Difference between revisions of "Optimization with absolute values"

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==Numerical Example==
 
==Numerical Example==
  
<math>\min{|x_1| + 2|x_2| + |x_3|} </math><br>
+
<math>\min{|x_1| + 2|x_2| + |x_3|} </math>
<math>\ s.t.      x_1 + x_2 - x_3 \le 10</math>
 
  
<math>   x_1 - 3x_2 + 2x_3= 12</math>
+
<math> \begin{align}
 +
\ s.t. x_1 + x_2 - x_3 \le 10 \\
 +
x_1 - 3x_2 + 2x_3= 12
 +
\end{align}</math>
  
 
We replace the absolute value quantities with a single variable:
 
We replace the absolute value quantities with a single variable:
Line 29: Line 31:
 
The problem has now been reformulated as a linear programming problem that can be solved normally:
 
The problem has now been reformulated as a linear programming problem that can be solved normally:
  
<math>\min{ U_1 + 2U_2 + U_3} </math><br>  
+
<math>\min{ U_1 + 2U_2 + U_3} </math>
  
<math>\             s.t.     x_1 + x_2 - x_3 \le 10</math>
+
<math> \begin{align}
 +
\ s.t. x_1 + x_2 - x_3 \le 10 \\
 +
x_1 - 3x_2 + 2x_3= 12
 +
\end{align}</math>
  
 
<math> x_1 - 3x_2 + 2x_3 = 12</math>
 
<math> x_1 - 3x_2 + 2x_3 = 12</math>

Revision as of 16:44, 20 November 2020

Authors: Matthew Chan (mdc297), Yilian Yin (), Brian Amado (ba392), Peter (pmw99), Dewei Xiao (dx58) - SYSEN 5800 Fall 2020

Steward: Fengqi You


Numerical Example

We replace the absolute value quantities with a single variable:

We must introduce additional constraints to ensure we do not lose any information by doing this substitution:

The problem has now been reformulated as a linear programming problem that can be solved normally:

The optimum value for the objective function is , which occurs when and and .