On the Minimal Displacement Vector of the Douglas-Rachford Operator

The Douglas-Rachford algorithm can be represented as the fixed point iteration of a firmly nonexpansive operator, which converges to a fixed point, provided it exists. When the operator has no fixed points, the algorithm's iterates diverge, but the difference between consecutive iterates converges to the minimal displacement vector, which can be used to certify infeasibility of an optimization problem. In this paper, we establish new properties of the minimal displacement vector, which allow us to generalize some existing results.


Introduction
The Douglas-Rachford algorithm is a powerful method for minimizing the sum of two convex functions and its asymptotic behavior is well-understood when the problem has a solution. While there exist some results studying feasibility problems involving two convex sets that do not intersect [BDM16,BM16,BM17], some recent works also study a more general setting in which the asymptotic behavior of the algorithm is characterized via the so-called minimal displacement vector. The authors in [BHM16] characterize this vector in terms of the domains of the functions, whose sum is to be minimized, and their Fenchel conjugates. This characterization is used in [RLY19] to show that a nonzero minimal displacement vector implies either primal or dual infeasibility of the problem, but there is an additional assumption imposed, which excludes the case of simultaneous primal and dual infeasibility. The authors in [BM19] derive a new convergence result on the algorithm applied to the problem of minimizing a convex function subject to a linear constraint, but they assume that the Fenchel dual problem is feasible. The analysis in [BGSB19,BL20] covers the case of simultaneous primal and dual infeasibility for a restricted class of problems and shows that the minimal displacement vector can be decomposed as the sum of two orthogonal vectors, one of which is a certificate of primal, and the other of dual infeasibility.
In this paper, we show that the orthogonal decomposition of the minimal displacement vector of the Douglas-Rachford operator established in [BGSB19,BL20] holds in the general case as well. We then consider a class of problems of minimizing a convex function subject to a convex constraint and show that the algorithm generates certificates of both primal and dual strong infeasibility. This allows us to recover the results reported in [BGSB19, BL20] as a special case of our analysis.
The paper is organized as follows. We introduce some definitions and notation in the sequel of Section 1, and some known results on the Douglas-Rachford algorithm in Section 2. Section 3 presents a decomposition of the minimal displacement vector and a new convergence result for a class of constrained convex minimization problems. Section 4 applies these new results to the problem of minimizing a convex quadratic function subject to a convex constraint. Finally, Section 5 concludes the paper.

Notation
All definitions introduced here are standard and can be found in [BC17], to which we also refer for basic results on convex analysis and monotone operator theory.
Let N denote the set of nonnegative integers, R the set of real numbers, and H, H 1 , H 2 be real Hilbert spaces with inner products · | · , induced norms · , and identity operators Id. The power set of H is denoted by 2 H . Let D be a nonempty subset of H with D being its closure. We denote the range of operator T : D → H by ran T and the kernel of a linear operator A by ker A. For a proper lower semicontinuous convex function f : H → ]−∞, +∞], we define its: For a nonempty closed convex set C ⊆ H, we define its: polar cone: indicator function: x / ∈ C.

Douglas-Rachford Algorithm
The Douglas-Rachford algorithm can be used to solve composite minimization problems of the form minimize where f : H → ]−∞, +∞] and g : H → ]−∞, +∞] are proper lower semicontinuous convex functions. Observe that (P) is feasible if 0 ∈ dom f −dom g and strongly infeasible if 0 / ∈ dom f − dom g. The Fenchel dual of (P) can be written as Starting from some s 0 ∈ H, the Douglas-Rachford algorithm applied to (P) generates the following iterates: which can be written compactly as s n = T n s 0 , where is a firmly nonexpansive operator [LM79]. It is easy to show from (1) that Note that T has a fixed point if and only if 0 ∈ ran(Id −T ). To deal with the potential lack of a fixed point of T , we define its minimal displacement vector as v = P ran(Id −T ) (0).
Since the set ran(Id −T ) is nonempty closed convex [Paz71, Lem. 4], the projection above is unique. We next show some useful relations among vector v, problem (P), and the Douglas-Rachford iterates, which hold regardless of the existence of a fixed point of T .
Fact 2.1. Let s 0 ∈ H and s n = T n s 0 . Then

Static Results
Although it is obvious that nonzero v P and v D imply strong infeasibility of (P) and (D), respectively, we next provide some useful identities.
Proposition 3.1. Vectors v P and v D satisfy the following equalities: Proof. Since the proofs of both equalities follow very similar arguments, we only provide a proof for the first. Using the definition of v P and [BC17, Prop. 6.47], we have Using [BC17,Thm. 16.29] and the facts that ι * D = σ D and ∂ι D = N D , the inclusion above is equivalent to where the second equality follows from σ C+D = σ C+D = σ C + σ D and σ −C = σ C • (− Id), and the third from [BC17, Prop. 13.49].
Proposition 3.2. The following relations hold between vectors v P , v D , and v: Proof.  ( Similarly, by (iii) we have v P ∈ rec ( dom g * ), hence (vii): Assuming that v P + v D = 0, the identity follows from Fact 2.1(i) and part (vi). We next assume that v P + v D = 0. Using [BC17,Thm. 3.16] together with the definitions of v P , v D , and v, we have Dividing the inequality by v P + v D = 0, we get v P + v D ≤ v . Combining this with Fact 2.1(i) and part (vi), we obtain the result.
Corollary 3.3. The following relations hold between vectors v, v P , and v D : Proof. Follows directly from Prop. 3.2 and [BC17, Cor. 6.31].
The authors in [RLY19] have also established connections between recession functions and the minimal displacement vector, but the equalities in Prop. 3.1 provide a tight characterization of the left-hand sides and improve the bounds given in [RLY19]. Also, if problem (P) is feasible, then 0 ∈ dom f − dom g and v P = 0, which according to Prop. 3.2(vii) implies v = v D ; similarly, if problem (D) is feasible, then v = v P . Although these identities were established in [RLY19], they follow as a special case of our analysis, which is also applicable when both (P) and (D) are infeasible.

