Mixed-integer linear fractional programming (MILFP): Difference between revisions
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<math>\begin{align} 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\\ | <math>\begin{align} 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\ge0,\quad \forall i \in I | m_i\ge0,\quad \forall i \in I\\ | ||
y_j\in {0,1},\quad \forall j \in J \end{align}</math> | |||
The properties of the objective function <math>\Q(x,y)</math> are shown as follows: | |||
# Numbered list item <math>\Q(x,y)</math> is (strictly) pseudoconcave and pseudoconvex over its domain. | |||
# Numbered list item The local optimal of <math>\Q(x,y)</math> is the same as its global optimal. |
Revision as of 20:15, 18 November 2020
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:
The properties of the objective function are shown as follows:
- Numbered list item is (strictly) pseudoconcave and pseudoconvex over its domain.
- Numbered list item The local optimal of is the same as its global optimal.