This paper proposes a period representation for modeling the multidrug HIV therapies and an Adaptive Multimeme Algorithm (AMmA) for designing the optimal therapy. The period representation offers benefits in terms of flexibility and reduction in dimensionality compared to the binary representation. The AMmA is a memetic algorithm which employs a list of three local searchers adaptively activated by an evolutionary framework. These local searchers, having different features according to the exploration logic and the pivot rule, have the role of exploring the decision space from different and complementary perspectives and, thus, assisting the standard evolutionary operators in the optimization process. Furthermore, the AMmA makes use of an adaptation which dynamically sets the algorithmic parameters in order to prevent stagnation and premature convergence. The numerical results demonstrate that the application of the proposed algorithm leads to very efficient medication schedules which quickly stimulate a strong immune response to HIV. The earlier termination of the medication schedule leads to lesser unpleasant side effects for the patient due to strong antiretroviral therapy. A numerical comparison shows that the AMmA is more efficient than three popular metaheuristics. Finally, a statistical test based on the calculation of the tolerance interval confirms the superiority of the AMmA compared to the other methods for the problem under study.