Inexact indefinite proximal ADMMs for 2-block separable convex programs and applications to 4-block DNNSDPs

This paper is concerned with two-block separable convex minimization problems with linear constraints, for which it is either impossible or too expensive to obtain the exact solutions of the subproblems involved in the proximal ADMM (alternating direction method of multipliers). Such structured convex minimization problems often arise from the two-block regroup settlement of three or four-block separable convex optimization problems with linear constraints, or from the constrained total-variation superresolution image reconstruction problems in image processing. For them, we propose an inexact indefinite proximal ADMM of step-size $\tau\in\!(0,\frac{\sqrt{5}+1}{2})$ with two easily implementable inexactness criteria to control the solution accuracy of subproblems, and establish the convergence under a mild assumption on indefinite proximal terms. We apply the proposed inexact indefinite proximal ADMMs to the three or four-block separable convex minimization problems with linear constraints, which are from the duality of the important class of doubly nonnegative semidefinite programming (DNNSDP) problems with many linear equality and/or inequality constraints. Numerical results indicate that the inexact indefinite proximal ADMM with the absolute error criterion has a comparable performance with the directly extended multi-block ADMM of step-size $\tau=1.618$ without convergence guarantee, whether in terms of the number of iterations or the computation time.


Introduction
Let X, Y and Z be the finite dimensional vector spaces endowed with the inner product ·, · and its induced norm · . Given closed proper convex functions f : X → (−∞, +∞] and g : Y → (−∞, +∞], we are concerned with the separable convex optimization problem min x∈X,y∈Y f (x) + g(y) where A : Z → X and B : Z → Y are the given linear operators, A * and B * denote the adjoint operators of A and B, respectively, and c ∈ Z is a given vector.
As well known, there are many important cases with the form of (1), which include the covariance selection problems and semidefinite least squares problems in statistics [1,30,39], the sparse plus low-rank recovery problem arising from the so-called robust PCA (principle component analysis) with noisy and incomplete data [34,32], the constrained total-variation image restoration and reconstruction problems [22,29], the simultaneous minimization of the nuclear norm and ℓ 1 -norm of a matrix arising from the low-rank and sparse representation for image classification and subspace clustering [40,36], and so on.
For the structured convex minimization problem (1), the alternating direction method of multipliers (ADMM for short), first proposed by Glowinski and Marrocco [11] and Gabay and Mercier [12], is one of the most popular methods. For any given σ > 0, let L σ : X × Y × Z → (−∞, +∞] denote the augmented Lagrangian function of problem (1) The ADMM, from an initial point (x 0 , y 0 , z 0 ) ∈ dom f × dom g × Z, consists of the steps            x k+1 ∈ arg min x∈X L σ (x, y k , z k ), y k+1 ∈ arg min y∈Y L σ (x k+1 , y, z k ), (2b) where τ ∈ (0, 1+ √ 5 2 ) is a constant to control the step-size in (2c). The iterative scheme of ADMM actually embeds a Gaussian-Seidel decomposition into each iteration of the classical augmented Lagraigan method of Hestenes-Powell-Rockafellar [14,25,28], so that the challenging task (i.e., the exact solution or the approximate solution with a high precision of the Lagrangian minimization problem) is relaxed to several easy ones.
