In many practical problems such as image processing, machine learning, and statistical modeling [1]-[3], the objective function often contains a coupling term H ( x , y ) . That is, for the nonconvex and nonseparable problem considered in this paper, the iterative scheme is as follows:

where f : ℝ n → ℝ ∪ { + ∞ } is a proper lower semicontinuous function, possibly nonsmooth and nonconvex, both g : ℝ m → ℝ and H : ℝ n × ℝ m → ℝ are continuously differentiable and possibly nonconvex functions, the gradient ∇ g is L g -Lipschitz continuous and the gradient ∇ H is L h -Lipschitz continuous. A ∈ ℝ m × n is a matrix and b ∈ ℝ m is a vector. In particular, when H ( x , y ) = 0 , problem (1) reduces to a separable optimization problem of the following form:

The alternating direction method of multipliers (ADMM) is one of the most effective approaches for solving problem (2). The convergence of ADMM has been extensively studied [4]-[9], and the convergence rate analysis is relatively well-established for problem (2). However, when H ( x , y ) ≠ 0 , for problem (1), the convergence results for ADMM are relatively limited. Gao et al. [10] proved the convergence of the proximal ADMM under the setting where H is smooth, f and g are convex, combined with the assumption that ∇ H is Lipschitz continuous. Chen et al. [11] proved the convergence of the extended ADMM when the coupling term H is a quadratic function. In recent years, researchers have begun to focus on solving (1) by using ADMM in non-convex settings, and have employed tools such as the Kurdyka-Łojasiewicz (KL) inequality to prove the convergence of the algorithm. Guo etal. [12][13] proved the convergence of the classic ADMM and extended it to the generalized ADMM (GADMM). Liu et al. [14] proposed a linearized ADMM and proved the convergence of the algorithm.

Regarding problem (2), it has been established in the literature that under convergence occurs, the symmetric ADMM often converges faster than the classical ADMM. Wu et al. [15] proposed a symmetric ADMM with one relaxation factor and verified its convergence. On this basis, some scholars have studied the symmetric ADMM with a relaxation factor and its variants for solving the nonseparable problem (1). For example, Dang et al. [16] proposed a linear proximal symmetric ADMM and verified that the sequence generated by the algorithm converges to a stationary point of the problem. The iterative scheme is as follows:

The parameters are selected as follows:

Dang et al. [17] proposed an inertial Bregman symmetric ADMM algorithm and established its global convergence. The iterative scheme is as follows:

where Δ ϕ i ( s , t ) , i = 1 , 2 represents the Bregman distances, and r ∈ ( − 1 , 1 ) is a relaxation factor. However, the convergence of the sequence generated by the algorithm depends on the strong convexity of the kernel function associated with the Bregman distance.

In recent studies on solving problem (2), we observe that Lu et al. [18] proposed a symmetric ADMM with two different relaxation factors, without introducing the Bregman distance, they proved its convergence with a broader range of parameters, numerical experiments verified that their algorithm converges faster than the algorithm proposed by [15]. Its iterative scheme is as follows:

Among them, ℒ β s is the augmented Lagrangian function associated with problem (2), defined as follows:

where β > 0 is the penalty parameter, α and s are two different relaxation factors.

The above research on symmetric ADMM algorithms for solving problem (1) is limited to those containing only one relaxation factor. However, inspired by the algorithm proposed by the work of Lu et al. [18], we consider introducing two relaxation factors to achieve wider applicability, faster convergence, and a more concise and unified algorithmic framework, this paper proposes using a symmetric ADMM with two different relaxation factors to solve the non-separable (1), the iterative scheme is as follows:

Compared with the algorithm proposed by Dang et al. [16], our algorithm has a wider range of parameter values, allowing it to be applied to more practical scenarios through appropriate parameter tuning. Moreover, unlike the algorithm proposed by Dang et al. [17], our method does not require the introduction of the Bregman distance, thereby providing a unified iterative framework for solving non-separable problems via ADMM. The optimality conditions are as follows:

In the case where H = 0 , when a = 0 and s = 1 , the proposed algorithm reduces to the classical ADMM. When a ≠ 0 and s = 1 , the algorithm reduces to the symmetric ADMM with a scaling factor. This demonstrates that our method provides an improved basic iterative framework for symmetric ADMM with relaxation factors.

