# Delayed Star Subgradient Methods for Constrained Nondifferentiable Quasi-Convex Optimization
## 4 Numerical Experiments  
In this section, we present experimental results when we apply DSSM-I to solve the Cobb-Douglas production efficiency problem. We performed all experiments on MATLAB (R2023b) on a MacBook Air 13.3-inch with Apple M1 chip processor and 8 GB memory. The Cobb-Douglas production efficiency problem [3; 6; 12] aims to maximize the ratio of the total profit represented by the Cobb-Douglas product function and the total cost represented by the linear function on the product factor subject to the funding level limitations constraint. More precisely, let \(a_{0},c_{0}>0\) and let \(a_{j},b_{ij},c_{j},p_{i}\geq 0\) for all \(j=1,2,\ldots,n\) and \(i=1,2,\ldots,m\), and the Cobb-Douglas production efficiency problem is defined by  
\[ \begin{array}{ll}\text{maximize}&f(\mathbf{x}):=\frac{a_{0}\prod_{j=1}^{n} x_{j}^{a_{j}}}{\sum_{j=1}^{n}c_{j}+c_{0}}\\ \text{subject to}&\sum_{j=1}^{n}b_{ij}x_{j}\geq p_{i},i=1,2,\ldots,m\\ &x_{j}\geq 0,j=1,2,\ldots,n,\end{array}  \tag{19}\]  
where \(\sum_{j=1}^{n}a_{j}=1\). Here, \(\mathbf{x}:=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}\) when \(x_{j}\) is the \(j\)-th product factor \((j=1,2,\ldots,n)\), and for each \(i=1,2,\ldots,m\), \(p_{i}\) is the profit to be obtained from project \(i\), and \(b_{ij}\) is the support of product factor \(j\) to project \(i\) to achieve \(p_{i}\). According to the literature [6; 12], we know that the Cobb-Douglas production efficiency problem (19) is a quasi-convex optimization problem. Note that the function \(f\) is continuous and quasi-concave [4] (Theorem 2.5.1), satisfying the Holder condition on \(\mathbb{R}^{n}\)[12] (Appendix B), and the constraint set \(C:=\bigcap_{i=1}^{m}\{\mathbf{x}\in\mathbb{R}^{n}:\langle\mathbf{b}_{i}, \mathbf{x}\rangle\geq p_{i}\}\cap[0,\infty)^{n}\) is the nonempty closed convex set, where \(\mathbf{b}_{i}:=(b_{1},\ldots,b_{in})^{\top}\) is a vector in \(\mathbb{R}^{n}\). Therefore, we can apply DSSM-I to solve the problem (19). However, since the exact optimal solution to problem (19) is not known, we are unable to apply DSSM-II to solve it. To ensure the boundedness of the constrained set, we set the constrained set to be the set \(D:=\{\mathbf{x}\in\mathbb{R}^{n}:\langle\mathbf{b}_{i},\mathbf{x}\rangle\geq p _{i}\}\cap[0.001,100]^{n}\) which is a subset of \(C\). Thus, the established convergence result from Theorem 2 guarantees that \(\lim_{k\to\infty}f(\mathbf{x}_{k})=f^{*}\). It is worth noting that the computational procedure of the metric projection onto the constrained set \(D\) is not an easy task since there is no explicit form of \(P_{D}\) in general. To deal with the situation, we utilized the classical Halpern iteration by performing the following inner loop: for arbitrary initial point \(\mathbf{u}_{l}\in\mathbb{R}^{n}\) and a sequence \(\{\lambda_{l}\}_{l=0}^{\infty}\subseteq(0,1)\) such that \(\lim_{l\to\infty}\lambda_{l}=0,\sum_{l=0}^{\infty}\lambda_{l}=\infty\), and \(\sum_{l=1}^{\infty}|\lambda_{l+1}-\lambda_{l}|<\infty\), we compute  
\[\mathbf{u}_{l+1}:=\lambda_{l}(\mathbf{x}_{k}-\alpha_{k}\mathbf{g}_{k-\tau_{k}}) +(1-\lambda_{l})P_{D_{m+1}}P_{D_{m}}P_{D_{m-1}}\cdots P_{D_{l}}\mathbf{u}_{l} \text{ for all }l\geq 1,\]  
where \(D_{m+1}=[0.001,100]^{n}\), \(D_{i}:=\{\mathbf{x}\in\mathbb{R}^{n}:\langle\mathbf{b}_{i},\mathbf{x}\rangle \geq p_{i}\}\) for all \(i=1,2,\ldots,m\). Then the sequence \(\{\mathbf{u}_{l}\}_{l=0}^{\infty}\) converges to the unique point \(P_{D}(\mathbf{x}_{k}-\alpha_{k}\mathbf{g}_{k-\tau_{k}})\), see [28] (Theorem 30.1) for further details. In our experiment, we put the inner initial point \(\mathbf{u}_{0}\) as the vector whose all coordinates are \(1\) and set \(\lambda_{l}:=\frac{1}{l+2}\) for all \(l\in\mathbb{N}_{0}\). We use the stopping criterion \(\frac{\|\mathbf{u}_{l+1}-\mathbf{u}_{l}\|}{\|\mathbf{u}_{l}\|+1}\leq 10^{-6}\) for the inner loop. To perform the numerical experiments, we randomly set the parameter \(a_{j},b_{ij}\in(0,1),a_{0},c_{0},c_{j}\in(0,10)\), and \(p_{i}\in\left(0,\frac{n}{2}\right)\).