Summary The problem is considered of maximizing a function in a convex region. to solve this problem a new method is developed, to be called “method of feasible directions”. It is a method of steep ascent. Starting with a feasible solution a sequence of feasible trial solutions with ever-increasing values for the objective function is obtained. This sequence will converge to a (local) maximum solution in all cases of importance. to obtain a new trial solution from an old one we have to determine (a) a direction in which the objective function increases and the solution remains feasible and (b) the length of the step to be taken in this direction. The method of feasible directions is first developed for problems with linear constraints and applied to the special cases of a linear and a quadratic objective function. In the latter two cases the number of steps is finite. An extension of the method to problems involving non-linear constraints is also considered.