The exact worst-case convergence rate of the alternating direction method of multipliers

Recently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak-Lojasiewicz (PL)inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two new scenarios.


Introduction
We consider the optimization problem where f : R n → R ∪ {∞} and g : R m → R ∪ {∞} are closed proper convex functions, 0 = A ∈ R r×n , 0 = B ∈ R r×m and b ∈ R r . Moreover, we assume that (x ⋆ , z ⋆ ) is an optimal solution of problem (1) and λ ⋆ is its corresponding Lagrange multipliers. Moreover, we denote the value of f and g at x ⋆ and z ⋆ with f ⋆ and g ⋆ , respectively. Problem (1) appears naturally (or after variable splitting) in many applications in statistics, machine learning and image processing to name but a few [6,20,25,37]. The most common method for solving problem (1) is the alternating direction method of multipliers (ADMM). ADMM is dual based approach that exploits separable structure and it may be described as follows.
ADMM was first proposed in [11,13] for solving nonlinear variational problems. We refer the interested reader to [14] for a historical review of ADMM. The popularity of ADMM is due to its capability to be implemented parallelly and hence can handle large-scale problems [6,19,30,40]. For example, it is used for solving inverse problems governed by partial differential equation forward models [28], and distributed energy resource coordinations [26], to mention but a few.
The convergence of ADMM has been investigated extensively in the literature and there exist many convergence results. However, different performance measures have been used for the computation of convergence rate; see [10,15,16,21,24,25,31,39]. In this paper, we consider the dual objective value as a performance measure.
Throughout the paper, we assume that each subproblem in steps 1 and 2 of Algorithm 1 attains its minimum. The Lagrangian function of problem (1) may be written as and the dual objective of problem (1) is also defined as D(λ) = min (x,z)∈R n ×R m f (x) + g(z) + λ, Ax + Bz − b .
We assume throughout the paper that strong duality holds for problem (1), that is max λ∈R r D(λ) = min Ax+Bz=b f (x) + g(z).
Note that we have strong duality when both functions f and g are real-valued. For extended convex functions, strong duality holds under some mild conditions; see e.g. [3,Chapter 15]. Some common performance measures for the analysis of ADMM are as follows, -Objective value: f (x N ) + g(z N ) − f ⋆ − g ⋆ ; -Primal and dual feasibility: Ax N + Bz N − b and A T B(z N − z N −1 ) ; -Dual objective value: D(λ ⋆ ) − D(λ N ); -Distance between (x N , z N , λ N ) and a saddle points of problem (2).
Note that the mathematical expressions are written in a non-ergodic sense for convenience. Each measure is useful in monitoring the progress and convergence of ADMM. The objective value is the most commonly used performance measure for the analysis of algorithms in convex optimization [3,4,33]. As mentioned earlier, ADMM is a dual based method and it may be interpreted as a proximal method applied to the dual problem; see [4,25] for further discussions and insights. Thus, a natural performance measure for ADMM would be dual objective value. In this study, we investigate the convergence rate of ADMM in terms of dual objective value and feasibility. It worth noting that most performance measures may be analyzed through the framework developed in Section 2.
Regarding dual objective value, the following convergence rate is known in the literature. This theorem holds for strongly convex functions f and g; recall that f is called strongly convex with modulus µ ≥ 0 if the function f − µ 2 · 2 is convex.
In this study we establish that Algorithm 1 has the convergence rate of O( 1 N ) in terms of dual objective value without assuming the strong convexity of g. Under this setting, we also prove that Algorithm 1 has the convergence rate of O( 1 N ) in terms of primal and dual residuals. Moreover, we show that the given bounds are exact. Furthermore, we study the linear and R-linear convergence.

Outline of our paper
Our paper is structured as follows. We present the semidefinite programming (SDP) performance estimation method in Section 2, and we develop the performance estimation to handle dual based methods including ADMM. In Section 3, we derive some new non-asymptotic convergence rates by using performance estimation for ADMM in terms of dual function, primal and dual residuals. Furthermore, we show that the given bounds are tight by providing some examples. In Section 4 we proceed with the study of the linear convergence of ADMM. We establish that ADMM enjoys a linear convergence if and only if the dual function satisfies the P L inequality when the objective function is strongly convex. Furthermore, we investigate the relation between the P L inequality and common conditions used by scholars to prove the linear convergence. Section 5 is devoted to the R-linear convergence. We prove that ADMM is R-linear convergent under two new scenarios which are weaker than the existing ones in the literature.