Notice that the subproblems (2a) and (2b) in the ADMM may have no closed-form solutions or even be difficult to solve. When the functions f and g enjoy a closed-form Moreau envelope, one usually introduces the proximal terms 1 2 x−x k P f and 1 2 y −y k Pg respectively into the subproblems (2a) and (2b) to cancel the operators AA * and BB * so as to get the exact solutions of proximal subproblems. This is the so-called proximal-ADMM which, for a chosen initial point (x 0 , y 0 , z 0 ) ∈ dom f × dom g × Z, consists of              x k+1 = arg min x∈X L σ (x, y k , z k ) + 1 2 x − x k P f , (3a) The existing works on the proximal ADMM mostly focus on the positive definite proximal terms (see, e.g., [15,35,41]). It is easy to see that the proximal subproblems with the positive definite proximal terms will have a big difference from the original subproblems of ADMM. In fact, as pointed out in the conclusion remarks of [15], "large and positive definite proximal terms will lead to easy solution of subproblems, but the number of iterations will increase. Therefore, for subproblems which are not extremely ill-posed, the proximal parameters should be small." In view of this, some researchers recently develop the semi-proximal or indefinite proximal ADMM [38,9,21] by using the positive semidefinite even indefinite proximal terms. The numerical experiments in [9] show that such tighter proximal terms display better numerical performance. In addition, it is worthwhile to emphasize that the ADMM itself is a semi-proximal (of course an indefinite proximal) ADMM, but is not in the family of positive definite proximal ADMMs.
In this paper we are interested in problem (1) in which the functions f and/or g may not have a closed-form Moreau envelope or the linear operators A and/or B have a large spectral norm (now the proximal subproblems with a positive definite proximal term are bad surrogates for those of the ADMM), for which it is impossible or too expensive to achieve the exact solutions of the proximal subproblems though they are unique. Such separable convex optimization problems arise directly from the constrained totalvariation superresolution image reconstruction problems [4,24] in image processing, and the two-block regroup of three or four-block separable convex minimization problems. Indeed, for the following four-block separable convex minimization problem where f i : X i → (−∞, +∞] for i = 1, 2, 3, 4 are closed proper convex functions, and A i : Z → X i for i = 1, 2, 3, 4 are linear operators, since the directly extended multi-block ADMM does not have the convergence guarantee (see the counterexamples in [3]), one may rearrange it as the form of (1) by reorganizing any two groups of variables into one group, and then apply the classical ADMM for solving the two-block regrouped problem. Clearly, the exact solution of each subproblem of ADMM for the two-block regrouped problem is difficult to obtain due to the cross of two classes of variables. In particular, the two-block regroup resolving of multi-block separable convex optimization also has a separate study value.
To resolve this class of difficult two-block separable convex minimization problems, we propose an inexact indefinite proximal ADMM with a step-size τ ∈ (0, √ 5+1 2 ), in which the proximal subproblems are solved to a certain accuracy with two easily implementable inexactness criteria to control the accuracy. Here, an indefinite proximal term, instead of a positive definite proximal term, is introduced into each subproblem of the ADMM to guarantee that each proximal subproblem has a unique solution as well as becomes a good surrogate for the original subproblem of the ADMM. For the proposed inexact indefinite proximal ADMM, we establish its convergence under a mild assumption on the indefinite proximal terms. To the best of our knowledge, this is the first convergent inexact proximal ADMM in which step-size τ may take the value in the interval (1, √ 5+1 2 ). We notice that a few existing research papers on inexact versions of the ADMM all focus on the unit step-size; see [8,15,24,13,5], and moreover, only the papers [24,13,5] develop truly implementable inexactness criteria in the exact solutions are not required. Our inexact indefinite proximal ADMM is using the same absolute error criterion and and a little different relative error from the one used in [24]. It is well known that the ADMM with τ = 1.618 requires less 20% to 50% iterations than the one with τ = 1, especially for those difficult SDP problems [33]. Thus, the proposed inexact indefinite proximal ADMMs with a large step-size is expected to have better performance.
In this work, we apply the inexact indefinite proximal ADMMs to the three and fourblock separable convex minimization problems with linear constraints, coming from the duality of the doubly nonnegative semidefinite programming (DNNSDP) problems with many linear equality and/or inequality constraints. Specifically, we solve the two-block regroupment for the dual problems of DNNSDPs with the inexact indefinite proximal ADMM. Observe that the iterates yielded by solving each subproblem in an alternating way can satisfy the optimality condition approximately. Hence, in the implementation of the inexact indefinite proximal ADMMs, we get the inexact solution of each subproblem by minimizing the two group of variables alternately. Numerical results indicate that the inexact indefinite proximal ADMM with the absolute error criterion is comparable with the directly extended multi-block ADMM with step-size τ = 1.618 whether in terms of the number of iterations or the computation time, while the one with the relative error criterion requires less outer-iterations but more computation time since the error criterion is more restrictive and requires more inner-iterations. Thus, the inexact indefinite proximal ADMM with the absolute error criterion provides an efficient tool for handling the three and four-block separable convex minimization problems.
