# Difference between revisions of "Duality"

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=== Definition: === | === Definition: === | ||

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'''Primal'''<blockquote>Maximize <math>z=\textstyle \sum_{j=1}^n \displaystyle c_j x_j</math> </blockquote><blockquote>subject to: | '''Primal'''<blockquote>Maximize <math>z=\textstyle \sum_{j=1}^n \displaystyle c_j x_j</math> </blockquote><blockquote>subject to: | ||

− | <math>\textstyle \sum_{j=1}^n \displaystyle a_{i,j} x_j\lneq b_j \qquad (i=1, 2, ... ,m) </math><math>x_j\gneq 0 \qquad (j=1, 2, ... ,n) </math></blockquote>'''Dual'''<blockquote> | + | <math>\textstyle \sum_{j=1}^n \displaystyle a_{i,j} x_j\lneq b_j \qquad (i=1, 2, ... ,m) </math></blockquote><blockquote><math>x_j\gneq 0 \qquad (j=1, 2, ... ,n) </math></blockquote><blockquote> |

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+ | </blockquote>'''Dual'''<blockquote> | ||

Minimize <math>v=\textstyle \sum_{i=1}^m \displaystyle b_i y_i</math> | Minimize <math>v=\textstyle \sum_{i=1}^m \displaystyle b_i y_i</math> | ||

## Revision as of 18:49, 7 November 2020

Author: Claire Gauthier, Trent Melsheimer, Alexa Piper, Nicholas Chung, Michael Kulbacki (SysEn 6800 Fall 2020)

Steward: TA's name, Fengqi You

## Introduction

Every linear programming optimization problem may be viewed either from the primal or the dual, this is the principal of **duality**. Duality develops the relationships between one linear programming problem and another related linear programming problem. For example in economics, if the primal optimization problem deals with production and consumption levels, then the dual of that problem relates to the prices of goods and services. The dual variables in this example can be referred to as shadow prices.

The shadow price of a constraint ...

## Theory, methodology, and/or algorithmic discussions

### Definition:

**Primal**

Maximize

subject to:

**Dual**

Minimize

subject to: