An approximate approach for fractional programming with absolute-value functions

This paper considers the following fractional programming with absolute-value functions: (FP-A):minZ=α+∑j=1ncj|xj|β+∑j=1ndj|xj|=N(x)D(x)subjecttoAx=b,where xT∈Rn is unrestricted; bT∈Rm; α and β are the scalars; A is an m∗n matrix; cjs and djs are unconstrained in sign. In some cases when some of cjs are positive and others are negative, adjacent extreme point (simplex-type) methods [Oper. Res. 19/1 (1971) 120; Eur. J. Oper. Res. 141 (2002) 233; Oper. Res. 13/6 (1965) 1029; Fractional Programming, Heldermann Verlag, Berlin, 1988] cannot be used to solve the problem (FP-A). In view of this, this paper proposes an approximate approach to reaching as close as possible an optimal solution of the problem (FP-A). First, the problem (FP-A) is converted into an equivalent non-linear quadratic mixed integer programming with absolute value. Then the model is linearized using piecewise logarithmic program with some linearization techniques. The whole problem is then solvable using the branch and bound method. The numerical example demonstrates that the proposed model can easily be applied to problem (FP-A).