Terminology and notation
In this subsection we review some definitions and concepts from convex analysis. The interested reader is referred to the classical text by Rockafellar [36] for more information. The n-dimensional Euclidean space is denoted by R n . We use ·, · and · to denote the Euclidean inner product and norm, respectively. The column vector e i represents the i-th standard unit vector and I stands for the identity matrix. For a matrix A, A i,j denotes its (i, j)-th entry, and A T represents the transpose of A. The notation A 0 means the matrix A is symmetric positive semidefinite. We use λ max (A) and λ min (A) to denote the largest and the smallest eigenvalue of symmetric matrix A, respectively. Moreover, the seminorm · A is defined as x A = Ax for any A ∈ R m×n . Suppose that f : R n → (−∞, ∞] is an extended convex function. The function f is called closed if its epi-graph is closed, that is {(x, r) : f (x) ≤ r} is a closed subset of R n+1 . The function f is said to be proper if there exists x ∈ R n with f (x) < ∞. We denote the set of proper and closed convex functions on R n by F 0 (R n ). The subgradients of f at x is denoted and defined as We call a differentiable function f L-smooth if for any Definition 1 Let f : R n → (−∞, ∞] be a closed proper function and let A ∈ R m×n . We say f is c-strongly convex relative to In the rest of the section, we assume that A ∈ R m×n . It is seen that any µ-strongly convex function is µ λmax(A T A) -strongly convex relative to . A . However, its converse does not necessarily hold unless A has full column rank. Hence, the assumption of strong convexity relative to . A for a given matrix A is weaker compared to the assumption of strong convexity. For further details on the strong convexity in relation to a given function, we refer the reader to [29]. We denote the set of c-strongly convex functions relative to . A on R n by F A c (R n ). We denote the distance function to the set X by d X (x) := inf y∈X y − x .
In the following sections we derive some new convergence rates for ADMM by using performance estimation. The main idea of performance estimation is based on interpolablity. Let I be an index set and let {( The next theorem gives necessary and sufficient conditions for F A c -interpolablity. for some x ⋆ ∈ ∂f * (−A T λ ⋆ ) and z ⋆ ∈ ∂g * (−B T λ ⋆ ). Equation (8) with (6) imply that (x ⋆ , z ⋆ ) is an optimal solution to problem (1). The optimality conditions for the subproblems of Algorithm 1 may be written as As

Performance estimation
In this section, we develop the performance estimation for ADMM. The performance estimation method introduced by Drori and Teboulle [9] is an SDPbased method for the analysis of first order methods. Since then, many scholars employed this strong tool to derive the worst case convergence rate of different iterative methods; see [2,23,38,41] and the references therein. Moreover, Gu and Yang [17] employed performance estimation to study the extension of the dual step length for ADMM. Note that while there are some similarities between our work and [17] in using performance estimation, the formulations and results are different. The worst-case convergence rate of Algorithm 1 with respect to dual objective value may be cast as the following abstract optimization problem, where f, g, A, B, b, z 0 , λ 0 , x ⋆ , z ⋆ , λ ⋆ are decision variables and N, t, c 1 , c 2 , ∆ are the given parameters.
By using Theorem 2 and the optimality conditions (9), problem (11) may be reformulated as the finite dimensional optimization problem, To handle problem (12), without loss of generality, we assume that the matrix A B has full row rank. Note this assumption does not employed in our arguments in the following sections. In addition, we introduce some new variables. As problem (1) is invariant under translation of (x, z), we may assume without loss of generality that b = 0 and (x ⋆ , z ⋆ ) = (0, 0). In addition, due to the full row rank of the matrix A B , we may assume that By using equality constraints of problem (12) and the newly introduced variables, we have for k ∈ {1, ..., N } It is worth noting that a pointx satisfying these conditions exists, as function f is strongly convex relative to A. In addition, one may considerz = z N by virtue of (10). For the sake of notation convenience, we introduce x N +1 =x. The reader should bear in mind that x N +1 is not generated by Algorithm 1.
. Hence, problem (12) may be written as In problem (15) 1 , x † , z † ,x, f ⋆ ,z, g ⋆ , z 0 are decision variables. By using the Gram matrix method, problem (15) may be relaxed as a semidefinite program as follows. Let By introducing matrix variable problem (15) may be relaxed as the following SDP, where the constant matrices L f i,j , L g i,j , L o , L 0 are determined according to the constraints of problem (15). In the following sections, we present some new convergence results that are derived by solving this kind of formulation.

