With the increasing size of optimization problems in various scientific and engineering fields, finding promising solutions for these large-scale optimization problems has become increasingly challenging. Dimension reduction-based evolutionary algorithms have emerged as one of the most efficient approaches to tackle these challenges. However, they still face difficulties in constructing appropriate subspaces and preserving the global optimum within those subspaces. To overcome these challenges, a multiform optimization framework with scaleadaptive subspace, called MFO-SAS, is proposed for large-scale optimization by taking full advantage of the subspace-assisted and multitasking mechanisms. To address the first challenge, a scale-adaptive switch strategy (SASS) is designed to switch the subspaces with different scales at different optimization stages, enabling efficient assistance of the optimization processes for the original problem with appropriate subspaces. To tackle the second challenge, a multiform optimization paradigm is adopted to conduct the simultaneous search over both the original problem space and the constructed subspaces in a multitasking scenario, thus facilitating the utilization of the search experiences from different problem spaces. Consequently, the MFO-SAS framework effectively leverages valuable knowledge obtained from different spaces to guide the search and dynamically balances the representation accuracy of the subspace with the computational cost of subspace search. Experimental results on the CEC2013 large-scale benchmark problems demonstrate the superiority of MFO-SAS over several state-of-the-art algorithms.

In solving multi-objective vehicle routing problems with time windows (MOVRPTW), most existing algorithms focus on the optimization of a single problem formulation. However, little effort has been devoted to exploiting valuable knowledge from the alternate formulations of MOVRPTW for better optimization performance. Aiming at this insufficiency, this study proposes a decomposition-based multi-objective multiform evolutionary algorithm (MMFEA/D), which performs the evolutionary search on multiple alternate formulations of MOVRPTW simultaneously to complement each other. In particular, the main characteristics of MMFEA/D are three folds. First, a multiform construction (MFC) strategy is adopted to construct multiple alternate formulations, each of which is formulated by grouping several adjacent subproblems based on the decomposition of MOVRPTW. Second, a transfer reproduction (TFR) mechanism is designed to generate offspring for each formulation via transferring promising solutions from other formulations, making that the useful traits captured from different formulations can be shared and leveraged to guide the evolutionary search. Third, an adaptive local search (ALS) strategy is developed to invest search effort on different alternate formulations as per their usefulness for MOVRPTW, thus facilitating the efficient allocation of computational resources. Experimental studies have demonstrated the superior performance of MMFEA/D on the classical Solomon instances and the real-world instances.