Optimum upper and lower flux bounds are sought for a general space-time reactor problem. The bounds are much narrower than previous bounds. Each bound is a sum of the products of known spatial modes and unknown time-dependent amplitude functions. To determine a bound, the amplitude functions must satisfy certain inequalities given by a comparison theorem of the Nagumo-Westphal type. An optimum bound is one that satisfies the inequalities and minimizes a “payoff function. In this paper, the payoff function is the weighted average of the magnitude of the bound at several points in the reactor. It is shown that an optimum bound can be determined by solving a linear programming problem at each time step. (Linear programming can be used even if there is feedback and the problem is nonlinear.) Using linear programming theory it is shown that an optimum bound always exists, although it may not be unique. Furthermore, an optimum bound satisfies the original space-time equation at each point in the reactor sampled by the payoff function. In an example, narrow bounds are determined for a difficult example in which the spatial shape of the flux changes radically with time.