We observe that there are several recent works [37,18,19,10] to regroup the multiblock separable convex minimization problems into two-block or several subblocks, and then solve each subblock simultaneously by introducing a positive definite proximal term related to the numbers of subproblems. Such procedures lead to easily solvable subproblems, but their performance becomes worse due to larger proximal terms.
The rest of this paper is organized as follows. Section 2 gives some notations and the main assumption. Section 3 describes the inexact indefinite proximal ADMMs and analyzes the properties of the sequence generated. The convergence of the inexact indefinite proximal ADMMs is established in Section 4. Section 5 applies the inexact indefinite proximal ADMMs for solving the duality of the doubly DNNSDPs with many linear equality and/or inequality constraints. Some concluding remarks are given in Section 6.

Notations and assumption
Notice that the functions f : X → (−∞, +∞] and g : Y → (−∞, +∞] are closed proper convex, and the subdifferential mappings of closed proper convex functions are maximal monotone [26,Theorem 12.17]. Hence, there exist self-adjoint operators Σ f 0 and Σ g 0 such that for all x, x ∈ dom f, u ∈ ∂f (x) and u ∈ ∂f ( x), and for all y, y ∈ dom g, v ∈ ∂g(y) and v ∈ ∂g( y), For a self-adjoint linear operator T : X → X, the notation T 0 (respectively, T ≻ 0) means that T is positive semidefinite (respectively, positive definite), that is, x, T x ≥ 0 for all x ∈ X (respectively, x, T x > 0 for all x ∈ X\{0}). Given a self-adjoint positive semidefinite linear operator T : X → X, we denote by · T the norm induced by T , i.e., Given a self-adjoint positive definite linear operator, we denote by λ max (T ) and λ min (T ) the largest eigenvalue and the smallest eigenvalue of T , respectively, and by D T (x, Ω) the distance induced by T from x to a closed set Ω, that is, When T is the identity operator, we suppress the notation T in D T (x, Ω) and write simply D(x, Ω). Clearly, for any positive definite linear operator T : X → X and γ > 0, In addition, for any u, v ∈ X and any self-adjoint linear operator T : X → X, the following two identities will be frequently used in the subsequent analysis: Throughout this paper, we make the following assumption for problem (1): has an optimal solution, to say (x * , y * ) ∈ dom f × dom g, and there exists a point ( where ∂f and ∂g are the subdifferential mappings of f and g, respectively. Moreover, any z * ∈ Z satisfying (9) is an optimal solution to the dual problem of (1). In the sequel, we call (x * , y * , z * ) ∈ dom f × dom g × Z a primal-dual solution pair of problem (1).

Inexact indefinite proximal ADMMs
In this section, we describe the iteration steps of the inexact indefinite proximal ADMMs for solving problem (1), and then analyze the properties of the sequence generated.
The iteration steps of our inexact indefinite proximal ADMMs are stated as follows.
The approximate optimality in (S.1) and (S.2) is measured by the following criteria: where F : X → X and G : Y → Y are self-adjoint positive definite linear operators with F −1 T f and G −1 T g ; where F and G are same as the one in (C2).
