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We formulate a materials design strategy combining a machine learning (ML) surrogate model with experimental design algorithms to search for high entropy alloys (HEAs) with large hardness in a model Al-Co-Cr-Cu-Fe-Ni system. We fabricated several alloys with hardness 10% higher than the best value in the original training dataset via only seven experiments. We find that a strategy using both the compositions and descriptors based on a knowledge of the properties of HEAs, outperforms that merely based on the compositions alone. This strategy offers a recipe to rapidly optimize multi-component systems, such as bulk metallic glasses and superalloys, towards desired properties.