## 4 Proposed Solution
### Main Characteristics of the Multi-Agent Gradient Descent Algorithm (MAGD)
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function Name** & **Formula** & **Domain Range** & \(f^{*}\) \\ \hline Rastrigin Function & \(f(x)=10n+\sum_{i=1}^{n}[x_{i}^{2}-10\cos(2\pi x_{i})]\) & \([-5.12,5.12]\) & 0 \\ \hline Ackley Function & \(f(x)=-20\exp\left(-02\sqrt{\frac{1}{2}\sum_{i=1}^{n}x_{i}^{2}}\right)-\exp \left(\frac{1}{2}\sum_{i=1}^{n}\cos(2\pi x_{i})\right)+20+\epsilon\) & \([-32.768,32.768]\) & 0 \\ \hline Goldstein-Price Function & \([1+(x+y+1)^{2}\cdot(19-14x+3x^{2}-14y+6 xy+3y^{2})]\cdot[30+(2x-3y)^{2}\cdot(18- 32x+12x^{2}+48y-36 xy+2y^{2})]\) & \([-2,2]\) & 3 \\ \hline Levy Function N.13 & \(f(x)=\sin^{2}(\pi x_{1})+\sum_{i=1}^{n}(x_{i}-1)^{2}[1+10\sin^{2}(\pi x_{1})+1 )+(u_{i}-1)^{2}[1+\sin^{2}(2\pi x_{1})]\) & \([-10.10]\) & 0 \\ \hline Himmelbau Function & \(f(x,y)=(x^{2}+y-11x^{2}+(x+y-7)^{2}\) & \([-5,5]\) & 0 \\ \hline Three-Hump Camel Function & \(f(x,y)=2x^{2}-10.5x^{4}+\frac{y}{5}+xy+y^{2}\) & \([-5,5]\) & 0 \\ \hline Griewank Function & \(f(x)=1+\frac{1}{800}\sum_{i=1}^{n}x_{i}^{2}-\frac{1}{11}\cos\left(\frac{y}{ 3}\right)\) & \([-600,600]\) & 0 \\ \hline Cross-in-Tray Function & \(f(x,y)=-0.0001\left(\left|\sin(x)\sin(y)\exp\left(\left|100-\sqrt{\frac{1}{2 }\pi^{2}}\right|\right)\right|+1\right)^{0}\) & \([-10,10]\) & \(-2.06261\) \\ \hline Eggholder Function & \(f(x,y)=-(y+4)\sin\left(\sqrt{\left|y/2+(y+4)\right|}\right)-x\sin\left(\sqrt{ \left|x-(y+47)\right|}\right)\) & \([-512,512]\) & \(-599.6407\) \\ \hline Michalewicz Function & \(f(x)=-\sum_{i=1}^{n}\sin(x)\left[\sin\left(\frac{y}{3}\right)\right]^{20k}\) & \([0,\pi]\) & \(-18.013\) (for n = 10) \\ \hline Alpine Function & \(f(x)=\sum_{i=1}^{n}\sin(x)+0.11\sin(1)\) & \([0,10]\) & 0 \\ \hline Schaffer Function N.2 & \(f(x,y)=0.5+\frac{\sin(y^{2}+y^{2})\cdot(1+\cos(y^{2}+y^{2}))}{(1+\cos(y^{2}+y^ {2}))^{2}}\) & \([-100,100]\) & 0 \\ \hline Schaffer Function N.4 & \(f(x,y)=0.5+\frac{\sin(y^{2}+y^{2})\cdot(1+\cos(y^{2}+y^{2}))}{(1+\cos(y^{2}+y^ {2}))^{2}}\) & \([-100,100]\) & 0 \\ \hline Easom Function & \(f(x,y)=-\cos(x)\cos(y)\exp\left(-\left(\left|x-\pi\right|^{2}+(y-\pi)^{2}\right)\right)\) & \([-100,100]\) & \(-1\) \\ \hline Shubert Function & \(f(x,y)=\sum_{i=1}^{n}\cos(\left|(1+1)+\left|\sum_{i=1}^{n}\cos(\left|(1+1)y+ \hat{\mu}\right|\right.\right)\) & \([-10,10]\) & \(-186.7309\) \\ \hline Schwefel Function & \(f(x)=418.9829n-\sum_{i=1}^{n}x_{i}\sin\left(\sqrt{\left|x\right|}\right)\) & \([-500,500]\) & 0 \\ \hline Zettl Function & \(f(x,y)=(x^{2}+y^{2}-2x)^{2}+\frac{y}{5}\) & \([-5,10]\) & \(-0.003791\) \\ \hline Leon Function & \(f(x,y)=100(y-x^{2})^{2}+(1-y)^{2}\) & \([-12,12]\) & 0 \\ \hline Drop-Wave Function & \(f(x,y)=-\frac{1+\cos(y^{2}+y^{2})}{(355(x^{2}+y^{2}))^{2}}\) & \([-512,512]\) & \(-1\) \\ \hline Langermann Function & \(f(x)=-\sum_{i=1}^{n}c_{i}\exp\left(-\frac{1}{n}\sum_{j=1}^{n}(j_{j}-a_{0})^{2} \right)\cos\left[\pi\sum_{j=1}^{n}(j_{j}-a_{ij})^{2}\right]\) & \([0,10]\) & \(-4.155\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark functions for optimization: formulas, ranges, and minimum values \(f^{*}\). The hyperparameters used for all algorithms in these experiments are summarized in Table 2. The learning rate of SBGD is initially set to 3, as recommended by the authors in their research, to ensure it is sufficiently large [15] (p. 9), as the specified value reflects the initial learning rate rather than its adaptive adjustments during the iterations. Tables 3 and 4 detail the hyperparameters specific to MAGD and SBGD, respectively. The exclusive SBGD hyperparameters are described as follows:

* \(tol_{\text{rms}}\in[0,1]\): This parameter determines the merging of agents that satisfy \(\|x_{i}-x_{j}\|_{1}<tol_{\text{merge}}\) at each iteration, where \(i,j=1,\ldots,N_{\text{agents}}\) and \(i\
eq j\). * \(tol_{m}\in[0,1]\): This threshold dictates the preservation of agents based on their _mass_. Agents with \(mass<(tol_{m}/N_{\text{agents}})\) are removed. Here, _mass_ is a term in SBGD proportional to the solution quality.