Worst-case convergence rate
In this section, we provide new convergence rates for ADMM with respect to some performance measures. Before we get to the theorems we need to present some lemmas.
and k denotes row number. If c > 0 is given, then Proof. As {t : E(t, c) 0} is a convex set, it suffices to prove the positive semidefiniteness of E(0, c) and E(c, c). Since E(0, c) is diagonally dominant, it is positive semidefinite. Now, we establish that the matrix K = E(1, 1) is positive definite. To this end, we show that all leading principal minors of K are positive. To compute the leading principal minors, we perform the following elementary row operations on K: i) Add the second row to the third row; ii) Add the second row to the last row; iii) Add the third row to the forth row; iv) For i = 4 : Hence, by performing these operations, we get an upper triangular matrix J with diagonal It is seen all first N diagonal elements of J are positive. We show that J N +1,N +1 is also positive. For i ≥ 4 we have So, which implies J N +1,N +1 > 0. Since we add a factor of i − th row to j − th row with i < j, all leading principal minors of matrices K and J are the same. Hence K is positive definite. As E(c, c) = cK, one can infer the positive definiteness of E(c, c) and the proof is complete.
In the upcoming lemma, we establish a valid inequality for ADMM that will be utilized in all the subsequent results presented in this section. where Proof. To establish the desired inequality, we demonstrate its validity by summing a series of valid inequalities. To simplify the notation, let f k = f (x k ) and g k = g(z k ) for k ∈ {1, . . . , N }. Note that b = 0 because x ⋆ = 0, z ⋆ = 0. By (4) and (9), we get the following inequality As λ k = λ k−1 + tAx k + tBz k , the inequality can be expressed as After performing some algebraic manipulations, we obtain which implies the desired inequality.
We may now prove the main result of this section.
Proof. As discussed in Section 2, we may assume that x ⋆ = 0 and z ⋆ = 0. By . By employing (4) and (9), we obtain By substituting v with Ax N in inequality (18) and summing it with (20), we get the following inequality after performing some algebraic manipulations where the positive semidefinite matrix E(t, c 1 ) is given in Lemma 1. As the inner product of positive semidefinite matrices is non-negative, inequality (21) implies that , and the proof is complete.
In comparison with Theorem 1, we could get a new convergence rate when only f is strongly convex, i.e. g does not need to be strongly convex. Also, the constant does not depend on λ 1 . One important question concerning bound (19) is its tightness, that is, if there is an optimization problem which attains the given convergence rate. It turns out that the bound (19) is exact. The following example demonstrates this point.
Example 1 Suppose that c 1 > 0, N ≥ 4 and t ∈ (0, c 1 ]. Let f, g : R → R be given as follows, It is seen that A = B = I in this problem. Note that (x ⋆ , z ⋆ ) = (0, 0) with Lagrangian multiplier λ ⋆ = 1 2 is an optimal solution and the optimal value is zero. One can check that Algorithm 1 with initial point λ 0 = −1 2 and z 0 = 0 generates the following points, , which shows the tightness of bound (19).
One important factor concerning dual-based methods that determines the efficiency of an algorithm is primal and dual feasibility (residual) convergence rates. In what follows, we study this subject under the setting of Theorem 3. The next theorem gives a convergence rate in terms of primal residual under the setting of Theorem 3.
Proof. The argument is similar to that used in the proof of Theorem 3. By setting v = Ax N in (18), one can infer the following inequality By employing (4) and (9), we have By summing (23) and (24), we obtain where the matrix D(t, c 1 ) is as follows, As the matrix D(t, c 1 ) is positive semidefinite, see Appendix A, inequality (25) implies that and the proof is complete.
The following example shows the exactness of bound (22).
In what follows, we study the convergence rate of ADMM in terms residual dual. To this end, we investigate the convergence rate of The next theorem provides a convergence rate for the aforementioned sequence.
Proof. Similar to the proof of Theorem 3, by setting v = Ax N in (18) for N −1 iterations, one can infer the following inequality By using (4) and (9), we have By summing (27) and (28), we obtain where the matrix F (t, c 1 ) is as follows, The rest of the proof proceeds analogously to the proof of Theorem 4.
The following example shows the tightness of this bound.

It can be seen that
, which shows that the bound is tight. Theorem 3 and 4 address the case that f is strongly convex relative to . A and g is convex. Based on numerical results by solving performance estimation problems including (15) we conjecture, under the assumptions of Theorem 3, if g is c 2 -strongly convex relative to . B , Algorithm 1 enjoys the following convergence rates We have verified these conjectures numerically for many specific values of the parameters.

