Modified Dinkelbach-type algorithm for generalized fractional programs with infinitely many ratios

In this paper, we extend the Dinkelbach-type algorithm of Crouzeix, Ferland, and Schaible to solve minmax fractional programs with infinitely many ratios. Parallel to the case with finitely many ratios, the task is to solve a sequence of continuous minmax problems, P(alpha(k))= min x is an element of X ( max t is an element of T [f(t) (x)- alpha(k)g(t) (x)]), until {a(k)} converges to the root of P(alpha)= 0. The solution of P(alpha(k)) is used to generate alpha(k+1). However, calculating the exact optimal solution of P(alpha(k)) requires an extraordinary amount of work. To improve, we apply an entropic regularization method which allows us to solve each problem P(alpha(k)) incompletely, generating an approximate sequence {alpha(k)}, while retaining the linear convergence rate under mild assumptions. We present also numerical test results on the algorithm which indicate that the new algorithm is robust and promising.