(b) When P f and P g are chosen to be self-adjoint positive semidefinite operators, the IEIDP-ADMMs with µ k ≡ ν k ≡ 0 reduce to the semi-proximal ADMM in [38,9]. Next we study the properties of the sequence generated by the IEIDP-ADMMs. For convenience, we let h(x, y) := A * x + B * y − c for (x, y) ∈ X × Y, and for each k ≥ 1 write Using these notations and noting that h(x * , y * ) = 0, we can rewrite Step (S.3) as Lemma 3.1 Let (x k , y k , z k ) be the sequence generated by the IEIDP-ADMMs with (x k , ξ k ) and (y k , η k ) satisfying equation (10) or (11). Suppose that Assumption 2.1 holds and the operator P g also satisfies P g + 3 8 Σ g 0. Then, for all k ≥ 0 we have Proof: From the expressions of φ k and ψ k and equations (10) and (11), it follows that Substituting the first identity in (12) into equations (13) and (14) respectively yields In view of inequalities (5) and (6), from the last two inclusions and equation (9) we have Adding the last two inequalities together and using equation (12) yields that Σg .
Next we deal with the term σ h(x k+1 , y k+1 ), B * ∆y k+1 in inequality (15). Notice that We first bound the first two terms in (15). From equations (14) and (12), it follows that Combining the last two inclusions with the second inequality in (6) yields that Using equation (8) and the given assumption P g + 3 8 Σ g 0, we have that where the last inequality is using 1 2 y k+1 −y k−1 2 Combining inequalities (18) and (17) with equation (16), we immediately obtain Now substituting inequality (19) into equation (15), we immediately obtain that By the first equality of (8) and equation (12), the term ∆z k+1 , z k+1 e can be written as Applying equation (8) to x k+1 e , P f ∆x k+1 and y k+1 e , (P g + σBB * )∆y k+1 yields Substituting the last three equalities into inequality (20), we have that Notice that x k+1 Σg ≥ 1 2 ∆y k+1 2 Σg . The last inequality implies the desired result. The proof is completed. ✷ The following lemma provides an upper bound for the term r k+1 in Lemma 3.1.
Similarly, we can prove that another two inequalities hold for all k ≥ 1. Summing up the three inequalities in (25) yields (22). When the criterion (C2') is used, for any γ > 0, Tg for all k ≥ 1. Indeed, when ν k = 0, the third inequality in (26) holds since now η k = 0; and when ν k = 0, from equation (7) and Tg . Similarly, one can prove that another two inequalities hold for all k ≥ 1. Summing up the three inequalities in (26) yields (23). The proof is completed. ✷ Based on the results of Lemmas 3.1 and 3.2, we obtain the following proposition.
Proposition 3.1 Let {(x k , y k , z k )} k≥1 be the sequence generated by the IEIPD-ADMMs. Suppose that Assumption 2.1 holds and the operator P g also satisfies P g + 3 8 Σ g 0. (a) If the criterion (C1) is used in (S.1) and (S.2), then for any given γ > 0 we have If the criterion (C2) is used for (S.1) and (S.2), then we have (c) If the criterion (C2') is used for (S.1) and (S.2), then for any given γ > 0 we have Proof: (a) From inequality (21) and the result of Lemma 3.1, it follows that where the second inequality is due to σBB * T g , the last inequality implies part (a).
(b) From inequality (22) and the result of Lemma 3.1, it follows that For the terms x k+1 e AA * and ∆x k+1 AA * , using equation (7) yields that x k+1 Combining the last inequalities with (29) and using σBB * T g yields part (b).
The proof of Part (c) is similar to that of part (b), we here omit it. ✷

Convergence analysis of the IEIDP-ADMMs
In this section we analyze the convergence of the IEIPD-ADMMs with the approximation criterion (C1) and (C2) respectively chosen for the minimization in (S.1) and (S.2).

Convergence of the IEIDP-ADMM with (C1)
For convenience, we write w k := (x k e ; y k e ; z k e ; ∆y k ; ∆z k ) for k ≥ 1, and let H : with the proximal operators P f and P g satisfying P f + Σ f 0 and P g + 3 4 Σ g 0.

