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Designs of write-head geometry and multilayer recording-media composition (10-30 design variables) were optimized using a multiobjective genetic algorithm (GA) so as to maximize/minimize read/write (R/W) performance measures such as signal-to-noise ratio (SNR), magnetic write width (MWW), and thermal stability of media. A head-media co-design system using a general-purpose graphic processing unit (GPGPU)-accelerated Landau-Lifshitz-Gilbert (LLG) simulator and conventional finite element method (FEM) electromagnetic analysis enables thousands of R/W simulations required for GA to be performed within a practical amount of time. GA optimization of the write head showed recording density limitations and the impact of vertically scaling media/head-disk interface (HDI) dimensions on it. GA optimization of recording media showed a tradeoff relationship (trilemma) among MWW, SNR, and the thermal stability of media. The effectiveness of increasing complexity in media design such as the number of layers and the variable range of material properties on overcoming the trilemma were clarified. Design variables for optimization results scatter over relatively wide ranges, suggesting a wide variety of head-media designs. Finally, results of head-media co-designing suggested the importance of matching between write head and recording media.