## 2 Preliminaries  
In this section, we recall some definitions and useful facts used in this work. We denote the symbols \(\mathbb{R}\) as the set of real numbers and \(\mathbb{N}_{0}\) as the set of nonnegative integers. Let \(\mathbf{u}=(x_{1},x_{2},\ldots,x_{p})\) and \(\mathbf{v}=(y_{1},y_{2},\ldots,y_{p})\) in \(\mathbb{R}^{p}\); the _inner products_ of \(\mathbf{u}\) and \(\mathbf{v}\), denoted by \(\langle\mathbf{u},\mathbf{v}\rangle\), is defined by \(\langle\mathbf{u},\mathbf{v}\rangle:=\sum_{i=1}^{p}x_{i}y_{i}\). Moreover, the _norm_ of \(\mathbf{u}\), denoted by \(\|\mathbf{u}\|\), is defined by \(\|\mathbf{u}\|:=\sqrt{\langle\mathbf{u},\mathbf{u}\rangle}\). If \(\mathcal{X}\) is a bounded set in \(\mathbb{R}^{p}\), then we denote the _diameter_ of \(\mathcal{X}\) by \(D_{\mathcal{X}}:=\sup\{\|\mathbf{u}-\mathbf{v}\|:\mathbf{u},\mathbf{v}\in \mathcal{X}\}\). Let \(\mathcal{X}\subset\mathbb{R}^{p}\) and \(\mathcal{Y}\subset\mathbb{R}^{r}\) be nonempty sets and \(F:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}\) be a function. We say that \(F\) is a _convex-concave_ function if the function \(F(\cdot,\mathbf{v}):\mathcal{X}\to\mathbb{R}\) is a convex function for each fixed \(\mathbf{v}\in\mathcal{Y}\), that is,  
\[F(\alpha\mathbf{x}+(1-\alpha)\mathbf{y},\mathbf{v})\leq\alpha F(\mathbf{x}, \mathbf{v})+(1-\alpha)F(\mathbf{y},\mathbf{v})\quad\text{for all }\mathbf{x},\mathbf{y}\in \mathcal{X}\text{ and }\alpha\in(0,1),\]  
and \(F(\mathbf{u},\cdot):\mathcal{Y}\to\mathbb{R}\) is a concave function for each fixed \(\mathbf{u}\in\mathcal{X}\), that is,  
\[F(\mathbf{u},\alpha\mathbf{w}+(1-\alpha)\mathbf{z})\geq\alpha F(\mathbf{u}, \mathbf{w})+(1-\alpha)F(\mathbf{u},\mathbf{z})\quad\text{for all }\mathbf{w},\mathbf{z}\in \mathcal{Y}\text{ and }\alpha\in(0,1).\]  
A vector pair \((\mathbf{u}^{*},\mathbf{v}^{*})\in\mathcal{X}\times\mathcal{Y}\) is said to be a _saddle point_ of \(F\) on \(\mathcal{X}\times\mathcal{Y}\) (with respect to minimizing in \(\mathbf{u}\) and maximizing in \(\mathbf{v}\)) if  
\[F(\mathbf{u}^{*},\mathbf{v})\leq F(\mathbf{u}^{*},\mathbf{v}^{*})\leq F( \mathbf{u},\mathbf{v}^{*}),\quad\text{for all }\mathbf{u}\in\mathcal{X},\mathbf{v}\in \mathcal{Y}.\]  
We call the value \(F^{*}:=F(\mathbf{u}^{*},\mathbf{v}^{*})\) the saddle value. Note that if \(\mathcal{X}\) and \(\mathcal{Y}\) are nonempty closed bounded convex sets and \(F:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}\) is a convex-concave function, then \(F\) has a saddle point on \(\mathcal{X}\times\mathcal{Y}\) (see [23], Corollary 37.6.2). Let \(f:\mathbb{R}^{p}\to\mathbb{R}\) be a function and let \(\mathbf{u}\in\mathbb{R}^{p}\). A vector \(\widehat{\
abla}f(\mathbf{u})\in\mathbb{R}^{p}\) is called a _subgradient_ of \(f\) at \(\mathbf{u}\) if  
\[f(\mathbf{v})\geq\langle\widehat{\
abla}f(\mathbf{u}),\mathbf{v}-\mathbf{u} \rangle+f(\mathbf{u})\quad\text{for all }\mathbf{v}\in\mathbb{R}^{p}.\]We denote the set of all subgradients of \(f\) at \(\mathbf{u}\) by \(\partial f(\mathbf{u})\) and refer to it as the _subdifferential_ of \(f\). The function \(f\) is called _subdifferentiable_ at \(\mathbf{u}\in\mathbb{R}^{p}\) if \(\partial f(\mathbf{u})\
eq\emptyset\). Note that if \(f:\mathbb{R}^{p}\to\mathbb{R}\) is a convex function and \(\mathbf{u}\in\mathbb{R}^{p}\), then \(f\) is subdifferentiable at \(\mathbf{u}\). If \(f:\mathbb{R}^{p}\to\mathbb{R}\) is a concave function, a vector \(\widetilde{\
abla}f(\mathbf{u})\in\mathbb{R}^{p}\) is called a _supergradient_ of \(f\) at \(\mathbf{u}\) if  
\[f(\mathbf{v})\leq\langle\widetilde{\
abla}f(\mathbf{u}),\mathbf{v}-\mathbf{u }\rangle+f(\mathbf{u})\ \ \ \text{for all }\mathbf{v}\in\mathbb{R}^{p}.\]  
The set of all supergradients of \(f\) at \(\mathbf{u}\) is denoted by \(\partial f(\mathbf{u})\) and is called the _superdifferential_ of \(f\). For simplicity, we also refer to them as the subgradient and subdifferential sets. Furthermore, we note that \(\partial f(\mathbf{u})=-\partial(-f)(\mathbf{u})\).