By the expressions of
2τ h(x k , y k ) 2 + ∆y k+1 2 Tg < ∞, the sequence {(x k , y k )} converges to an optimal solution of (1) and the sequence {z k } converges to an optimal solution to the dual problem of (1).
By equality (12), the limits in (36) imply that the sequence { x k+1 e AA * } is bounded. By the definition of H and the boundedness of {Hw k }, the sequence { x k+1 e P f +Σ f } is also bounded. Thus, the sequence { x k+1 e T f } is bounded. In addition, the boundedness of {Hw k } also implies that the sequences { y k+1 e Tg } and { z k+1 e } are bounded. Together with the positive definiteness of T f and T g , it follows that {(x k , y k , z k )} is bounded. So, there exists a convergent subsequence, to say {(x k , y k , z k )} K . Without loss of generality, we assume {(x k , y k , z k )} K → (x ∞ , y ∞ , z ∞ ). Since lim k→∞ z k − z k−1 = 0, we have h(x ∞ , y ∞ ) = 0. In addition, taking the limit k → ∞ with k ∈ K on the both sides of (13) and (14), and using the closedness of the graphs of ∂f and ∂g (see [27]), we have −A * z ∞ ∈ ∂f (x ∞ ) and −B * z ∞ ∈ ∂g(y ∞ ).
Along with (9), (x ∞ , y ∞ ) is an optimal solution of (1) and z ∞ is the associated multiplier.
This means that for any ε > 0, there exists a sufficiently large k 0 ∈ K such that Hw k 0 W k 0 < ε 2K 1 and ∞ l=k 0 ̟ l < ε 2K 1 . By Lemma 4.1, for any k ≥ k 0 we have This, by the positive definiteness of W, shows that lim k→∞ Hw k+1 = 0. Consequently, Combining lim k→∞ y k+1 −y ∞ Tg = 0 with T g ≻ 0 yields lim k→∞ y k+1 −y ∞ = 0. In addition, the second limit in (36) implies lim k→∞ h(x k , y k ) = 0. Together with By the positive definiteness of T f , it follows that lim k→∞ x k −x ∞ = 0. Thus, lim k→∞ x k = x ∞ , lim k→∞ y k = y ∞ and lim k→∞ z k = z ∞ . That is, (x ∞ , y ∞ , z ∞ ) is the unique limit point of {(x k , y k , z k )}. ✷ Remark 4.1 Theorem 4.1 shows that one can establish the convergence of {(x k , y k , z k )} generated by the IEIDP-ADMM with (C1) if P f and P g are chosen such that In fact, using the same arguments, one can get the convergence of {(x k , y k , z k )} generated by the IEIDP-ADMM with (C1) if P f and P g are chosen such that for some a ∈ [ 1 2 , 1),

Convergence of the IEIDP-ADMM with (C2)
For each k ≥ 1, we write w k := (x k e ; y k e ; z k e ; ∆y k ; ∆y k ; ∆z k ), and let H :

the block diagonal linear operator defined by
for the proximal operators P f and P g satisfying P f +Σ f 0 and P g + 3 4 Σ g 0. To establish the convergence of the IEIDP-ADMM with (C2), we need the following lemma.
Theorem 4.2 Let {(x k , y k , z k )} k≥1 be the sequence generated by the IEIDP-ADMM with the criterion (C2) and max(µ k , ν k ) ≤ min(0.1, 2−τ 4 ). Suppose that Assumption 2.1 holds and the operators P f and P g also satisfy P f + 3 8 Σ f 0 and P g + 3 8 Σ g 0. Then, for h(x k , y k ) 2 + ∆y k+1 2 Tg < ∞, the sequence {(x k , y k )} converges to an optimal solution of problem (1) and the sequence {z k } converges to an optimal solution to the dual problem of (1).
