A NEW CONVERGENCE PROOF OF AUGMENTED LAGRANGIAN-BASED METHOD WITH FULL JACOBIAN DECOMPOSITION FOR STRUCTURED VARIATIONAL INEQUALITIES

. In the work, we present a new proof for global convergence of a classical method, augmented Lagrangian-based method with full Jacobian decomposition, for a special class of variational inequality problems with a separable structure. This work can be regarded as an improvement to work [14]. The convergence result of the work is established under more general conditions and proven in a new way.

Clearly, (1)-(2) is called as a specially structured variational inequality problem in that each g i is a function of only variable y i . The separable variational inequality problem arises naturally from many applications such as economics, transportation, 46 XI-HONG YAN and engineering [1,8,13]. Moreover, the problem plays an important role in optimization, which has been demonstrated in the literature where many optimization problems can be reformulated as separable variational inequality problems. For example, a separable convex minimization problem with linear constraints defined as min y1,y2,...,ym can be reformulated as the variational inequality (1)-(2), where g i is corresponding to a subgradient of φ i and λ is a Lagrangian multiplier to the linear constraints Due to wide applications, a significant portion of optimization research was dedicated towards designing algorithms for solving the problem (1)-(2), see [2,3,5,11,10,9,12,7]. While many strategies exist for such problem, the alternating direction method of multipliers has been shown to subsume and dominate them since it was developed by Gabay and Mercier [4]. The process of the alternating direction algorithm is as follows: For a given λ k , sequentially solve the following subproblems (3) in the order of i = 1, 2, · · · , m to obtain y k+1 i , respectively, Then, improve λ by formula (4), The scheme (3)-(4) can be comprehended as an augmented Lagrangian-based method with full Gauss-Seidel decomposition and its global convergence has not been proven until the most recent work by Han and Yuan [6] which shows that the scheme (3)-(4) with involved strongly monotone functions g i , (i = 1, 2, · · · , m) is globally convergent. Another famous approach is the augmented Lagrangian-based method with full Jacobian decomposition which was exploited by He et al. [9]. The approach has computational attractiveness due to its iterative scheme described as follows: For a given λ k , simultaneously obtain y k+1 i such that Then update λ through (4). The global convergence of the augmented Lagrangianbased method with full Jacobian decomposition is stated by Wang et al. [14] where it is required that each involved function g i is strongly monotone with modulus µ gi and each µ gi is larger than a constant. Actually, the requirements on modulus µ gi are strict since the coefficient µ gi is always small for a general strongly monotone operator and hard to be estimated in practice. To avoid this weakness, we provide a new proof of the global convergence of the scheme (5)-(4) for the separable variational inequality problem (1)- (2). In brief, the main contribution of the paper is that I provide a new proof of the global convergence of the scheme (5)-(4) for the separable variational inequality problem (1)-(2) by reducing the requirement of each modulus µ gi . The rest of the paper is broken down as follows. Next section includes some notations and fundamental concepts concerned in our study. Section 3 describes the process of the augmented Lagrangian-based method with full Jacobian decomposition and provides the convergence result of the concerned algorithm. Finally, Section 4 gives the conclusions of this work.

Preliminaries.
To present a remarkably reader-friendly convergence analysis, we first explain some basic definitions and notations. In the sequent sections, R n means an n-dimensional Euclidean space, T means the transpose, · means standard Euclidean norm, and δ max (B) means the maximum eigenvalue of square matrix B. All vectors in the following analysis are column vectors. Hence, let x T y be the standard inner product for any two vectors x, y ∈ R n . We are interested in analyzing the global convergence of the method in the sense of the matrix norm so we define M -norm as y M := y T M y, where M is a symmetric and positive definite matrix. In the present work, we define where I is the identity matrix in R l×l . Then, it gives access to an important result In the present work, it is assumed that each g i is strongly monotone with modulus µ gi (i = 1, 2, · · · , m).
3. Convergence analysis. In the section, for the completeness of the paper, firstly, the process of the augmented Lagrangian-based method with Jacobian decomposition for the separable variational inequality problem (1)-(2) is stated. Then, we conduct an analysis on the convergence of the method in a new manner. Augmented Lagrangian-based method with full Jacobian decomposition for (1)- (2) S0. Choose a starting point v 0 = (y 0 1 , ..., y 0 m , λ 0 ) ∈ V and β. Set k = 0.

Remark 1.
In the paper, an easily implementable stopping criterion is given by The stopping criterion (9) was justified by Wang et al. [14]. Now, we concentrate on the convergence analysis. In order to get the convergence result, we first present some useful lemmas.
Since v * = (y * 1 , ..., y * m , λ * ) ∈ V is a solution of the problem, we have Moreover, set y i = y * i (i = 1, 2, ..., m) in each inequality of (7) such that (12) Using the sum taken over all the inequalities included in (11) and (12) and the Noticing the strong monotonicity of all the functions g i (i = 1, 2, ..., m) and the fact m i=1 B i y * i = b, we rewrite the above inequality as Note that b can be replaced by where the last equation of above formula is obtained by the formula for square of sum of m numbers. Hence proved.
Proof. Using the result of Lemma 3.1 and the equation (8), we get Now, we investigate the terms β||B i y k+1 A NEW CONVERGENCE PROOF

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The sum of all the inequalities and equations in (13) and (14) gives that where the last equation of (15) is deduced based on the formula for square of sum of two numbers. Since from (15), it follows that The proof is completed.
where the matrix M is given by (6).
With the above analysis, the convergence theorem of the method is ready to be presented. Theorem 3.3. With the following hypotheses: For each i ∈ {1, 2, ..., m}, g i (y i ) is continuous and µ gi -strongly monotone, for any , the sequence {v k } generated by the augmented Lagrangian-based method with full Jacobian decomposition converges to a global optimal point of the problem (1)-(2).
Along with (22) and (20), we can conclude that the sequence generated is globally convergent. This completes the proof.

4.
Conclusion. The augmented Lagrangian-based method with full Jacobian decomposition developed by He et al. [9] is of great interest for solving the separable variational inequality problem (1)- (2). It allows us to get a solution in a parallel perspective. The convergence of the method is ensured with the hypotheses on the strongly monotone modulus u gi . In fact, it is practically difficult to satisfy and verify the hypotheses. In this work, a new convergence result is shown by the relaxation of the hypotheses. This result makes a contribution to convergence of the the augmented Lagrangian-based method with full Jacobian decomposition under only strongly monotone assumptions on the involved functions.