Spiral optimization algorithm

The spiral optimization (SPO) algorithm is an uncomplicated search concept inspired by spiral phenomena in nature.

Metaphor

The motivation for focusing on spiral phenomena was due to the insight that the dynamics that generate logarithmic spirals share the diversification and intensification behavior. The diversification behavior can work for a global search (exploration) and the intensification behavior enables an intensive search around a current found good solution (exploitation).

Algorithm

Spiral Optimization (SPO) Algorithm

The SPO algorithm is a multipoint search algorithm that has no objective function gradient, which uses multiple spiral models that can be described as deterministic dynamical systems. As search points follow logarithmic spiral trajectories towards the common center, defined as the current best point, better solutions can be found and the common center can be updated. The general SPO algorithm for a minimization problem under the maximum iteration $k_{\max }$ (termination criterion) is as follows:

0) Set the number of search points $m\geq 2$ and the maximum iteration number $k_{\max }$.
1) Place the initial search points $x_{i}(0)\in \mathbb {R} ^{n}~(i=1,\ldots ,m)$ and determine the center $x^{\star }(0)=x_{i_{\text{b}}}(0)$, $\displaystyle i_{\text{b}}=\mathop {\text{argmin}} _{i=1,\ldots ,m}\{f(x_{i}(0))\}$,and then set $k=0$.
2) Decide the step rate $r(k)$ by a rule.
3) Update the search points: $x_{i}(k+1)=x^{\star }(k)+r(k)R(\theta )(x_{i}(k)-x^{\star }(k))~~~(i=1,\ldots ,m).$
4) Update the center: $x^{\star }(k+1)={\begin{cases}x_{i_{\text{b}}}(k+1)~{\big (}{\text{if}}~f(x_{i_{\text{b}}}(k+1))<f(x^{\star }(k)){\big )},\\x^{\star }(k)~~~~~~~~~{\big (}{\text{otherwise}}{\big )},\\\end{cases}}$ where $\displaystyle i_{\text{b}}=\mathop {\text{argmin}} _{i=1,\ldots ,m}\{f(x_{i}(k+1))\}$.
5) Set $k:=k+1$. If $k=k_{\max }$ is satisfied then terminate and output $x^{\star }(k)$. Otherwise, return to Step 2).

Setting

The search performance depends on setting the composite rotation matrix $R(\theta )$, the step rate $r(k)$, and the initial points $x_{i}(0)~(i=1,\ldots ,m)$. Some effective settings with high theoritical backgrounds have been proposed to this date.

Setting 1 (Spiral Descent Direction Setting)

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

• Set $R(\theta )$ as follows:$R(\theta )={\begin{bmatrix}0_{n-1}^{\top }&-1\\I_{n-1}&0_{n-1}\\\end{bmatrix}}$ where $I_{n-1}$ is the $(n-1)\times (n-1)$ identity matrix and $0_{n-1}$ is the $(n-1)\times 1$ zero vector.
• Place the initial points $x_{i}(0)\in \mathbb {R} ^{n}$ $(i=1,\ldots ,m)$ at random to satisfy the following condition: $\min _{i=1,\ldots ,m}\{\max _{j=1,\ldots ,m}{\bigl \{}{\text{rank}}{\bigl [}d_{j,i}(0)~R(\theta )d_{j,i}(0)~~\cdots ~~R(\theta )^{2n-1}d_{j,i}(0){\bigr ]}{\bigr \}}{\bigr \}}=n$ where $d_{j,i}(0)=x_{j}(0)-x_{i}(0)$. Note that this condition is almost all satisfied by a random placing and thus no check is actually fine.
• Set $r(k)$ at Step 2) as follows:$r(k)=r={\sqrt[{k_{\max }}]{\delta }}~~~~{\text{(constant value)}}$ where a sufficiently small $\delta >0$ such as $\delta =1/k_{\max }$ or $\delta =10^{-3}$.

Setting 2 (Convergence Setting)

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

• Set $r(k)$ at Step 2) as follows:$r(k)={\begin{cases}1~~(k^{\star }\leqq k\leqq k^{\star }+2n-1),\\h~~(k\geqq k^{\star }+2n),\end{cases}}$ where $k^{\star }$ is an iteration when the center is newly updated at Step4 and $h={\sqrt[{2n}]{\delta }},\delta \in (0,1)$ such as $\delta =0.5$. Thus we have to add the following rules about $k^{\star }$ to the Algorithm:

  •(Step 1) $k^{\star }=0$.

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

Future works

• The algorithms with the above settings are deterministic. Thus, incorporating some random operations would make this algorithm powerful for the global optimization.
• To find an appropriate balance between diversification and intensification spirals depending on the target problem class including $k_{\max}$ is important to enhance the performance.