Remark 4.2 Theorem 4.2 shows that one can establish the convergence of {(x k , y k , z k )} generated by the IEIDP-ADMM with (C2) if P f and P g are chosen such that In fact, using the same arguments, one can get the convergence of {(x k , y k , z k )} generated by the IEIDP-ADMM with (C2) if P f and P g are chosen such that for some a 1 , a 2 ∈ [ 1 2 , 1),

Convergence of the IEIDP-ADMM with (C2')
Let w k for k ≥ 1 be same as the one of the last subsection. Define the block diagonal linear operator H : with the proximal operators P f and P g satisfying P f + Σ f 0 and P g + 3 4 Σ g 0. ) for some constant γ ≥ 360. Suppose that Assumption 2.1 holds and the operators P f and P g also satisfy P f + 3 8 Σ f 0 and P g + 3 8 Σ g 0. Then, when τ ∈ (0, 2), the following inequality holds for all k ≥ 1 where .

Now by the given condition
Together with the expressions of V and W k , it is not difficult to verify that Combining this relation with (41) and the condition max(µ 2 ) 1 γ , we obtain the desired result. The proof is completed. ✷ By Lemma 4.3 we can establish the following convergence result of the IEIDP-ADMM with the criterion (C2'). Since the proof is similar to that of Theorem 4.1, we omit it. Theorem 4.3 Let {(x k , y k , z k )} k≥1 be the sequence generated by the IEIDP-ADMM with the criterion (C2') and max(µ 2 ) for some constant γ ≥ 360. Suppose that Assumption 2.1 holds and P f and P g also satisfy P f + 3 8 Σ f 0 and P g + Tg < ∞, the sequence {(x k , y k )} converges to an optimal solution of (1) and the sequence {z k } converges to an optimal solution to the dual problem of (1).
To close this section, we want to point out that the convergence of the inexact positive definite proximal ADMM [24] with (C1) and a special (C2') is only established for τ = 1, while the convergence results of Theorem 4.1 and Theorem 4.3 extend it to the inexact indefinite proximal ADMM with τ ∈ (0, √ 5+1 2 ) and τ ∈ (0, 1.6], respectively.

Applications to doubly nonnegative SDPs
Let S n + be the cone of n × n positive semidefinite matrices in the vector space S n of n ×n real symmetric matrices, endowed with the Frobenius inner product ·, · and its induced norm · . The doubly nonnegative SDP problem is described as follows: where A E : S n → R m E and A I : S n → R m I are the linear operators, b E ∈ R m E and b I ∈ R m I are the given vectors, and X − M ∈ K means that every entry of X − M is nonnegative. We always assume that A E is surjective. The dual of (44) has the form where K * is the positive dual cone of K. For the four-block separable convex minimization problem (45), one may use the multi-block ADMM with Gaussian back substitution [16,17] or the proximal ADMM [31] to solve. In this section, we apply the IEIDP-ADMMs for (45) by viewing (y I , Z) as a block and (y E , S) as a block (respectively, viewing (Z, y E ) as a block and S as a block when m I = 0). Notice that, by introducing a slack variable, problem (44) can be equivalently written as and an elementary calculation yields the dual problem of (46) as follows Problem (47) is still a four-block separable convex minimization since (z, Z) can be solved simultaneously. Hence, in this section we also apply the IEIDP-ADMMs for solving (47) by viewing (y I , z, Z) as a block and (y E , S) as a block.
Throughout this section, instead of using the constraint qualification (CQ) in Assumption 2.1, we use the following more familiar Slater's CQ for problem (45): Assumption 5.1 (a) For problem (44), there exists a point X ∈ S n such that . By [2,Corollary 5.3.6], under Assumption 5.1, the strong duality for (44) and (45) holds, and the following Karush-Kuhn-Tucker (KKT) condition has nonempty solutions: and propose the following partial IEIDP-ADMM for problem (45) with three blocks, where for a given σ > 0, the augmented Lagrangian function of (45) is defined as (S.3) Update the Lagrange multiplier X k+1 via the following formula (S.4) Let k ← k + 1, and go to Step (S.1).
