The semi-autonomous particle swarm optimizer (SAPSO) [1] is a relatively recent algorithm for global continuous optimization based on gradient direction and diversity controlling approach, providing autonomy for the particles and the swarm for exploiting regions in the search space, and preserving exploration during the whole search process. In the first study, although SAPSO algorithm holds the property rotational invariance in which it normally brings a lack of directional diversity in PSO context, the algorithm has shown very good performance in comparison to other PSO-like algorithms. In this paper, an improved version of SAPSO, named rotationally invariant SAPSO (RI-SAPSO), is proposed, which still holds the same property, but now it incorporates a rotation matrix generated by an exponential map to maintain directional diversity. A mathematical proof to prove that the RI-SAPSO algorithm is rotationally invariant is given. RI-SAPSO was evaluated on test functions extracted from CEC 2017 benchmark problems with six other PSO-like algorithms, along with its previous version. The comparative study was strengthened with a non-parametric Friedman's hypothesis test for 1 × k comparisons and p-values were adjusted in the post-hoc procedure. Simulation results showed that the proposed RI-SAPSO, in most problems, was able to find much better solutions and statistical significances were also observed.

半自主粒子群优化器（SAPSO）[1]是一种相对较新的算法，该算法基于梯度方向和分集控制方法进行全局连续优化，为粒子和粒子群提供自治权，以利用搜索空间中的区域并保留探索在整个搜索过程中。在第一项研究中，尽管SAPSO算法拥有属性旋转不变性在通常情况下，它在PSO上下文中缺乏方向分集，与其他类似PSO的算法相比，该算法表现出非常好的性能。在本文中，提出了一种改进的SAPSO版本，称为旋转不变SAPSO（RI-SAPSO），它仍然具有相同的属性，但是现在它合并了一个由指数图生成的旋转矩阵，以保持方向分集。给出了证明RI-SAPSO算法具有旋转不变性的数学证明。通过从CEC 2017基准测试问题中提取的测试功能以及其他六个PSO类似算法对RI-SAPSO进行了评估。通过非参数弗里德曼假设检验对1× k进行了比较研究 比较和p值在事后程序中进行了调整。仿真结果表明，所提出的RI-SAPSO在大多数问题中都能够找到更好的解决方案，并且还具有统计学意义。