections (ukm+1, ukm+2, . . . , ukm+m) for k ∈ N under mild assumptions converge weakly to an ε-cycle (uε

1, uε 2, . . . , uε m) ∈ Hm, such that

uε

1 = uε m + ε(ΠC1(uε m) − uε m),

uε

2 = uε 1 + ε(ΠC2(uε 1) − uε 1),

(3) . . . . . . uε

m = uε m−1 + ε(ΠCm(uε m−1) − uε m−1).

It was shown by De Pierro in 2001 that the under-relaxed cyclic projections converge weakly to an ε-cycle for any starting point if and only if the set of solutions to (3) is nonempty. We focus on the convergence of such ε-cycles as ε ↓ 0, u0

i := Lim sup ε↓0

uε

i,

i ∈ {1, . . . , m}, (4) 9/18

Limit B

In addition to the ε-limit cycles, the following iterative process can be considered. We let ukm+i = ukm+i−1 + λkm+iΠCm(ukm+i−1 − ukm+i−1), where λp ↓ 0,

p λp = +∞. We are then interested in the limits

¯ ui = lim

k→∞ ukm+i,

i ∈ {1, . . . , m}. (5) 10/18

The conjecture

Conjecture 1 (De Pierro). The least squares solution S = Arg min

u∈H m

• i=1

min

xi∈Ci

u − xi2 exists if and only if both limits A and B exist and solve this least squares problem. Bauschke and Edwards proved the conjecture for families of closed affine subspaces satisfying a metric regularity condition, while Baillon, Combettes and Cominetti in 2014 described sev- eral geometric conditions under which the conjecture is true. We disprove the conjecture by constructing a system of compact convex sets in R3 for which the limit A does not exist. 11/18

A misleading example

C1 = co {(−2, 2, 1), (−2, 2, −1)}, C2 = co {(2, 2, 1), (2, 2, −1)}, C3 = {(x, y, z) | x2 + y2 ≤ 1, |z| ≤ 1}, S =

• 0, 5

3, z

• : |z| ≤ 1
• .

S C1 C2 C3 u0 z0=0.5 S C1 C2 C3 u0 z0=-0.5

Under-relaxed projections for ε = 0.5 and different starting points. 12/18

A misleading example

Fix u0 and let ε ↓ 0, then the limit cycle shrinks towards the point in the least squares segment S at the height z0. Thus, the initial point u0 serves as an ‘anchor’ that provides some hope for the limit cycles uε to converge as ε ↓ 0.

S C1 C2 C3 u0 z0=0

The iterative process for ε = 3

4 and ε = 1 4.

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The counterexample

As before, we consider C1 = co {(−2, 2, 1), (−2, 2, −1)}, C2 = co {(2, 2, 1), (2, 2, −1)}, but replace C3 with a compact convex subset of the cylinder C3 = co {pk | k ∈ N}, pk = (cos tk, sin tk, (−1)k) where {tk} is a monotonically increasing sequence with t1 = π

4

and tk → π

2 as k → ∞.

The least squares solution is the same, S = {(0, 5

3, z) : |z| ≤ 1}.

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The counterexample

C1 C2 C3 p1 p3 p2 p4

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Reduction to two dimensions

Proposition 1. Let C1, C2, C3 be defined as before and let C′

1, C′ 2, C′ 3

be their xy-projections. Then the triple (u1, u2, u3) is an ε-cycle for C1, C2, C3 with relevant projections (w1, w2, w3) if and only if the following two properties hold: (i) the points u1, u2, u3, w1, w2 and w3 lie in a plane orthogonal to the z-axis; (ii) the projections (u′

1, u′ 2, u′ 3) on the xy-plane are an ε-cycle for

the two-dimensional sets C′

1, C′ 2, C′ 3 with projections (w′ 1, w′ 2, w′ 3).

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Reduction to two dimensions

The projections w′

2 of the two-dimensional cycles correspond to

a zigzag path of w2’s in 3D. As ε → 0, limit cycles ‘follow’ this path, and hence there is no convergence for limit A.

a}=C1 { ' b}=C2 { ' v1 v2 v3 C3 '

C1 C2 C3 p1 p3 p2 p4

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Some references

Alvaro R. De Pierro. From parallel to sequential projection methods and vice versa in convex feasibility: results and conjectures. In Inherently par- allel algorithms in feasibility and optimization and their applications (Haifa, 2000), volume 8 of Stud. Comput. Math., pages 187–201. North-Holland, Amsterdam, 2001. H.H. Bauschke and M.R. Edwards. A conjecture by De Pierro is true for translates of regular subspaces.

• J. Nonlinear Convex Anal., 6(1):93–116,

2005. J.-B. Baillon, P. L. Combettes, and R. Cominetti. There is no variational characterization of the cycles in the method of periodic projections. J. Funct. Anal., 262(1):400–408, 2012. J.B. Baillon, P.L. Combettes, and R. Cominetti. Asymptotic behavior of com- positions of under-relaxed nonexpansive operators. J. Dyn. Games, 1(3):331– 346, 2014.

• R. Cominetti, V. Roshchina, A. Williamson, A counterexample to De Pierro’s

conjecture on the convergence of under-relaxed cyclic projections, January 2018, arXiv:1801.03216

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