For the approximate optimal solution (Z k+1 , y k+1 E ) of subproblem (50), one may get it by solving the problem min Z,y E φ k (Z, y E ) in an alternating way.
From the expression of the function φ k (·, ·), it is immediate to obtain that

Comparing this system with the optimality condition of min
. This means that (Z k j , y k j E ) satisfies the criterion (C1) with ν k ≡ 0 when ξ k j ≤ µ k+1 and ∞ k=0 µ k+1 < ∞. In addition, let By using equation (49) and [20,Theorem 7.7.6], it is not difficult to verify that This means that (Z k j , y k j E ) satisfies the criterion (C2) with F = δ −1 I and ν k ≡ 0 once since the right hand side of the first inequality is less than In the sequel, we call Algorithm 5.1 with the subproblems in (50) solved alternately by the criteria (C1) and (C2) IEIDP-ADMM1 and IEIDP-ADMM2, respectively.
We apply IEIDP-ADMM1 and IEIDP-ADMM2 for the doubly nonnegative SDPs without inequality constraint A I X ≥ b I , and compare their performance with that of the 3-block ADMM of step-size τ = 1.618 (for short, ADMM3d). Among others, the doubly nonnegative SDP test examples can be found in [31]. We have implemented IEIDP-ADMM1, IEIDP-ADMM2 and ADMM3d in MATLAB, where ε = 10 −5 and µ k = min(0.1, 1 k 1.001 ) for k ≥ 1 are used for IEIDP-ADMM1 and IEIDP-ADMM2. Notice that when (Z k j −Z k ; y k j E −y k E ) T f < 1, the criterion (C2) is more restrictive than (C1). Moreover, the criterion (C2) will require much more inner iterations as the primal and dual infeasibility becomes smaller since (Z k j −Z k ; y So, in the implementation of IEIDP-ADMM2, we modify the criterion (C2) into where η P and η D are defined below. In addition, the implementation of ADMM3d here is different from that of [33] since the former uses the solution order Z → y E → S, while the latter uses the order y E → Z → S. The computational results for all DNNSDPs are obtained on a Windows system with Intel(R) Core(TM) i3-2120 CPU@3.30GHz.
In the implementation of the three solvers, the penalty parameter σ is dynamically adjusted according to the progress of the algorithms, and the idea to adjust σ is to balance the progress of primal feasibilities (η P , η S , η K ) and dual feasibilities (η D , η S * , η K * ). The exact details on the adjustment strategies are not given here. In addition, all the solvers also adopt some kind of restart strategies to ameliorate slow convergence. During the testing, we use the same adjustment strategy of σ and restart strategy for all the solvers. Figure 1 shows the performance profiles of IEIDP-ADMM1, IEIDP-ADMM2 and ADMM3d in terms of number of iterations and computing time, respectively, for the total 605 (including BIQ(165), RCP(120), θ + (113), FAP (13) and QAP(95)) tested problems. We recall that a point (x, y) is in the performance profiles curve of a method if and only if it can solve (100y)% of all tested problems no slower than x times of any other methods. We see that IEIDP-ADMM1 and ADMM3d need the comparable iterations and computing time. Among others, IEIDP-ADMM2 requires the least number of iterations for 60% test problems, but it needs the most computing time which is about 1.5 times that of IEIDP-ADMM1 and ADMM3d for about 80% test problems.
The iterates (y k j E , S k j ) for j ≥ 1 yielded by minimizing ψ k (y E , S) alternately satisfy . Comparing the last system with the optimality condition of problem min y E ,S ψ k (y E , S), we have (η k j , 0) ∈ ∂ψ k (y k j E , S k j ). This means that (y In addition, let By using equation (56) and [20,Theorem 7.7.6], it is not difficult to verify that This means that (y We call Algorithm 5.2 with the two subproblems in (S.1) solved alternately by the criteria (C1) and (C2) IEIDP-ADMM1 and IEIDP-ADMM2, respectively.