In this section, we provide the necessary definitions and properties required for the following study.

Notations:

ℝ , ℝ n , ℝ m × n : real numbers, n -dimensional real vectors, m × n -real matrices. 〈 ⋅ , ⋅ 〉 and ‖   ⋅   ‖ : inner product and associated norm.

Set-valued mapping F : ℝ n ⇉ ℝ m :

Domain: dom   F : = { x ∈ ℝ n | F ( x ) ≠ ∅ } . Graph: Gra   F : = { ( x , y ) ∈ ℝ n × ℝ m : y ∈ F ( x ) } .

Distance: For S ⊆ ℝ n and x ∈ ℝ n , d ( x , S ) : = inf { ‖ y − x ‖ | y ∈ S } , with d ( x , S ) : = + ∞ .

Definition 1. [19] The domain of function f : ℝ n → ℝ ∪ { + ∞ } is denoted by

then f ( x ) is called a proper function.

Definition 2. [19] A function f : ℝ n → ℝ ∪ { + ∞ } is said to be lower semicontinuous at x ∈ ℝ n if it satisfies

If this holds for every point in dom   f , then f is said to be a lower semicontinuous function.

Definition 3. [20] Let f : ℝ n → ℝ ∪ { + ∞ } be a proper lower semicontinuous function.

(i) The Fréchet subdifferential of f at x ∈ dom   f , written ∂ ^ f ( x ) , is the set of vectors x * ∈ ℝ n that satisfy

When x ∉ dom   f , we set ∂ ^ f ( x ) = ∅ .

(ii) The limiting-subdifferential of f at x ∈ dom   f , written ∂ f ( x ) , is defined as follows

Lemma 1. [21] Let g : R m → R be a continuously differentiable function and suppose that ∇ g is L -Lipschitz continuous. Then

Definition 4. ([22], Kurdyka-Lojasiewicz inequality) Let f : ℝ n → ℝ ∪ { + ∞ } be a proper lower semicontinuous function. For − ∞ < η 1 < η 1 ≤ + ∞ , set

We say that function f has the KL property at x * ∈ dom ( ∂ f ) if there exist η ∈ ( 0 , + ∞ ] , a neighbourhood U of x * , and a continuous concave function φ : [ 0 , η ) → ℝ + , such that

(i) φ ( 0 ) = 0 ;

(ii) φ is C 1 on ( 0 , η ) and continuous at 0;

(iii) φ ′ ( x ) > 0 ,   ∀ x ∈ ( 0 , η ) ;

(iv) for all x in U ∩ [ f ( x * ) < f < f ( x * ) + η ] , the Kurdyka-Lojasiewicz inequality holds

where d ( x , ∂ f ( x ) ) = inf y ∈ ∂ f ( x ) ‖ y − x ‖ is the distance from x to ∂ f ( x ) .

Lemma 2. ([23], Uniformized KL property) Let Ω be a compact set and f : ℝ n → ℝ ∪ { + ∞ } be a proper and lower semicontinuous function. Assume that f is constant on Ω and satisfies the KL property at each point of Ω . Then, there exist ϵ > 0 , η > 0 , and φ ∈ Φ η such that for all x ¯ ∈ Ω and for all x in the following intersection:

one has

Lemma 3. [24] Suppose that F ( x , y ) = f ( x ) + g ( x ) , where f : ℝ n → ℝ ∪ { + ∞ } and g : ℝ m → ℝ ∪ { + ∞ } are proper lower semicontinuous functions. Then for all ( x , y ) ∈ dom   F = dom   f × dom   g , we have

Definition 5. ([24], Kurdyka-Lojasiewicz function) If f satisfies the KL property at each point of dom ( ∂ f ) , then f is called a KL function.

