The generation of magic squares of any size has been researched since ancient times. However，due to the need to solve complex combinatorial optimization problems，many existing methods are rule-based，and therefore can only generate a limited number of special magic squares. One of the non-rule-based generation methods is evolutionary computation，and this paper proposes a multi-stage evolutionary strategy that hierarchically classifies the constraints of magic squares into three stages of rows，columns，and diagonals，and uses the individuals satisfying each layer’s constraints as the initial individuals for the next layer. A magic square is a state in which the sum of each element in a square matrix in rows，columns，and diagonals is equal to a constant value. A square matrix that satisfies only the constraints in rows and columns is called a semi-magic square. In this research，the semi-magic square was divided into two layers，and by adding a lower layer that generates individuals satisfying only the row constraints，the need for exploration in the row direction was eliminated，resulting in a significant reduction in the number of searches. Experiments compared the necessary time and generation number between prior research and the proposed method，and achieved a significant reduction. Specifically，it was possible to generate a 90x90 magic square in about 232 times faster than prior research，with a successful generation in an average of 6 minutes and 57 seconds (10 trials).

On December 21, 2023, Open Space Discussion 2023 (OSD2023) was held as a 1st-day event of the 2023 Symposium on Evolutionary Computation. This event was motivated to provide an opportunity to share, discuss, and create future directions in evolutionary computation. This paper offers an event report for OSD2023, including a summary of the discussions and participant feedback.

A global multiobjective design optimization method for three-dimensional design optimization based on topology optimization is presented. The proposed method employs a technique that compresses three-dimensional images using octree for encoding to reduce the number of design variables and improve optimization efficiency. The resulting octree structure is optimized through evolutionary algorithm with operators inspired from genetic programming. To validate the proposed method, an optimization problem to reproduce the target shape is solved. The results demonstrate that the proposed method effectively captures design features with a minimal number of solution evaluations. Furthermore, to validate the efficacy of the proposed method for aerodynamic design optimization problems, a design optimization of wingtip design for micro aerial vehicles is presented. It is confirmed that the proposed method can reproduce various geometries. The non-dominated solution obtained captured the trade-off relationship between the two objective functions. In addition, the non-dominated solution exhibited a lift coefficient improvement of up to 55% and a drag coefficient reduction of up to 2% compared to the baseline design. Geometry and aerodynamic performance revealed cant-up and cant-down shapes specific to the design conditions.

A Kriging-assisted Reference Vector Guided Evolutionary Algorithm (K-RVEA) is one of the most successful surrogate-assisted multiobjective evolutionary algorithms (SAMOEAs) for solving expensive multiobjective optimization problems (EMOPs). In K-RVEA, an evolutionary search algorithm is conducted on Kriging models approximating objective functions, to identify better solutions while saving the expensive function evaluations. Thus, the maximum number of generations for this model-based search process is crucial in improving the performance of K-RVEA. Although many works have attempted to improve the K-RVEA framework, there has been little attention in this regard. Accordingly, this paper proposes an extended K-RVEA that can adaptively select an appropriate number of generations for the model-based search process. To this end, this paper starts by conducting an analysis to understand an effective strategy to improve the K-RVEA performance in terms of the number of generations. Subsequently, we introduce an adaptive selection mechanism based on the analytical results, and integrate it to the K-RVEA framework. Experimental results on benchmark problems in their multi/many-objective settings reveal that our extended K-RVEA outperforms the original K-RVEA on many experimental cases.

This paper proposes Tracking Swarm Optimization based on PSO and CMA-ES (TSOPC), which aims to continuously track optima in dynamic optimization problems with frequent environmental changes. The proposed algorithm tracks optima by roughly estimating the direction of optima movement by Tracking CMA-ES (TCMA-ES) and locally searching optima by Tracking PSO (TPSO), each of which is improved from CMA-ES and PSO. In TSCPC, TPSO and TCMA-ES cooperate with each other by exchanging their own superior solutions to find better solutions and by relocating their solutions when losing to track optima. Through the intensive experiments on four evaluation functions in two and ten dimensions based on the Moving Peaks Benchmark (MPB), the following implications have revealed: (1) the offline-error (which value becomes small when tracking solutions are close to optima) of TSOPC is smaller than that of TPSO and TCMA-ES; (2) the offline-error of TSOPC is affected by the ratio of TPSO and TCMA-ES, and an increase of the ratio of TCMA-ES contributes to decreasing the offline-error in the case of difficulty in tracking optima.