# Delayed Star Subgradient Methods for Constrained Nondifferentiable Quasi-Convex Optimization
## 1 Introduction  
In this work, we consider the quasi-convex optimization problem:  
\[ \begin{array}{ll}\text{minimize}&f(\mathbf{x})\\ \text{subject to}&\mathbf{x}\in X,\end{array}  \tag{1}\]  
where the objective function \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) is quasi-convex continuous (possibly) and nondifferentiable, and the constrained set \(X\) is nonempty closed and convex. Throughout this work, we will denote the set of all optimal solutions and the optimal value of the problem (1) as \(X^{*}:=\operatorname*{argmin}_{\mathbf{x}\in X}f(\mathbf{x})\) and \(f^{*}:=\min_{\mathbf{x}\in X}f(\mathbf{x})\), respectively. We also assume that \(X^{*}\
eq\emptyset\). The problem (1) not only forms a fundamental setting for optimization as every convex function is quasi-convex but also generalizes to several practical situations in many fields such as economics, engineering, and others. An important special setting is the problem of minimizing the ratios of two functions [1] (Lemma 4), [2] (Theorem 2.3.8), for example, optimizing the ratio of outputs and inputs for the highest-quality productivity assessments in economics, such as debt-to-equity ratio analysis in corporate financial planning, inventory-to-sales analysis in production planning, and nurse-patient ratio analysis in hospital planning. This kind of problem is known as fractional programming [3, 4, 5]. As mentioned above, various kinds of iterative methods have been developed to solve the quasi-convex optimization problem (1); see [1, 6, 7, 8, 9, 10, 11, 12, 13, 14] and references therein. If the function \(f\) is convex, it is well-known that the classical method to deal with the constrained convex optimization problem is the projected subgradient method: for an initial point \(\mathbf{x}_{0}\in X\), compute  
\[\mathbf{x}_{k+1}:=P_{X}(\mathbf{x}_{k}-\alpha_{k}\widetilde{\
abla}f(\mathbf{x} _{k})),\qquad\forall k\in\mathbb{N}_{0},\]  
where \(\widetilde{\
abla}f(\mathbf{x}_{k})\in\{\mathbf{g}\in\mathbb{R}^{n}:\langle \mathbf{g},\mathbf{y}-\mathbf{x}\rangle\leq f(\mathbf{y})-f(\mathbf{x})\) for all \(\mathbf{y}\in\mathbb{R}^{n}\}\) is the Fenchel subgradient of \(f\) at \(\mathbf{x}_{k}\). Because of the simplicity, this method has been continuously developed in the literature, see for instance [8; 15]. However, for the quasi-convex case, it should be noted that the Fenchel subgradient may not exist, in general. In this situation, we need a more general subgradient concept, which is the so-called star subgradient proposed in [16; 17]. By utilizing the notion of a star subgradient, [PERSON] [1] proposed the star subgradient method to solve the constrained quasi-convex optimization problem (1): for an initial point \(\mathbf{x}_{0}\in X\),  
\[\mathbf{x}_{k+1}:=P_{X}\Bigg{(}\mathbf{x}_{k}-\alpha_{k}\frac{\widetilde{ \
abla}^{*}f(\mathbf{x}_{k})}{\|\widetilde{\
abla}^{*}f(\mathbf{x}_{k})\|} \Bigg{)},\qquad\forall k\in\mathbb{N}_{0}, \tag{2}\]  
where \(\widetilde{\
abla}^{*}f(\mathbf{x}_{k})\in\partial^{*}f(\mathbf{x}_{k}) \setminus\{\mathbf{0}\}\) is the nonzero star subgradient of \(f\) at \(\mathbf{x}_{k}\). After that, [PERSON] et al. [6] proposed an inexact version of the star-subgradient method by considering the approximated star subgradient with the presence of noises. By imposing the Holder condition, they presented the convergence results in various aspects for both constant and diminishing step sizes. Subsequently, several methods have been investigated for solving more general problem settings, for instance, a conditional star subgradient method [8], a stochastic star subgradient method [7], and an incremental star subgradient method [9]. As we can notice, the above-mentioned methods require computing the star subgradient in every single iteration; however, there are some practical situations such as large-scale problems in which the exact star subgradients may be computationally expensive.