[Paper Review] Asymptotic behavior of compositions of under-relaxed nonexpansive operators

Jean-Bernard Baillon, Patrick L. Combettes+1 more|arXiv (Cornell University)|Apr 26, 2013

Optimization and Variational Analysis21 references4 citations

TL;DR

This paper establishes an asymptotic connection between under-relaxed compositions of nonexpansive operators and the fixed points of their average in Hilbert spaces. It proves that as relaxation parameter ε→0, the limit cycles of under-relaxed cyclic iterations converge weakly (and strongly under regularity) to the minimizers of the average squared distance function, thereby confirming De Pierro’s conjecture for projections onto affine subspaces and more general convex sets.

ABSTRACT

In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.

Motivation & Objective

• To resolve the long-standing open problem of connecting the fixed points of under-relaxed compositions of nonexpansive operators to the fixed points of their average.
• To establish a general asymptotic principle linking limit cycles of under-relaxed iterations to the minimizers of the average squared distance function.
• To prove De Pierro’s conjecture for projections onto closed convex sets, extending prior results from affine subspaces to broader classes of convex sets.
• To characterize conditions under which weak or strong convergence of cycles to the average operator’s fixed points occurs.

Proposed method

• Introduces under-relaxed compositions $ R^ u $ as $ R^ u = ( ext{Id} + u(T_m - ext{Id})) igcirc ext{Id} + u(T_1 - ext{Id})) $, where $ u o 0^+ $, generalizing relaxation in iterative methods.
• Analyzes the weak limit cycles $ (x_i^ u) $ of the sequence generated by periodic under-relaxed iterations $ y_{k u}^ u $, showing convergence to solutions of coupled fixed-point equations.
• Applies the theory of nonexpansive operators and monotone operator splitting in Hilbert spaces, leveraging properties of resolvents and projections.
• Uses the concept of regularity of subspaces $ (E_i) $, defined by $ \max_i d_{E_i}(y_k) \to 0 \Rightarrow d_E(y_k) \to 0 $, to ensure strong convergence.
• Reduces the problem to a subspace $ y_0 + E^ot $, where $ E = \bigcap E_i $, and applies contraction arguments to prove convergence.
• Establishes convergence via the contraction property of the average operator $ T = \frac{1}{m}\sum T_i $ and the under-relaxed composition $ R^ u $ in the relevant affine subspace.

Experimental results

Research questions

• RQ1 Is there a relationship between the fixed points of under-relaxed compositions of nonexpansive operators and the fixed points of their average?
• RQ2 Does De Pierro’s conjecture—that under-relaxed cyclic projections converge to the least-squares solution—hold for general closed convex sets beyond affine subspaces?
• RQ3 Under what conditions does the limit cycle of under-relaxed iterations converge strongly to the minimizer of the average squared distance?
• RQ4 Can the asymptotic behavior of under-relaxed compositions be characterized uniformly across different classes of nonexpansive operators?

Key findings

• For projections onto closed convex sets $ C_i $, the limit cycle $ (x_i^ u) $ of under-relaxed iterations converges weakly to $ \overline{x} = P_S y_0 $, where $ S $ is the set of minimizers of $ \Phi(x) = \frac{1}{2m}\sum d_{C_i}^2(x) $, as $ \nu \to 0^+ $.
• Under a regularity condition on the normal spaces $ E_i $, the convergence of $ x_i^ u $ to $ \overline{x} $ is strong, not just weak.
• The result confirms De Pierro’s conjecture for projections onto affine subspaces and extends it to unbounded convex cylinders $ B_i + E_i $, where $ B_i $ is bounded and $ E_i $ is a closed subspace.
• The fixed point set of the average operator $ T $ is shown to coincide with the set of minimizers of $ \Phi $, and this set is affine: $ S = z + E $, where $ E = \bigcap E_i $.
• The convergence of $ x_i^ u $ to $ \overline{x} $ is established via contraction arguments in the affine subspace $ y_0 + E^ot $, under the condition that $ \|L \circ P_{E^ot}\| < 1 $, where $ L = \frac{1}{m}\sum P_{E_i} $.
• An alternative proof of the main result in [7] is derived from Corollary 3.9, showing that regularity of $ (E_i) $ implies both existence of the cycle and convergence to the least-squares solution.