## 1 Introduction  
In this work, we focus on developing an iterative method for solving a finite-sum nonsmooth convex-concave minimax optimization problem in a decentralized setting:  
\[\min_{\mathbf{u}\in\mathcal{X}}\max_{\mathbf{v}\in\mathcal{Y}}\sum_{i=1}^{m}F_{ i}(\mathbf{u},\mathbf{v}), \tag{1}\]  
where the component coupling function \(F_{i}:\mathbb{R}^{p}\times\mathbb{R}^{r}\rightarrow\mathbb{R}\) for each \(i=1,\ldots,m\) is a convex-concave function and the constrained sets \(\mathcal{X}\subset\mathbb{R}^{p}\) and \(\mathcal{Y}\subset\mathbb{R}^{r}\) are compact convex sets. The convex-concave optimization problem (1) has been extensively studied, with numerous methods proposed in the literature. It arises in various areas of convex optimization. In addition, it has been widely applied in multiple fields, namely, generative adversarial networks (GANs) [1, 2], adversarial robust learning [3], classification [4, 5, 6], and image processing [7, 8]. When \(m=1\), the gradient descent-ascent method, a well-known approach introduced by [PERSON], [PERSON], and [PERSON] [9], is commonly used to solve the problem in the smooth case. Furthermore, for cases where the coupling function is not necessarily smooth, [PERSON] and [PERSON] [10] proposed the subgradient method, which uses subgradient computation in the place of gradient computation. Since the coupling objective function is a convex-concave real-valued function, the existence of subgradients for both variables is guaranteed. A common difficulty with gradient or subgradient methods for convex optimization is that while subgradients exist, they are often hard to compute. To address this issue, the concept of delay techniques for calculating subgradients was introduced [11, 12, 13, 14]. Specifically, the next iteration can be updated using information from stale iterates rather than computing the subgradient in the current iterate, which saves computational runtime. Recently,Arunrat and [PERSON] [15] proposed a method based on the subgradient method, together with the delayed technique to give more strategies in order to calculate the subgradient for solving problem (1) when \(m=1\). In this approach, the iterates are updated simultaneously using delayed subgradients at stale iterates of their respective variables, combined with the current iterates of the other variable. The method is referred to as the delayed subgradient method and can be explicitly formulated as follows: For a current iterate \((\mathbf{u}[k],\mathbf{v}[k])\in\mathcal{X}\times\mathcal{Y}(k\geq 0)\), compute  
\[\mathbf{u}[k+1]:=P_{\mathcal{X}}\Big{[}\mathbf{u}[k]-\gamma\widehat{\
abla}_{ \mathbf{u}}F(\mathbf{u}[k-\tau_{k}],\mathbf{v}[k])\Big{]},\]  
and  
\[\mathbf{v}[k+1]:=P_{\mathcal{Y}}\Big{[}\mathbf{v}[k]+\gamma\widehat{\
abla}_{ \mathbf{v}}F(\mathbf{u}[k],\mathbf{v}[k-\mu_{k}])\Big{]},\]  
where \(\widetilde{\
abla}_{\mathbf{u}}F(\mathbf{u}[k-\tau_{k}],\mathbf{v}[k])\) is a subgradient of \(F\) with respect to \(\mathbf{u}\) at a stale iterate \(\mathbf{u}[k-\tau_{k}]\); \(\widetilde{\
abla}_{\mathbf{v}}F(\mathbf{u}[k],\mathbf{v}[k-\mu_{k}])\) is a subgradient of \(F\) with respect to \(\mathbf{v}\) at a stale iterate \(\mathbf{v}[k-\mu_{k}]\); and \(P_{\mathcal{X}}\) and \(P_{\mathcal{Y}}\) are metric projections on \(\mathcal{X}\) and \(\mathcal{Y}\), respectively. In some situations, the objective functions can be expressed as a finite sum or within a stochastic setting. Focusing on each coordinate individually rather than on the entire function can often yield better results. Several methods were also developed to address this type of convex-concave optimization problem [16; 17; 18; 19; 20]. On the other hand, many authors are interested in studying convex minimization problems where the objective function is expressed as a finite sum. This problem is known as the additive convex optimization problem.