Definition 6. ( x * , y * , λ * ) is a stationary point of the augmented Lagrangian function ℒ β s ( ⋅ ) for problem (1), if and only if

Lemma 4. Let the iterative sequence generated by the algorithm (5) be denoted as { w k : ( x k , y k , λ k ) } . Then, the following holds

Proof. Combining the second and fourth equations in (6) yields

subtracting the above two equations, we obtain

and thus

According to the optimality condition (6) of the x -subproblem, we have

putting (9) into (11), we get

According to the third and fourth equations in (6), we obtain

This completes the proof. □

Assumption A. let f : ℝ n → ℝ ∪ { + ∞ } be a proper lower semicontinuous function, and let g : ℝ m → ℝ be a continuously differentiable function with an L g -Lipschitz continuous gradient ∇ g , and H : ℝ m × ℝ n → ℝ be a continuously differentiable function with an L h -Lipschitz continuous gradient ∇ H . Also assume that

{ ( α , s ) ∈ ℝ 2 | α < s < 0 } , { β ∈ ℝ | β > − 1 s ( L g + L h ) }

Lemma 5. Let the iterative sequence generated by the algorithm (5) be denoted

as { w k : ( x k , y k , λ k ) } . Suppose that this sequence is bounded and that Assumption A holds. Then the following conclusion holds

where η > 0 (see Remark (0.1)).

Proof. The polarization identity implies

Due to ∇ g is L g -Lipschitz continuous and ∇ H is L h -Lipschitz continuous, we combining the lemma 1, then we have

From the fourth equation in the optimality condition (6) of the problem, we obtain

We obtain λ k + 1 = ∇ g ( y k + 1 ) + ∇ y H ( x k + 1 , y k + 1 ) in (13) and combining the Lipschitz continuities of ∇ g and ∇ H , we have

Recall (9) and (10), we get

From the definition of the augmented Lagrangian function ℒ β s ( ⋅ ) in (4), and in combination with (15), we have

Then, combining it with (16) and (17) yields

Substituting (13) into the above expression and simplifying, we obtain

Note that, by using (4), (6), (20) and (19), we have

Summing the inequalities (23), (24) and the first equation in Section (5) we obtain

Thus

there η : = min { s β − L g − L h 2 − 2 ( L g 2 + L h 2 ) + α s β 2 ( α + s ) β , − 2 L h 2 ( α + s ) β } . This completes the proof. □

Remark3.1. Thus, as long as η > 0 , the sequence ℒ β s ( w k + 1 ) has sufficient descent properties, which means that ℒ β s ( w k + 1 ) is monotonically nonincreasing. Therefore, we can achieve η > 0 by appropriately choosing the parameters ( α , s , β ) . The constraints on these parameters are as follows:

{ ( α , s ) ∈ ℝ 2 | α < s < 0 } , { β ∈ ℝ | β > − 1 s ( L g + L h ) }

Lemma 6. Let the iterative sequence generated by the algorithm (5) be denoted as { w k : ( x k , y k , λ k ) } . Suppose that this sequence is bounded, that Assumption A holds and η > 0 . Then we have

Proof. The boundedness of { w k } implies the existence of a subsequence { w k j } converging to w * . Moreover, the lower semicontinuity of f ( x ) together with the continuity of g ( y ) ensures that ℒ β s ( ⋅ ) is lower semicontinuous. Hence,

Hence, { ℒ β s ( w k j ) } is bounded below. The fact that it is also nonincreasing implies its convergence. Since { ℒ β s ( w k ) } is monotonic and contains a convergent subsequence, it follows that { ℒ β s ( w k ) } itself converges, satisfying ℒ β s ( w k ) ≥ ℒ β s ( w * ) . Finally, invoking Equation (14) yields

By summing over k = 0 , ⋯ , n , and observing that ℒ β s ( w 0 ) < + ∞ , we arrive at

The condition η > 0 yields ∑ k = 0 ∞ ‖ y k + 1 − y k ‖ 2 < + ∞ and ∑ k = 0 ∞ ‖ x k + 1 − x k ‖ 2 < + ∞ . Using (19), we further obtain ∑ k = 0 ∞ ‖ λ k + 1 − λ k ‖ 2 < + ∞ . Hence, ∑ k = 0 + ∞ ‖ w k + 1 − w k ‖ 2 < + ∞ . This completes the proof. □

Lemma 7. Let the iterative sequence generated by the algorithm (5) be denoted as { w k : ( x k , y k , λ k ) } . Suppose that this sequence is bounded and that Assumption A holds. We define

Hence, it holds that ε k + 1 : = ( ε 1 k + 1 , ε 2 k + 1 , ε 3 k + 1 ) ∈ ∂ ℒ β s ( w k + 1 ) , and there exists a constant δ > 0 such that

Proof. From the definition of the function ℒ β s ( ⋅ ) in (4), the following system of equations holds