Linear convergence of ADMM
In this section we study the linear convergence of ADMM. The linear convergence of ADMM has been addressed by some authors and some conditions for linear convergence have been proposed, see [8,18,19,22,27,34,42]. Two common types of assumptions employed for proving the linear convergence of ADMM are error bound property and L-smoothness. To the best knowledge of authors, most scholars investigated the linear convergence of the sequence {(x k , z k , λ k )} to a saddle point and there is no result in terms of dual objective value for ADMM. In line with the previous section, we study the linear convergence in terms of dual objective value and we derive some formulas for linear convergence rate by using performance estimation. It is noteworthy to mention that the term "Q-linear convergence" is also employed to describe the linear convergence in the literature. As mentioned earlier, error bound property is used by scholars for establishing the linear convergence; see e.g. [18,22,27,35,42]. Let stands for augmented dual objective for the given a > 0 and Λ ⋆ denotes the optimal solution set of the dual problem. Note that function D a is an 1 a -smooth function on its domain without assuming strong convexity; see [22,Lemma 2.2].

Definition 2 The function D a satisfies the error bound if we have
for some τ > 0.
Hong et al. [22] established the linear convergence by employing error bound property (30).
Recently, some scholars established the linear convergence of gradient methods for L-smooth convex functions by replacing strong convexity with some mild conditions, see [1,5,32] and references therein. Inspired by these results, we prove the linear convergence of ADMM by using the so-called P L inequality. Concerning differentiability of dual objective, by (7), we have Note that inclusion (31) holds as an equality under some mild conditions, see e.g. [3,Chapter 3].

