ALGORITHMS FOR QUADRATIC FRACTIONAL PROGRAMMING PROBLEMS

Abstract

Consider the nonlinear fractional programming problem max{f(x)/g(x)|xεS}, where g(x) > 0 for all xεS. Jagannathan and Dinkelbach have shown that the maximuJn of this problem is equal to ξ^0 if and only if max{f(x)-ξg(x)|xεS} is 0 for ξ=ξ^0. Based on this result, we treat here a special case: f(x)=1/2 x^t Cx +r^t x +s, g(x)=1/2x^t Dx +p^t x +q and S is a polyhedron, where C is negative definite and D is positive semidefinite. Two algorithms are proposed; one is a straightforward application of the parametric programming technique of quadratic programming, and the other is a modification of the Dinkelbach's method. It is proved that both are finite algorithms. In the computational experiment performed for the case of D=0, the followings are observed: (i) The parametric programming approach is slightly faster than the Dinkelbachts, but there is no significant difference, and (ii) the quadratic fractional programming problems as above can usually be solved in computation time only slightly greater (about 10-20%) than that required by the ordinary (concave) quadratic programming problems.