We apply the IEIDP-ADMM1 and IEIDP-ADMM2 for solving the extended BIQ problems described in Section 4.2 of [31], and compare its performance with the fourblock proximal ADMM of step-size τ = 1.618 (although without convergent guarantee) by adding a proximal term σ 2 y I − y k I 2 T for the y I part, where T = A I A * I I −A I A * I . We call this method PADMM4d. The computational results for all the extended BIQ problems are obtained on the same desktop computer as before.
The three solvers IEIDP-ADMM1 and IEIDP-ADMM2 and PADMM4d were stopped whenever η < 10 −6 or the number of iteration is over k max = 40000. Figure 2 plots the performance profiles of IEIDP-ADMM1, IEIDP-ADMM2 and PADMM4d in terms of the number of iterations and computing time, respectively, for the total 165 extended BIQ tested problems. It can be seen from this figure that IEIDP-ADMM1, IEIDP-ADMM2 and PADMM4d are comparable in terms of the iterations and computing time, IEIDP-ADMM1 and IEIDP-ADMM2 need the least number of iterations for at least 80% tested problems, which is about 90% that of PADMM4d, and PADMM4d requires the least computing time for about 70% tested problems, which is about 90% that of IEIDP-ADMM2. We apply the IEIDP-ADMM1 for solving the extended BIQ problems described in Section 4.2 of [31], and compare its performance with the previous PADMM4d and the four-block ADMM of step-size τ = 1.618 (although without convergent guarantee). We call the latter ADMM4d. The computational results for all the extended BIQ problems are obtained on the same desktop computer as before. We measure the accuracy of an approximate optimal solution (X, y I , z, Z, y E , S) for (44) and (47) by the relative residual η = max η P , η D , η S , η K , η S * , η K * , η C 1 , η C 2 , η I , η I * , where η P , η S , η K , η S * , η K * , η C 1 , η C 2 are defined as before. The solvers IEIDP-ADMM1 and PADMM4d and ADMM4d were terminated whenever η < 10 −6 or the number of iteration is over k max = 40000.   Figure 3 plots the performance profiles of IEIDP-ADMM1, PADMM4d and ADMM4d in terms of the number of iterations and computing time, respectively, for the total 165 extended BIQ tested problems. We see that, when applying the IEIDP-ADMM for solving the dual problem (47), the number of iterations and the computing time of the IEIDP-ADMM1 are still comparable with those of ADMM4d, but PADMM4d requires more 4 times iterations than IEIDP-ADMM1 and ADMM4d as do for at least 80% test problems. This means that a small proximal term as possible is the key to the performance of proximal-type ADMMs. The computing time of PADMM4d is a little less than that of IEIDP-ADMM1 and ADMM4d since the latter solves an m I × m I linear system with the conjugate gradient method, where m I may attain 374250.

Conclusion
We developed an inexact indefinite proximal ADMM of step-size τ ∈ (0, √ 5+1 2 ) with two easily implementable inexactness criteria for the two-block separable convex minimization problems with linear constraints, for which it is either impossible or too expensive to obtain the exact solutions of the subproblems involved in the proximal ADMM. Numerical results for the DNNSDPs with many linear equality and/or inequality constraints show that the inexact indefinite proximal ADMMs are effective for this class of difficult three or four block separable separable convex optimization problems with linear constraints. Among others, the inexact indefinite proximal ADMM with the absolute error criterion (C1) is comparable with the directly extended ADMM of step-size τ = 1.618, whether in terms of the number of iterations or computing time, and is superior to the one with the relative error criterion (C2) by weighing the number of iterations and the computing time since the latter is very restrictive and requires too many iterations for the solution of subproblems. In our future research work, we will explore other easily implementable inexact criteria like relaxing µ k+1 and ν k+1 in (C2) to be a constant, and study the nonergodic convergence [6,7] for the inexact indefinite proximal ADMMs.