Fast growing E. coli cells in glucose-aerobic conditions excrete fermentation by-products such as acetate. This phenomenon is known as overflow metabolism and can pose a major problem in industrial bio-processes. In this paper, we study optimal control strategies for feeding a fed-batch reactor subject to overflow metabolism. We consider that acetate has an inhibitor effect on the glucose uptake, and we also consider the cost associated to process duration. In our approach, using the Pontryagin Maximum Principle and numerical solutions we describe the optimal feeding policy that maximizes biomass productivity and minimizes the cost duration of the process. We show that a singular regime is possible, in which cells grow at a slow rate to prevent acetate formation. If the cost associated to the process is too high, only bang-bang solutions are allowed.