Definition 3
The function D is said to satisfy the P L inequality if there exists an L p > 0 such that for any λ ∈ R r we have Note that if f and g are strongly convex, then −D is an L-smooth convex . This follows from the duality between smoothness and strong convexity and In the next proposition, we show that definitions (30) and (32) are equivalent.
Proposition 1 Let L a = 1 a denote the Lipschitz constant of ∇D a , where D a is given in (29).
i) If D a satisfies the error bound (30), then D satisfies the P L inequality with L p = 1 Laτ 2 . ii) If D satisfies the P L inequality, then D a satisfies the error bound (30) with τ = Lp 1+aLp .
Proof. First we prove i). Suppose λ ∈ R r and ξ ∈ b − A∂f * (−A T λ) − B∂g * (−B T λ). By identity (6), we have ξ = b − Ax − Bz for some (x,z) ∈ argmin f (x) + g(z) + λ, Ax + Bz − b . Due to the smoothness of D a and (30), we get where As we assume strong duality, we have D a (λ ⋆ ) = D(λ ⋆ ). By the definitions of x,ȳ, we get By [22, Lemma 2.1], we have ∇D a (ν) = Ax + Bz − b. This equality with (33) imply and the proof of i) is complete. Now we establish ii). Let λ be in the domain of ∇D a . By [22,Lemma 2 . By the P L inequality, we have where the equality follows from D(ν) = D a (λ)+ a 2 Ax + Bz − b 2 and D a (λ ⋆ ) = D(λ ⋆ ). Hence, This inequality says that D a satisfies the P L inequality. On the other hand, the P L inequality implies the error bound with the same constant, see [5], and the proof is complete.
In what follows, we employ performance estimation to derive a linear convergence rate for ADMM in terms of dual objective when the P L inequality holds. To this end, we compare the value of dual problem in two consecutive iterations, that is, D(λ ⋆ )−D(λ 2 ) D(λ ⋆ )−D(λ 1 ) . The following optimization problem gives the worst-case convergence rate, s. t. {x 2 , z 2 , λ 2 } is generated by Algorithm 1 w.r.t. f, g, A, B, b, λ 1 , z 1 (35) (x ⋆ , z ⋆ ) is an optimal solution and its Lagrangian multipliers is λ ⋆ D satisfies the P L inequality Analogous to our discussion in Section 2, we may assume without loss of In addition, we assume thatx 1 ∈ argmin f (x)+ λ 1 , Ax andx 2 ∈ argmin f (x)+ λ 2 , Ax . Hence, Moreover, by (36) and (31), we get On the other hand, λ 2 = λ 1 + tAx 2 + tBz 2 . Therefore, by using Theorem 2, problem (35) may be relaxed as follows, By deriving an upper bound for the optimal value of problem (37) in the next theorem, we establish the linear convergence of ADMM in the presence of the P L inequality.
Theorem 6 Let f ∈ F A c1 (R n ) and g ∈ F B c2 (R m ) with c 1 , c 2 > 0, and let D satisfies the P L inequality with L p . Suppose that t ≤ √ c 1 c 2 .
Proof. The argument is based on weak duality. Indeed, by introducing suitable Lagrangian multipliers, we establish that the given convergence rates are upper bounds for problem (37). First, we prove (i). Assume that α denotes the right hand side of inequality (38). As 2c 1 c 2 − t 2 > 0 and 4c 1 c 2 − c 2 t − 2t 2 > 0, we have 0 < α < 1. With some algebra, one can show that Hence, we get for any feasible point of problem (35) and the proof of the first part is complete. For (ii), we proceed analogously to the proof of (i), but with different Lagrange multipliers. Let β denote the right hand side of inequality (39), i.e. .
It is seen that 0 < β < 1. By doing some calculus, we have The rest of the proof is similar to that of the former case.
We computed the bounds in Theorem 6 by selecting suitable Lagrangian multipliers and solving the semidefinite formulation of problem (37) by hand. The semidefinite formulation is formed analogous to problem (16). Note that the optimal value of problem (37) may be smaller than the bounds introduced in Theorem 6. Indeed, our aim was to provide a concrete mathematical proof for the linear convergence rate. However, the linear convergence rate factor is not necessarily tight. Needless to say that the optimal value of problem (37) also does not necessarily give the tight convergence factor as it is just a relaxation of problem (35).
Recently the authors showed that the P L inequality is necessary and sufficient conditions for the linear convergence of the gradient method with constant step lengths for L-smooth function; see [1,Theorem 5]. In what follows, we establish that the P L inequality is a necessary condition for the linear convergence of ADMM. Firstly, we present a lemma that is very useful for our proof.
Proof. Without loss of generality we assume that c 1 = c 2 = 0. By optimality conditions, we have By using these inequities, we get Hence, we have which completes the proof.
The next theorem establishes that the P L inequality is a necessary condition for the linear convergence of ADMM.
Theorem 7 Let f ∈ F A c1 (R n ) and g ∈ F B c2 (R m ). If Algorithm 1 is linearly convergent with respect to the dual objective value, then D satisfies the P L inequality.
Another assumption used for establishing linear convergence is L-smoothness; see for example [7,8,12,34]. Deng et al. [8] show that the sequence {(x k , z k , λ k )} is convergent linearly to a saddle point under Scenario 1 and 2 given in Table  1.
It is worth mentioning that Scenario 1 or Scenario 2 implies strong convexity of the dual objective function and therefore the P L inequality is resulted, see [1]. Hence, Theorem 6 implies the linear convergence in terms of dual value under Scenario 1 or Scenario 2. Deng et al. [8] studied the linear convergence under Scenario 3, but they just proved the linear convergence of the sequence {(x k , Bz k , λ k )}. In the next section, we investigate the R-linear convergence without assuming L-smoothness of f . Indeed, we establish the R-linear convergence when f is strongly convex, g is L-smooth and B has full row rank.
Note that the P L inequality does not imply necessarily Scenario 1 or Scenario 2. Indeed, consider the following optimization problem, Hence, the P L inequality holds for L p = 1 2 while neither f nor g is L-smooth. As mentioned earlier the performance estimation problem including the P L inequality at finite set of points is a relaxation for computing the worstcase convergence rate. Contrary to Theorem 6, we could not manage to prove the linear convergence of primal and dual residuals under the assumptions of Theorem 6 by employing performance estimation.

