Inexact indefinite proximal ADMMs for 2block separable convex programs and applications to 4block DNNSDPs
Abstract
This paper is concerned with twoblock 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 twoblock regroup of three or fourblock separable convex optimization problems with linear constraints, or from the constrained totalvariation superresolution image reconstruction problems in image processing. For them, we propose an inexact indefinite proximal ADMM of stepsize 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 fourblock separable convex minimization problems with linear constraints, which come from 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 multiblock ADMM of stepsize without convergence guarantee, whether in terms of the number of iterations or the computation time.
Keywords: Separable convex optimization, inexact proximal ADMM, DNNSDPs
1 Introduction
Let and be the finite dimensional vector spaces endowed with the inner product and its induced norm . Given closed proper convex functions and , we are concerned with the separable convex optimization problem
(1) 
where and are the given linear operators, and denote the adjoint operators of and , respectively, and 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 lowrank recovery problem arising from the socalled robust PCA (principle component analysis) with noisy and incomplete data [34, 32], the constrained totalvariation image restoration and reconstruction problems [22, 29], the simultaneous minimization of the nuclear norm and norm of a matrix arising from the lowrank 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 , let denote the augmented Lagrangian function of problem (1)
The ADMM, from an initial point , consists of the steps
(2a)  
(2b)  
(2c) 
where is a constant to control the stepsize in (2c). The iterative scheme of ADMM actually embeds a GaussianSeidel decomposition into each iteration of the classical augmented Lagraigan method of HestenesPowellRockafellar [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 closedform solutions or even be difficult to solve. When the functions and enjoy a closedform Moreau envelope, one usually introduces the proximal terms and respectively into the subproblems (2a) and (2b) to cancel the operators and so as to get the exact solutions of proximal subproblems. This is the socalled proximalADMM which, for a chosen initial point , consists of
(3a)  
(3b)  
(3c) 
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 illposed, the proximal parameters should be small.” In view of this, some researchers recently develop the semiproximal 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 semiproximal (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 and/or may not have a closedform Moreau envelope or the linear operators and/or 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 twoblock regroup of three or fourblock separable convex minimization problems. Indeed, for the following fourblock separable convex minimization problem
(4) 
where for are closed proper convex functions, and for are linear operators, since the directly extended multiblock 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 twoblock regrouped problem. Clearly, the exact solution of each subproblem of ADMM for the twoblock regrouped problem is difficult to obtain due to the cross of two classes of variables. In particular, the twoblock regroup resolving of multiblock separable convex optimization also has a separate study value.
To resolve this class of difficult twoblock separable convex minimization problems, we propose an inexact indefinite proximal ADMM with a stepsize , 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 stepsize may take the value in the interval . We notice that a few existing research papers on inexact versions of the ADMM all focus on the unit stepsize; 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 requires less to iterations than the one with , especially for those difficult SDP problems [33]. Thus, the proposed inexact indefinite proximal ADMMs with a large stepsize 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 twoblock 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 multiblock ADMM with stepsize whether in terms of the number of iterations or the computation time, while the one with the relative error criterion requires less outeriterations but more computation time since the error criterion is more restrictive and requires more inneriterations. Thus, the inexact indefinite proximal ADMM with the absolute error criterion provides an efficient tool for handling the three and fourblock 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 twoblock 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.
2 Notations and assumption
Notice that the functions and are closed proper convex, and the subdifferential mappings of closed proper convex functions are maximal monotone [26, Theorem 12.17]. Hence, there exist selfadjoint operators and such that for all and ,
(5) 
and for all and ,
(6) 
For a selfadjoint linear operator , the notation (respectively, ) means that is positive semidefinite (respectively, positive definite), that is, for all (respectively, for all ). Given a selfadjoint positive semidefinite linear operator , we denote by the norm induced by , i.e.,
Given a selfadjoint positive definite linear operator, we denote by and the largest eigenvalue and the smallest eigenvalue of , respectively, and by the distance induced by from to a closed set , that is, When is the identity operator, we suppress the notation in and write simply . Clearly, for any positive definite linear operator and ,
(7) 
In addition, for any and any selfadjoint linear operator , the following two identities will be frequently used in the subsequent analysis:
(8) 
Throughout this paper, we make the following assumption for problem (1):
Assumption 2.1
Problem (1) has an optimal solution, to say , and there exists a point such that .
Under Assumption 2.1, from [27, Corollary 28.2.2 & 28.3.1] and [27, Theorem 6.5 & 23.8], it follows that there exists a Lagrange multiplier such that
(9) 
where and are the subdifferential mappings of and , respectively. Moreover, any satisfying (9) is an optimal solution to the dual problem of (1). In the sequel, we call a primaldual solution pair of problem (1).
3 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.
IEIDPADMM (Inexact indefinite proximal ADMM for (1)) (S.0) Let be given. Choose selfadjoint linear operators and such that and . Choose an initial point . Set . (S.1) Find . (S.2) Find . (S.3) Update the Lagrange multiplier via the following formula (S.4) Let , and go to Step (S.1).
The approximate optimality in (S.1) and (S.2) is measured by the following criteria:

and ;

, and , where and are selfadjoint positive definite linear operators with and ;

, and , where and are same as the one in (C2).
Notice that (C1) is an absolute error criterion, while (C2) and (C2’) are a relative error criterion. Clearly, when the approximate optimality of and is measured by (C1), (S.1) and (S.2) are equivalent to finding and such that
(10) 
If the approximate optimality of and is measured by (C2) or (C2’), (S.1) and (S.2) are equivalent to finding and such that with or ,
(11) 
Remark 3.1
(a) When the proximal operators and are chosen as for a constant and the stepsize is set to be , the IEIDPADMM with (C1) reduces to the IADM1 in [24]. If, in addition, taking , the IEIDPADMM with (C2’) requires
whereas the LADM2 in [24] is actually requiring that and satisfy
Since and , the above inexact criterion (C2’) is looser than Criterion 2 used in [24].
(b) When and are chosen to be selfadjoint positive semidefinite operators, the IEIDPADMMs with reduce to the semiproximal ADMM in [38, 9].
(c) For the selfadjoint positive definite linear operators and in (C2) and (C2’), an immediate choice is and . Since and are easy to estimate, such a choice is convenient for the numerical implementation.
Next we study the properties of the sequence generated by the IEIDPADMMs. For convenience, we let for , and for each write
Using these notations and noting that , we can rewrite Step (S.3) as
(12) 
Lemma 3.1
Proof: From the expressions of and and equations (10) and (11), it follows that
(13)  
(14) 
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
(15)  
Next we deal with the term in inequality (15). Notice that
(16)  
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
(17) 
Using equation (2) and the given assumption , we have that
(18) 
where the last inequality is using Combining inequalities (3) and (3) with equation (16), we immediately obtain
(19) 
Now substituting inequality (3) into equation (15), we immediately obtain that
(20) 
By the first equality of (2) and equation (12), the term can be written as
Applying equation (2) to and yields
Substituting the last three equalities into inequality (3), we have that