In scenarios such as therapeutic modeling or pest control, one aims to suppress infective agents or maximize crop yields while minimizing the side-effects of interventions, such as cost, environmental impact, and toxicity. Here, we consider the eradication of persistent microbes (e.g., Escherichia coli, multiply resistant Staphylococcus aureus (MRSA-“superbug”), Mycobacterium tuberculosis, Pseudomonas aeruginosa) through medication. Such microbe populations consist of metabolically active and metabolically inactive (persistent) subpopulations. It turns out that, for efficient medication strategies, the two goals, eradication of active bacteria on one hand and eradication of inactive bacteria on the other, are in conflict. Using multiobjective optimization, we obtain a survey of the full spectrum of best solutions. We find that, if treatment time is limited and the total medication dose is constant, the application of the medication should be concentrated both at the beginning and end of the treatment. If the treatment time is increased, the medication should become increasingly spread out over the treatment period until it is uniformly spread over the entire period. The transition between short and long overall treatment times sees optimal medication strategies clustered into groups.