A 100 kW regenerative Brayton heat engine driven by the hybrid of fossil fuel and solar energy was considered for optimization based on multiple criteria. A thermodynamic model of such hybrid system was developed so that the power output, thermal efficiency and dimensionless thermo-economic performance with the imperfect performance of parabolic dish solar collector, the external irreversibility of Brayton heat engine and conductive thermal bridging loss could be obtained. Evolutionary algorithm based on NSGA-II (Elitist Non-dominated Sorting Genetic Algorithm) was employed to optimize triple-objective and dual-objective functions, where the temperatures of hot reservoir, cold reservoir and working fluid, the effectiveness of hot-side heat exchanger, cold-side heat exchanger and regenerator were considered as design variables. Using decision makings, including Shannon Entropy, LINMAP and TOPSIS methods, the final optimal solutions were selected from Pareto frontier obtained by NSGA-II. The results show that there exists an appropriate working fluid temperature to cause optimal solution under each given condition. The comparisons of triple-objective and dual-objective optimization with single-objective optimization indicate that multi-objective optimization can yield the more suitable results due to the lower deviation index from the ideal solution. In the analysis of triple-objective optimization, an expected result is obtained that the optimal values of the power out, efficiency and dimensionless thermo-economic performance of solar-dish Brayton system (68.65 kW, 0.2331 and 0.3077) are 22.6%, 34.9% and 18.4% respectively less than that of convectional Brayton heat engine. Finally, a range of functional relationship between the optimized objectives in Pareto frontier is fitted to provide more detailed insight into the optimal design of solar-dish Brayton system.