Dynamic Results
Proposition 3.4. Let (c n ) n∈N be a sequence in H satisfying 1 n c n → c and D ⊆ H a nonempty closed convex set. Then 1 n P D c n → P rec D c. , we can only establish the weak convergence, i.e., 1 n a n := 1 n P D c n ⇀ P rec D c =: a. Using Moreau's decomposition [BC17, Thm. 6.30], it follows that 1 n b n := 1 n (Id −P D )c n ⇀ P (rec D) ⊖ c =: b and c 2 = a 2 + b 2 . For an arbitrary vector z ∈ D, [BC17, Thm. 3.16] yields c n − z 2 ≥ a n − z 2 + b n 2 , ∀n ∈ N.
Dividing the inequality by n 2 and taking the limit superior, we get lim 1 n c n 2 ≥ lim ( 1 n a n 2 + 1 n b n 2 ) ≥ lim 1 n a n 2 + lim 1 n b n 2 , and thus lim 1 n a n 2 ≤ lim 1 n c n 2 − lim 1 where the second inequality follows from [BC17, Lem. 2.42]. The inequality above yields lim 1 n a n ≤ a , which due to [BC17, Lem. 2.51] implies 1 n a n → a.
The proposition above generalizes [BL20, Prop. 3.2(iii)], in which a similar result is shown, but with an additional assumption that lim 1 n P D c n exists. We next consider a restricted class of problem (P) in which f is the indicator function of a nonempty closed convex set.
Corollary 3.5. Let s 0 ∈ H and (x n ,x n , ν n ) n∈N be the sequences generated by (1) with f = ι D , where D ⊆ H is a nonempty closed convex set. Then Proof.

Constrained Minimization of a Quadratic Function
Consider the following convex optimization problem: with Q : H 1 → H 1 a monotone self-adjoint bounded linear operator, q ∈ H 1 , A : H 1 → H 2 a bounded linear operator, and C a nonempty closed convex subset of H 2 ; we assume that ran Q and ran A are closed. The objective function of the problem is convex, continuous, and Fréchet differentiable [BC17, Prop. 17.36(i)]. (i) If there existsμ ∈ (rec C) ⊖ such that A * μ = 0 and σ C (μ) < 0, then problem (2) is strongly infeasible. (ii) If there existsz ∈ H 1 such that Qz = 0, Az ∈ rec C, and q |z < 0, then the dual of problem (2) is strongly infeasible.
When H 1 and H 2 are finite-dimensional Euclidean spaces and C has some additional structure, problem (2) reduces to the one considered in [BGSB19], where the Douglas-Rachford algorithm (which is equivalent to the alternating direction method of multipliers) was shown to generate certificates of primal and dual strong infeasibility. This result was generalized in [BL20] to the case where H 1 and H 2 are real Hilbert spaces and C is a general nonempty closed convex set.
The following proposition was first proven in [BGSB19] and then extended in [BL20] to a more general setting. We next show that the same result is a direct consequence of our analysis presented in Section 3. We use the notation where the first and second components are elements of H 1 and H 2 , respectively.
Proof. Let (p n , r n ) be the Douglas-Rachford iterates corresponding to s n in (1) so that (p n+1 , r n+1 ) = T (p n , r n ). As Prox f = P D with D = H 1 × C, we have (z n , y n ) = P D (p n , r n ) = (p n , P C r n ).  Due to Prop. 3.1, we obtain (viii)&(ix): Using the identity rec f * = σ dom f , it is easy to show that the recession functions of those in (4) are given by Due to Prop. 3.1, we obtain Prop. 4.1 and Prop. 4.2 imply that, if −v ′′ P is nonzero, then problem (2) is strongly infeasible, and similarly, if −v ′ D is nonzero, then its dual is strongly infeasible. Moreover, these infeasibility certificates are limits of the sequences ( 1 n µ n ) n∈N and ( 1 n z n ) n∈N . Remark 4.3. Using Fact 2.1(iii), the identity in (5), and the structure of g in (3b), it is easy to show that (z n − z n+1 , y n − y n+1 , µ n − µ n+1 ) → (v ′ D , v ′′ D , v ′′ P ). We do not know whether x n − x n+1 → v D and ν n − ν n+1 → v P hold in a more general setting.

Conclusions
We have presented some useful properties of the minimal displacement vector of the Douglas-Rachford operator applied to the problem of minimizing the sum of two convex functions. In particular, we showed that the minimal displacement vector can be decomposed as the sum of two orthogonal vectors, one of which is a certificate of primal, and the other of dual strong infeasibility of the problem. Moreover, we showed that these infeasibility certificates can be obtained as the limits of sequences constructed from the Douglas-Rachford iterates, which allowed us to recover and generalize some existing results. It would be interesting to explore whether these convergence results hold in a more general case in which one of the functions is not necessarily the indicator function of a convex set.