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 1/18

Infeasible problem 2/18

Convergence of alternating projections For two closed convex sets C 1 , C 2 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 � x 1 − x 2 � (1) x 1 ∈ C 1 x 2 ∈ C 2 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? C 2 C 1 C 3 4/18

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 C 1 , C 2 , . . . , C m closed convex sets in a Hilbert space H u km +1 = Π C 1 ( u km ) , u km +2 = Π C 2 ( u km +1 ) , . . . . . . u km + m = Π C m ( u km + m − 1 ) , where by Π C ( u ) we denote the projection of a point u ∈ H onto a closed convex set C , Π C ( u ) = arg min � x − u � . x ∈ C 5/18

There is no variational characterisation C 2 C 1 C 3 It was shown by Baillon, Combettes and Cominetti in 2012 that for m ≥ 3 there is no function Φ : H m → R such that for any collection of compact convex sets C 1 , C 2 , . . . , C m ∈ H such limit cycles are precisely the solutions to the minimisation problem min Φ( x 1 , x 2 , . . . , x m ) . x i ∈ C i 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 C 1 , C 2 , . . . , C m ⊆ H and an initial point u 0 ∈ H , we let u km +1 = u km + ε (Π C 1 ( u km ) − u km ) , u km +2 = u km +1 + ε (Π C 2 ( u km +1 ) − u km +1 ) , (2) . . . . . . u km + m = u km + m − 1 + ε (Π C m ( u km + m − 1 ) − u km + m − 1 ) , 7/18

Under-relaxed projections 8/18

Limit A Limits of tuples of under-relaxed projections ( u km +1 , u km +2 , . . . , u km + m ) for k ∈ N under mild assumptions converge weakly to an ε -cycle ( u ε 1 , u ε 2 , . . . , u ε m ) ∈ H m , such that u ε 1 = u ε m + ε (Π C 1 ( u ε m ) − u ε m ) , u ε 2 = u ε 1 + ε (Π C 2 ( u ε 1 ) − u ε 1 ) , (3) . . . . . . u ε m = u ε m − 1 + ε (Π C m ( u ε m − 1 ) − u ε 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, u 0 u ε i := Lim sup i ∈ { 1 , . . . , m } , (4) i , ε ↓ 0 9/18

Limit B In addition to the ε -limit cycles, the following iterative process can be considered. We let u km + i = u km + i − 1 + λ km + i Π C m ( u km + i − 1 − u km + i − 1 ) , where λ p ↓ 0, � p λ p = + ∞ . We are then interested in the limits ¯ u i = lim i ∈ { 1 , . . . , m } . (5) k →∞ u km + i , 10/18

The conjecture Conjecture 1 (De Pierro) . The least squares solution m � u − x i � 2 � S = Arg min min x i ∈ C i u ∈H i =1 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 R 3 for which the limit A does not exist. 11/18

A misleading example C 1 = co { ( − 2 , 2 , 1) , ( − 2 , 2 , − 1) } , C 2 = co { (2 , 2 , 1) , (2 , 2 , − 1) } , C 3 = { ( x, y, z ) | x 2 + y 2 ≤ 1 , | z | ≤ 1 } , 0 , 5 �� � � S = : | z | ≤ 1 3 , z . C 3 C 3 u 0 z 0 =0.5 S C 1 u 0 z 0 =-0.5 C 1 S C 2 C 2 Under-relaxed projections for ε = 0 . 5 and different starting points. 12/18

A misleading example Fix u 0 and let ε ↓ 0, then the limit cycle shrinks towards the point in the least squares segment S at the height z 0 . Thus, the initial point u 0 serves as an ‘anchor’ that provides some hope for the limit cycles u ε to converge as ε ↓ 0. C 3 u 0 z 0 =0 C 1 S C 2 The iterative process for ε = 3 4 and ε = 1 4 . 13/18

The counterexample As before, we consider C 1 = co { ( − 2 , 2 , 1) , ( − 2 , 2 , − 1) } , C 2 = co { (2 , 2 , 1) , (2 , 2 , − 1) } , but replace C 3 with a compact convex subset of the cylinder p k = (cos t k , sin t k , ( − 1) k ) C 3 = co { p k | k ∈ N } , where { t k } is a monotonically increasing sequence with t 1 = π 4 and t k → π 2 as k → ∞ . The least squares solution is the same, S = { (0 , 5 3 , z ) : | z | ≤ 1 } . 14/18

The counterexample p 2 p 4 C 1 C 3 p 3 p 1 C 2 15/18

Reduction to two dimensions Proposition 1. Let C 1 , C 2 , C 3 be defined as before and let C ′ 1 , C ′ 2 , C ′ 3 be their xy -projections. Then the triple ( u 1 , u 2 , u 3 ) is an ε -cycle for C 1 , C 2 , C 3 with relevant projections ( w 1 , w 2 , w 3 ) if and only if the following two properties hold: (i) the points u 1 , u 2 , u 3 , w 1 , w 2 and w 3 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 ) . 16/18

Reduction to two dimensions The projections w ′ 2 of the two-dimensional cycles correspond to a zigzag path of w 2 ’s in 3D. As ε → 0, limit cycles ‘follow’ this path, and hence there is no convergence for limit A. ' ' { a }= C 1 { b }= C 2 p 2 p 4 v 3 C 1 v 2 C 3 v 1 ' C 3 p 3 p 1 C 2 17/18

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 18/18