Simplex-like sequential methods for a class of generalized fractional programs

Abstract

A sequential method for a class of generalized fractional programming problems is proposed. The considered objective function is the ratio of powers of affine functions and the feasible region is a polyhedron, not necessarily bounded. Theoretical properties of the optimization problem are first established and the maximal domains of pseudoconcavity are characterized. When the objective function is pseudoconcave in the feasible region, the proposed algorithm takes advantage of the nice optimization properties of pseudoconcave functions; the particular structure of the objective function allows to provide a simplex-like algorithm even when the objective function is not pseudoconcave. Computational results validate the nice performance of the proposed algorithm.

Appendix

In this appendix a detailed description of the algorithm is presented. In particular, the main procedure is aimed to initialize the process and to split the visit of the feasible region with respect to the sign of the linear function $c^Tx+c_0$. As we need to follow different steps according to the generalized convexity property of f, we will distinguish two subprocedures: Visit1 finds maximum points when f is pseudoconcave, while Visit2 deals with the pseudoconvex case. It’s worth remarking that we consider the case $\alpha \ne \beta$, since for $\alpha = \beta$, solution methods have been already proposed (see for instance Cambini and Sodini 2010a).

Denote by $x_b$ an incumbent maximum point of P and by $U_b$ the incumbent optimal value of P. Set:

$$\begin{aligned} C_{\max }= & {} \underset{x\in X}{\sup }(c^{T}x+c_{0}) \quad \quad \quad \quad C_{\min }=\underset{x\in X}{\inf }(c^{T}x+c_{0}) \\ \theta _{\min }= & {} \underset{x \in X}{\min \;} (d^Tx+d_0)\quad \quad \quad \quad \theta _{\max }=\underset{x \in X}{\sup \;} d^Tx+d_0 \\ X_{<}= & {} \{x \in X: c^Tx+c_0 < 0\} \end{aligned}$$

We are ready to present the main algorithm and the two subprocedures Visit1 and Visit2; to get them more readable, some comment rows have been added.