# Difference between revisions of "Mixed-integer linear fractional programming (MILFP)"

Author: Xiang Zhao (SysEn 6800 Fall 2020)

Steward: Allen Yang, Fengqi You

## Introduction

The mixed-integer linear fractional programming (MILFP) is a kind of mixed-integer nonlinear programming (MINLP) that is widely applied in chemical engineering, environmental engineering, and their hybrid field ranging from cyclic-scheduling problems to the life cycle optimization (LCO). Specifically, the objective function of the MINFP is shown as a ratio of two linear functions formed by various continuous variables and discrete variables. However, the pseudo-convexity and the combinatorial nature of the fractional objective function can cause computational challenges to the general-purpose global optimizers, such as BARON, to solve this MILFP problem. In this regard, we introduce the basic knowledge and solution steps of three algorithms, namely the Parametric Algorithm, Reformulation-Linearization method, and Branch-and-Bound with Charnes-Cooper Transformation Method, to efficiently and effectively tackle this computational challenge.

## Standard Form and Properties

Consider such standard form of the MILFP:

${\displaystyle \max \quad \mathbb {Q} (x,y)={c_{0}+\sum _{i}c_{1,i}m_{i}+\sum _{j}c_{2,j}y_{j} \over d_{0}+\sum _{i}d_{1,i}m_{i}+\sum _{j}d_{2,j}y_{j}}}$

{\displaystyle {\begin{aligned}s.t.\quad \ a_{0,k}+\sum _{i}a_{1,i}m_{i}+\sum _{j}a_{2,j}y_{j}=0,\quad \forall k\in K\\m_{i}\geq 0,\quad \forall i\in I\\y_{j}\in {0,1},\quad \forall j\in J\end{aligned}}}

The properties of the objective function ${\displaystyle \mathbb {Q} (x,y)}$ are shown as follows:

1. Numbered list item ${\displaystyle \mathbb {Q} (x,y)}$ is (strictly) pseudoconcave and pseudoconvex over its domain.
2. Numbered list item The local optimal of ${\displaystyle \mathbb {Q} (x,y)}$ is the same as its global optimal.