Tensorial Evolutionary Computation for Spatial Optimization Problems

Spatial optimization problems (SOPs) refer to a class of problems where the decision variables require spatial organization. Existing methods based on evolutionary algorithms (EAs) fit conventional evolutionary operators by flattening the spatial representations of SOPs to one dimension, which hence loses crucial spatial structures of the potential solutions. To address this issue, this article proposes a tensorial evolutionary algorithm (TEA) by formulating a tensor space of the SOPs, where the population of candidate solutions is represented as a third-order tensor. Accordingly, we design a tensor-product crossover operator and a two-phase mutation consisting of a local convolutional mutation operator and a global Gaussian mutation operator in the tensor space. These operators are all implemented based on tensor operations. Under the consideration of spatial context on individual solutions, we devise three types of evolutionary tensors, namely, the crossover tensor, the convolutional mutant tensor, and the Gaussian mutant tensor for the three tensorial evolutionary operators correspondingly. The resultant TEA is tested using two typical real-world SOPs, namely, the facility layout problem and the image steganography problem. Comparative experimental studies validate the effectiveness and robustness of TEA.