Setting 1 (Periodic Descent Direction Setting)

This setting is an effective setting for high dimensional problems under the maximum iteration ${\displaystyle k_{\max }}$. The conditions on ${\displaystyle R(\theta )}$ and ${\displaystyle x_{i}(0)~(i=1,\ldots ,m)}$ together ensure that the spiral models generate descent directions periodically. The condition of ${\displaystyle r(k)}$ works to utilize the periodic descent directions under the search termination ${\displaystyle k_{\max }}$.

• Set ${\displaystyle R(\theta )}$ as follows:${\displaystyle R(\theta )={\begin{bmatrix}0_{n-1}^{\top }&-1\\I_{n-1}&0_{n-1}\\\end{bmatrix}}}$ where ${\displaystyle I_{n-1}}$ is the ${\displaystyle (n-1)\times (n-1)}$ identity matrix and ${\displaystyle 0_{n-1}}$ is the ${\displaystyle (n-1)\times 1}$ zero vector.
• Place the initial points ${\displaystyle x_{i}(0)\in \mathbb {R} ^{n}}$ ${\displaystyle (i=1,\ldots ,m)}$ at random to satisfy the following condition:

${\displaystyle \min _{i=1,\ldots ,m}\{\max _{j=1,\ldots ,m}{\bigl \{}{\text{rank}}{\bigl [}d_{j,i}(0)~R(\theta )d_{j,i}(0)~~\cdots ~~R(\theta )^{2n-1}d_{j,i}(0){\bigr ]}{\bigr \}}{\bigr \}}=n}$ where ${\displaystyle d_{j,i}(0)=x_{j}(0)-x_{i}(0)}$. Note that this condition is almost all satisfied by a random placing and thus no check is actually fine.

• Set ${\displaystyle r(k)}$ at Step 2) as follows:${\displaystyle r(k)=r={\sqrt[{k_{\max }}]{\delta }}~~~~{\text{(constant value)}}}$ where a sufficiently small ${\displaystyle \delta >0}$ such as ${\displaystyle \delta =1/k_{\max }}$ or ${\displaystyle \delta =10^{-3}}$.^[3]

Setting 2 (Convergence Setting)

This setting ensures that the SPO algorithm converges to a stationary point under the maximum iteration ${\displaystyle k_{\max }=\infty }$. The settings of ${\displaystyle R(\theta )}$ and the initial points ${\displaystyle x_{i}(0)~(i=1,\ldots ,m)}$ are the same with the above Setting 1. The setting of ${\displaystyle r(k)}$ is as follows.

• Set ${\displaystyle r(k)}$ at Step 2) as follows:${\displaystyle r(k)={\begin{cases}1&(k^{\star }\leqq k\leqq k^{\star }+2n-1),\\h&(k\geqq k^{\star }+2n),\end{cases}}}$ where ${\displaystyle k^{\star }}$ is an iteration when the center is newly updated at Step 4) and ${\displaystyle h={\sqrt[{2n}]{\delta }},\delta \in (0,1)}$ such as ${\displaystyle \delta =0.5}$. Thus we have to add the following rules about ${\displaystyle k^{\star }}$ to the Algorithm:

•(Step 1) ${\displaystyle k^{\star }=0}$.

•(Step 4) If ${\displaystyle x^{\star }(k+1)\neq x^{\star }(k)}$ then ${\displaystyle k^{\star }=k+1}$.^[4]