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

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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,\quad  y_j\in {0,1},\quad \forall j \in J \end{align}</math>
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       m_i\ge0,\quad \forall i \in I\\
 +
     
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      y_j\in {0,1},\quad \forall j \in J \end{align}</math>
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The properties of the objective function <math>\Q(x,y)</math> are shown as follows:
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# Numbered list item <math>\Q(x,y)</math> is (strictly) pseudoconcave and pseudoconvex over its domain.
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# 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:

  1. Numbered list item is (strictly) pseudoconcave and pseudoconvex over its domain.
  2. Numbered list item The local optimal of is the same as its global optimal.