# Delayed Star Subgradient Methods for Constrained Nondifferentiable Quasi-Convex Optimization
## 3 Algorithms and Convergence Results
### Delayed Star Subgradient Method II  
In order to obtain the convergence of the generated sequence, we need a modified version of DDSM-I which is defined as the following Algorithm 2. **Initialization**: Given a stepsize \(\{\alpha_{k}\}_{k=0}^{\infty}\subset(0,\infty)\), the delays \(\{\tau_{k}\}_{k=0}^{\infty}\subset\mathbb{N}_{0}\), and initial points \(\mathbf{x}_{0},\mathbf{x}_{-1},\mathbf{x}_{-2},\ldots,\mathbf{x}_{-\tau}\in X\). **Iterative Step**: For a current point \(\mathbf{x}_{k}\in X\), if \(f(\mathbf{x}_{k})=f^{*}\), we set  
\[\mathbf{x}_{k+1}:=\mathbf{x}_{k}\]  
and then stop. If \(f(\mathbf{x}_{k})>f^{*}\), we compute  
\[\mathbf{x}_{k+1}:=P_{X}(\mathbf{x}_{k}-\alpha_{k}\mathbf{g}_{k-\tau_{k}}),\]  
where \(\mathbf{g}_{k-\tau_{k}}\in\partial^{*}f(\mathbf{x}_{k-\tau_{k}})\cap\mathcal{S}\) is a unit star subgradient of \(f\) at \(\mathbf{x}_{k-\tau_{k}}\). **Update**: \(k:=k+1\). which together with the nonincreasing property of \(\{\alpha_{k}\}_{k=0}^{\infty}\) implies  
\[\sum_{k=0}^{N}\alpha_{k}\|\mathbf{x}_{k}-\mathbf{x}_{k-\tau_{k}}\| \leq\sum_{i=0}^{\tau}\sum_{k=0}^{N}\alpha_{k}^{2}=(\tau+1)\sum_{k=0}^{N}\alpha_{ k}^{2}. \tag{15}\]  
By approaching \(N\to\infty\) in (15) and using the assumption that \(\sum_{k=0}^{\infty}\alpha_{k}^{2}<\infty\), we obtain that \(\sum_{k=0}^{\infty}\alpha_{k}\|\mathbf{x}_{k}-\mathbf{x}_{k-\tau_{k}}\|<\infty\), and so  
\[\sum_{k=0}^{\infty}\Bigl{(}\alpha_{k}^{2}+2\alpha_{k}\|\mathbf{x}_{k}-\mathbf{ x}_{k-\tau_{k}}\|\Bigr{)}<\infty.\]  
Therefore, the sequence \(\{x_{k}\}_{k=0}^{\infty}\) is quasi-Fejer monotone. By applying Fact 2, we obtain that the sequence \(\{\mathbf{x}_{k}\}_{k=0}^{\infty}\) is bounded and \(\lim_{k\to\infty}\|\mathbf{x}_{k}-\mathbf{x}^{*}\|\) exists. Since the sequence \(\{\mathbf{x}_{k}\}_{k=0}^{\infty}\) is bounded and \(\liminf_{k\to\infty}f(\mathbf{x}_{k})=f^{*}\) (Theorem 3), there exists a subsequence \(\{\mathbf{x}_{k}\}_{i=0}^{\infty}\) of \(\{\mathbf{x}_{k}\}_{k=0}^{\infty}\) such that  
\[\lim_{i\to\infty}f(\mathbf{x}_{k_{i}})=\liminf_{k\to\infty}f(\mathbf{x}_{k})= f^{*}.\]  
Since \(\{\mathbf{x}_{k_{i}}\}_{i=0}^{\infty}\) is also bounded, there exists a subsequence \(\{\mathbf{x}_{k_{i}}\}_{j=0}^{\infty}\) of \(\{\mathbf{x}_{k_{i}}\}_{i=0}^{\infty}\) such that \(\lim_{j\to\infty}\mathbf{x}_{k_{i}}=\bar{\mathbf{x}}\in\mathbb{R}^{n}\). Thus, the continuity of \(f\) yields  
\[f(\bar{\mathbf{x}})=\lim_{j\to\infty}f(\mathbf{x}_{k_{j}})=\lim_{i\to\infty}f( \mathbf{x}_{k_{i}})=f^{*}.\]  
Moreover, the closeness of \(X\) implies \(\bar{\mathbf{x}}\in X\), and hence we obtain that \(\bar{\mathbf{x}}\in X^{*}\). Since \(\lim_{k\to\infty}\|\mathbf{x}_{k}-\bar{\mathbf{x}}\|\) exists and \(\lim_{j\to\infty}\|\mathbf{x}_{k_{j}}-\bar{\mathbf{x}}\|=0\), we conclude that \(\lim_{k\to\infty}\|\mathbf{x}_{k}-\bar{\mathbf{x}}\|=0\). Moreover, invoking the continuity of \(f\), we also have that \(\lim_{k\to\infty}f(\mathbf{x}_{k})=f^{*}\). The proof is complete. We end the convergence analysis of DSSM-II by investigating the finite convergence of the generated sequence as the following theorem. Theorem 5: _Suppose that Assumptions 1 and 2 hold and \(\mathcal{B}(\mathbf{x}^{*};\delta)\subset X^{*}\) for some \(\delta>0\). Let \(\{\mathbf{x}_{k}\}_{k=0}^{\infty}\) be a sequence generated by DSSM-II. If one of the following statements holds:_  
_(i)_ \(\alpha_{k}=\alpha\in\left(0,\frac{2\delta}{2\tau+3}\right)\) _for all_ \(k\in\mathbb{N}_{0}\)_;_  
_(ii)_ \(\{\alpha_{k}\}_{k=0}^{\infty}\) _is nonincreasing with_ \(\lim_{k\to\infty}\alpha_{k}=0\) _and_ \(\sum_{k=0}^{\infty}\alpha_{k}=\infty\)_,_  
_then there exists \(K\in\mathbb{N}_{0}\) such that \(\mathbf{x}_{K}\in X^{*}\)._  
Proof: We suppose by contradiction that \(\mathbf{x}_{k}\
otin X^{*}\) for all \(k\in\mathbb{N}_{0}\). This implies \(\mathbf{x}_{k-\tau_{k}}\
otin X^{*}\) for all \(k\in\mathbb{N}_{0}\).