The vertebrate immune system generates high-affinity antibodies to external antigens through a process of somatic hypermutation that takes place in germinal centers formed in the secondary lymphoid tissues. B cells proliferating in these germinal centers experience random mutations in the genes encoding the variable region of their immunoglobulin molecules and are subsequently selected for high-affinity binding to antigen. These germinal center reactions last for only about 2 weeks, yet in that time typically produce multiple point mutations resulting in affinity increases of factors of ten to a hundred or more. We have attempted to understand this extraordinary effectiveness by casting the problem of affinity maturation as an optimization problem in which a quantity that we call the total affinity is maximized as a functional of (\mu(t)), the mutation rate as a function of time. We have developed a single-compartment model for the process and an optimization algorithm based on the Pontryagin maximum principle. Our results show that the optimum mutation schedule is one with brief bursts of high mutation rates interspersed between periods of mutation-free growth. Though this result at first seems highly non-physiological, we show that, in fact, it provides a framework within which the anatomy and kinetics of the germinal center reaction can be understood.