A Multifactorial Optimization Framework Based on Adaptive Intertask Coordinate System

The searching ability of the population-based search algorithms strongly relies on the coordinate system on which they are implemented. However, the widely used coordinate systems in the existing multifactorial optimization (MFO) algorithms are still fixed and might not be suitable for various function landscapes with differential modalities, rotations, and dimensions; thus, the intertask knowledge transfer might not be efficient. Therefore, this article proposes a novel intertask knowledge transfer strategy for MFOs implemented upon an active coordinate system that is established on a common subspace of two search spaces. The proper coordinate system might identify some common modality in a proper subspace to some extent. In this article, to seek the intermediate subspace, we innovatively introduce the geodesic flow that starts from a subspace, reaching another subspace in unit time. A low-dimension intermediate subspace is drawn from a uniform distribution defined on the geodesic flow, and the corresponding coordinate system is given. The intertask trial generation method is applied to the individuals by first projecting them on the low-dimension subspace, which reveals the important invariant features of the multiple function landscapes. Since intermediate subspace is generated from the major eigenvectors of tasks’ spaces, this model turns out to be intrinsically regularized by neglecting the minor and small eigenvalues. Therefore, the transfer strategy can alleviate the influence of noise led by redundant dimensions. The proposed method exhibits promising performance in the experiments.

I. Introduction

Problems seldom exist in separate tasks. Human beings can easily tackle multiple relative tasks at the same time by capturing the related characteristics of the tasks and utilizing the previous knowledge [1]. Thereby, the optimization tasks can benefit from the exchange of the shared problem-solving experience of multiple tasks. Many multitask schemes invoke a number of distinct types of domains. The roles of tasks in the multitask scheme are symmetric, namely, some tasks can be the targets of transferred knowledge, while the other tasks are the sources and vice versa. The source tasks are usually assumed to have underlying complementaries for the target tasks, although the distinction between the tasks is supposed to be considered. The main objective of multitasking is to efficiently utilize the shared problem-solving experience to attain good performance on every task [2]. In optimization, underlying complementaries among the different tasks can lie in either the distribution of solutions or the search directions, in some cases, both of them [3]–[5]. Hence, it is important to properly deal with the knowledge transfer between different domains. It is worth noting that we have no a priori concerning the explicit relationship of the tasks [6] in the black-box optimization. Some works [5], [7] analyzed task complementarity and gave some analytical views on the benefit of the intertask knowledge transfer in multitask optimization.