Successive upper approximation methods for generalized fractional programs

The majorization approximation procedure consists in replacing the resolution of a nonlinear optimization problem by solving a sequence of simpler ones, whose objective and constraint functions upper estimate those of the original problem. For generalized fractional programming, i.e., constrained minimization programs whose objective functions are maximums of finite ratios of functions, we propose an adapted scheme that simultaneously upper approximates parametric functions formed by the objective and constraint functions. For directionally convex functions, that is, functions whose directional derivatives are convex with respect to directions, we will establish that every cluster point of the generated sequence satisfies Karush-Kuhn-Tucker type conditions expressed in terms of directional derivatives. The proposed procedure unifies several existing methods and gives rise to new ones. Numerical problems are solved to test the efficiency of our methods, and comparisons with different approaches are given.