# Delayed Star Subgradient Methods for Constrained Nondifferentiable Quasi-Convex Optimization
## 2 Preliminaries  
This section contains important symbols, definitions, and tools utilized in this work. Throughout this paper, let \(\mathbb{R}\) be the set of all real numbers, \(\mathbb{N}_{0}\) the set of all non-negative integers, and \(\mathbb{N}\) the set of all positive integers. We denote with \(\mathbb{R}^{n}\) a Euclidean space with an inner product \(\langle\cdot,\cdot\rangle\) and an induced norm \(\|\cdot\|\). We denote the unit sphere by the set \(\mathcal{S}:=\{\mathbf{z}\in\mathbb{R}^{n}:\|\mathbf{z}\|=1\}\) and the open ball centered at the \(\mathbf{x}\in\mathbb{R}^{n}\) with radius \(\delta>0\) by the set \(\mathcal{B}(\mathbf{x};\delta):=\{\mathbf{z}\in\mathbb{R}^{n}:\|\mathbf{z}- \mathbf{x}\|<\delta\}\). Let \(X\subset\mathbb{R}^{n}\) be a nonempty set. The symbol \(\mathrm{cl}(X)\) is the closure of \(X\). The distance (function) between \(\mathbf{y}\in\mathbb{R}^{n}\) and \(X\) is the function \(\mathrm{dist}:\mathbb{R}^{n}\to\mathbb{R}\) defined by  
\[\mathrm{dist}(\mathbf{y},X):=\inf_{\mathbf{x}\in X}\|\mathbf{y}-\mathbf{x}\|.\]  
The metric projection of \(\mathbf{y}\in\mathbb{R}^{n}\) onto \(X\) is the point \(P_{X}(y)\in X\) such that \(\|P_{X}(\mathbf{y})-\mathbf{y}\|\leq\|\mathbf{x}-\mathbf{y}\|\) for all \(\mathbf{x}\in X\). Note that the metric projection \(P_{X}(\mathbf{y})\) exists and is unique for all \(\mathbf{y}\in\mathbb{R}^{n}\) whenever \(X\) is nonempty, closed, and convex [27] (Theorem 1.2.3). The metric projection onto a nonempty closed convex set \(X\) has a nonexpansive property, i.e., \(\|P_{X}(\mathbf{x})-P_{X}(\mathbf{y})\|\leq\|\mathbf{x}-\mathbf{y}\|\) for all \(\mathbf{x},\mathbf{y}\in\mathbb{R}^{n}\)[27] (Theorem 2.2.21). Let \(f:\mathbb{R}^{n}\to\mathbb{R}\) be a function, and let \(\alpha\in\mathbb{R}\). The strictly sublevel and sublevel set of \(f\) corresponding to \(\alpha\) are defined by  
\[S^{<}_{f,\alpha}:=\{\mathbf{x}\in\mathbb{R}^{n}:f(\mathbf{x})<\alpha\}\text{ and }S^{<}_{f,\alpha}:=\{\mathbf{x}\in\mathbb{R}^{n}:f(\mathbf{x})\leq\alpha\},\]  
respectively. The function \(f\) is called upper semicontinuous on \(\mathbb{R}^{n}\) if \(S^{<}_{f,\alpha}\) is an open set for all \(\alpha\in\mathbb{R}\). The function \(f\) is called lower semicontinuous on \(\mathbb{R}^{n}\) if \(S^{<}_{f,\alpha}\) is a closed set for all \(\alpha\in\mathbb{R}\). The function \(f\) is continuous on \(\mathbb{R}^{n}\) if it is both upper semicontinuous and lower semicontinuous on \(\mathbb{R}^{n}\). The function \(f\) is called quasi-convex on \(\mathbb{R}^{n}\) if \(S^{<}_{f,\alpha}\) is convex for all \(\alpha\in\mathbb{R}\). For any \(\mathbf{x}\in\mathbb{R}^{n}\), the star subdifferential [16; 17] of \(f\) at \(\mathbf{x}\) is the set  
\[\partial^{\star}f(\mathbf{x}):=\Big{\{}\mathbf{g}\in\mathbb{R}^{n}:\langle \mathbf{g},\mathbf{y}-\mathbf{x}\rangle\leq 0\text{ for all }\mathbf{y}\in S^{<}_{f,f(\mathbf{x})}\Big{\}}.\]  
The element \(\mathbf{g}\in\partial^{\star}f(\mathbf{x})\) is called the star subgradient of \(f\) at \(\mathbf{x}\), and it is denoted by \(\widetilde{\
abla}^{\star}f(\mathbf{x})\). It is obvious that \(\mathbf{0}\in\partial^{\star}f(\mathbf{x})\) for all \(\mathbf{x}\in\mathbb{R}^{n}\). The following fact provides the basic properties of the star subgradient, which can be found in [16] (Proposition 30) and [6] (Lemma 2.1). **Fact 1**. : _Let \(f:\mathbb{R}^{n}\to\mathbb{R}\) be a quasi-convex, and let \(\mathbf{x}\in\mathbb{R}^{n}\). Then the following statements hold:_  
\((i)\): \(\partial^{\star}f(\mathbf{x})\setminus\{\mathbf{0}\}\) _is a nonempty set;_  
\((ii)\): \(\partial^{\star}f(\mathbf{x})\) _is a closed, convex cone._  
Next, we will recall the definition and properties of the quasi-Fejer monotone sequence. Let \(X\) be a nonempty subset of \(\mathbb{R}^{n}\).