An algorithmic approach is developed for solving the minimax continuous resource allocation problem with multimodal cost functions. Unlike previous research in the same area which developed solutions for the same problem by imposing restrictions on the cost functions, such as the assumptions of monotoniciy or convexity, this approach is applicable to problems with multimodal functions with a finite number of local extrema. Another significant advantage demonstrated by this approach is that it provides all the optimal solutions to the problem; in contrast to previous algorithms which provided a single optimal solution. When a further level of optimization using a second objective function is desired, one needs the entire set of optimal solutions as provided by the procedures of this thesis.