R-linear convergence of ADMM
In this section, we study the R-linear convergence of ADMM. Recall that ADMM enjoys R-linear convergent in terms of dual objective value if for some ρ ≥ 0 and γ ∈ [0, 1).
We investigate the R-linear convergence under the following scenarios: -(S1): f ∈ F A c1 (R n ) is L-smooth with c 1 > 0 and A has full row rank; -(S2): f ∈ F A c1 (R n ) with c 1 > 0, g is L-smooth and B has full row rank. Our technique for proving the R-linear convergence is based on establishing the linear convergence of the sequence {V k } given by Note that V k is called Lyapunov function for ADMM; see [6].
First we consider the case that f is L-smooth and c 1 -strongly convex relative to A. The following proposition establishes the linear convergence of {V k }.
Proposition 2 Let f ∈ F A c1 (R n ) be L-smooth with c 1 > 0, g ∈ F 0 (R m ) and let A has full row rank. If t < c1L λmin(AA T ) , then where d = L λmin(AA T ) . Proof. We may assume without loss of generality that x ⋆ , z ⋆ and b are zero; see our discussion in Section 2. By optimality conditions, we have for some η k ∈ ∂g(z k+1 ) and η ⋆ ∈ ∂g(z ⋆ ). Let α = 2t c 2 1 d 2 +2c1dt 2 −4c 2 1 t 2 +t 4 . By Theorem 2, we get As A T λ 2 ≥ L d λ 2 and λ k+1 = λ k + tAx k+1 + tBz k+1 , we obtain the following inequality after performing some algebraic manipulations The above inequality implies that and the proof is complete.
Note that one can improve bound (42) under the assumptions of Proposition 2 and the µ-strong convexity of f by employing the following known inequality Indeed, we employed the given inequality but we could not manage to obtain a closed form formula for the convergence rate. The next theorem establishes the R-linear convergence of ADMM in terms of dual objective value under the assumptions of Proposition 2.
Theorem 8 Let N ≥ 4 and let A has full row rank. Suppose that f ∈ F c1 (R n ) is L-smooth with c 1 > 0 and g ∈ F 0 (R m ). If t < min{c 1 , Proof. By Theorem 3 and Proposition 2, one can infer the following inequalities, which shows the desired inequality.
Nishihara et al. [34] showed the R-linear convergence of ADMM in terms of {x k , z k , λ k } under the following conditions: i) The function f is L-smooth and µ-strong with µ > 0; ii) The matrix A is invertible and that B has full column rank.
In Theorem 8, we obtain the R-linear convergence under weaker assumptions. Indeed, we replace condition ii) with the matrix A having full row rank.
In the sequel, we investigate the R-linear convergence under the hypotheses of scenario (S2). The next proposition shows the linear convergence of {V k }.
Proposition 3 Let f ∈ F A c1 (R n ) with c 1 > 0 and let g ∈ F 0 (R m ) be L-smooth. Suppose that B has full row rank and k ≥ 1. If t ≤ min{ c1 2 , L 2λmin(BB T ) }, then Proof. Analogous to the proof of Proposition 2, we assume that x ⋆ = 0, z ⋆ = 0 and b = 0. Due to the optimality conditions, we have for some ξ k+1 ∈ ∂f (x k+1 ) and ξ ⋆ ∈ ∂f (x ⋆ ). Suppose that d = L λmin(BB T ) and α = 2dt d+t . By Theorem 2, we obtain By employing B T λ 2 ≥ L d λ 2 and λ k+1 = λ k + tAx k+1 + tBz k+1 , the aforementioned inequality can be expressed as follows after some algebraic manipulation, Hence, we have and the proof is complete.
As the sequence {V k } is not increasing [6, Convergence Proof], we have V 1 ≤ V 0 . Thus, by using Theorem 3 and Proposition 3, one can infer the following theorem.
Theorem 9 Let f ∈ F A c1 (R n ) with c 1 > 0 and let g ∈ F 0 (R m ) be L-smooth. Assume that N ≥ 5 and B has full row rank. If t < min{ c1 2 , L 2λmin(BB T ) }, then In the same line, one can infer the R-linear convergence in terms of primal and dual residuals under the assumptions of Theorem 8 and Theorem 9. In this section, we proved the linear convergence of {V k } under two scenarios (S1) and (S2). By (7), it is readily seen that function −D is strongly convex under the hypotheses of both scenarios (S1) and (S2). Therefore, both scenarios imply the P L inequality. One may wonder that if the P L inequality and the strong convexity of f imply the linear of {V k }. By using performance estimation, we could not establish such an implication.
As mentioned above, function −D under both scenarios are µ-strongly convex. Hence, the optimal solution set of the dual problem is unique and one can infer the R-linear convergence of λ N by using Theorem 8 (Theorem 9) and the known inequality,

Concluding remarks
In this paper we developed performance estimation framework to handle dualbased methods. Thanks to this framework, we could obtain some tight convergence rates for ADMM. This framework may be exploited for the analysis of other variants of ADMM in the ergodic and non-ergodic sense. Moreover, similarly to [23], one can apply this framework for introducing and analyzing new accelerated ADMM variants. Moreover, most results hold for any arbitrary positive step length, t, but we managed to get closed form formulas for some interval of positive numbers.