This study presents a novel multi-constraint and multi-objective optimization based approach that applies genetic algorithms (GAs) for developing high-frequency transformer (HFT) designs for dual active bridge converters (DABs) in solid-state transformers (SSTs). SSTs are increasingly adopted in modern power systems due to their higher efficiency, compact structure, and improved operational reliability when compared with conventional transformers. Developing HFTs for SSTs involves several challenges, particularly the need to balance competing objectives such as improving efficiency, limiting losses, and reducing the area product while satisfying multiple design constraints. To address these challenges, this work applies a constrained multi-objective GA implemented in MATLAB to optimize the design of an HFT for a DAB converter. The methodology allows for the simultaneous optimization of multiple design objectives while taking into consideration restrictions like efficiency, leakage inductance, temperature limits, core winding area, and sizes. Our comparison with particle swarm optimization (PSO) indicates that the GA achieves more consistent convergence and consistently lower total losses. The case studies reinforce this observation, giving compact and high-performance HFT designs tailored for SST applications. The optimization approach provides a reliable and scalable method for developing thermally robust and space-efficient HFTs suitable for next-generation SST platforms and renewable-energy applications.