Social cognitive optimization

Social cognitive optimization (SCO) is a population-based metaheuristic optimization algorithm which was developed in 2002.^[1] This algorithm is based on the social cognitive theory, and the key point of the ergodicity is the process of individual learning of a set of agents with their own memory and their social learning with the knowledge points in the social sharing library. It has been used for solving continuous optimization,^[2][3] integer programming,^[4] and combinatorial optimization problems. It has been incorporated into the NLPSolver extension of Calc in Apache OpenOffice.

Algorithm

Let $f(x)$ be a global optimization problem, where $x$ is a state in the problem space $S$. In SCO, each state is called a knowledge point, and the function $f$ is the goodness function.

In SCO, there are a population of $N_c$ cognitive agents solving in parallel, with a social sharing library. Each agent holds a private memory containing one knowledge point, and the social sharing library contains a set of ${\displaystyle N_{L}}$ knowledge points. The algorithm runs in T iterative learning cycles. By running as a Markov chain process, the system behavior in the tth cycle only depends on the system status in the (t − 1)th cycle. The process flow is in follows:

• [1. Initialization]：Initialize the private knowledge point $x_{i}$ in the memory of each agent $i$, and all knowledge points in the social sharing library $X$, normally at random in the problem space $S$.
• [2. Learning cycle]： At each cycle $t$ ${\displaystyle (t=1,\ldots ,T)}$：
  • [2.1. Observational learning] For each agent $i$ ${\displaystyle (i=1,\ldots ,N_{c})}$：
    • [2.1.1. Model selection]：Find a high-quality model point ${\displaystyle x_{M}}$ in $X(t)$ , normally realized using tournament selection, which returns the best knowledge point from randomly selected $\tau _{B}$ points.
    • [2.1.2. Quality Evaluation]：Compare the private knowledge point $x_{i}(t)$ and the model point ${\displaystyle x_{M}}$，and return the one with higher quality as the base point ${\displaystyle x_{Base}}$，and another as the reference point ${\displaystyle x_{Ref}}$。
    • [2.1.3. Learning]：Combine ${\displaystyle x_{Base}}$ and ${\displaystyle x_{Ref}}$ to generate a new knowledge point ${\displaystyle x_{i}(t+1)}$. Normally ${\displaystyle x_{i}(t+1)}$ should be around ${\displaystyle x_{Base}}$，and the distance with ${\displaystyle x_{Base}}$ is related to the distance between ${\displaystyle x_{Ref}}$ and ${\displaystyle x_{Base}}$, and boundary handling mechanism should be incorporated here to ensure that ${\displaystyle x_{i}(t+1)\in S}$.
    • [2.1.4. Knowledge sharing]：Share a knowledge point, normally ${\displaystyle x_{i}(t+1)}$, to the social sharing library $X$.
    • [2.1.5. Individual update]：Update the private knowledge of agent $i$, normally replace ${\displaystyle x_{i}(t)}$ by ${\displaystyle x_{i}(t+1)}$. Some Monte Carlo types might also be considered.
  • [2.2. Library Maintenance]：The social sharing library using all knowledge points submitted by agents to update $X(t)$ into ${\displaystyle X(t+1)}$. A simple way is one by one tournament selection: for each knowledge point submitted by an agent, replace the worse one among ${\displaystyle \tau _{W}}$ points randomly selected from $X(t)$.
• [3. Termination]：Return the best knowledge point found by the agents.

SCO has three main parameters, i.e., the number of agents $N_c$, the size of social sharing library ${\displaystyle N_{L}}$, and the learning cycle $T$. With the initialization process, the total number of knowledge points to be generated is ${\displaystyle N_{L}+N_{c}*(T+1)}$, and is not related too much with ${\displaystyle N_{L}}$ if $T$ is large.

Compared to traditional swarm algorithms, e.g. particle swarm optimization, SCO can achieving high-quality solutions as $N_c$ is small, even as ${\displaystyle N_{c}=1}$. Nevertheless, smaller $N_c$ and ${\displaystyle N_{L}}$ might lead to premature convergence. Some variants ^[5] were proposed to guaranteed the global convergence.

References

1. Xie, Xiao-Feng; Zhang, Wen-Jun; Yang, Zhi-Lian (2002). Social cognitive optimization for nonlinear programming problems. International Conference on Machine Learning and Cybernetics (ICMLC), Beijing, China: 779-783.
2. Xie, Xiao-Feng; Zhang, Wen-Jun (2004). Solving engineering design problems by social cognitive optimization. Genetic and Evolutionary Computation Conference (GECCO), Seattle, WA, USA: 261-262.
3. Xu, Gang-Gang; Han, Luo-Cheng; Yu, Ming-Long; Zhang, Ai-Lan (2011). Reactive power optimization based on improved social cognitive optimization algorithm. International Conference on Mechatronic Science, Electric Engineering and Computer (MEC), Jilin, China: 97-100.
4. Fan, Caixia (2010). Solving integer programming based on maximum entropy social cognitive optimization algorithm. International Conference on Information Technology and Scientific Management (ICITSM), Tianjing, China: 795-798.
5. Sun, Jia-ze; Wang, Shu-yan; chen, Hao (2014). A guaranteed global convergence social cognitive optimizer. Mathematical Problems in Engineering: Art. No. 534162.