Outer-approximation (OA): Difference between revisions
| Line 17: | Line 17: | ||
<math display=block>0 \leq x_{2}\leq 4</math> | <math display=block>0 \leq x_{2}\leq 4</math> | ||
<math display=block>y_{1},y_{2} \in \big\{0,1\big\} </math> | <math display=block>y_{1},y_{2} \in \big\{0,1\big\} </math> | ||
'''Solution'''<br> | '''''Solution'''''<br> | ||
''Step 1a:'' Start from | '''Step 1a:''' Start from | ||
<math display=inline>y_{1}=y_{2}=1</math> and solve the NLP below: <br> | <math display=inline>y_{1}=y_{2}=1</math> and solve the NLP below: <br> | ||
''Minimize'' <math display=block> f= 2+ \big(x_{1}\big)^{2} +\big(x_{2}\big)^{2} </math> | ''Minimize'' <math display=block> f= 2+ \big(x_{1}\big)^{2} +\big(x_{2}\big)^{2} </math> | ||
| Line 29: | Line 29: | ||
<math display=block>0 \leq x_{1}\leq 4</math> | <math display=block>0 \leq x_{1}\leq 4</math> | ||
<math display=block>0 \leq x_{2}\leq 4</math> | <math display=block>0 \leq x_{2}\leq 4</math> | ||
''Solution:''<math display=inline>x_{1}=2, x_{2}=1, Upper Bound=7 | ''Solution: ''<math display=inline>x_{1}=2, x_{2}=1</math>, Upper Bound = 7 <br> | ||
==Conclusion== | ==Conclusion== | ||
==References== | ==References== | ||
Revision as of 06:30, 26 November 2021
Author: Yousef Aloufi (CHEME 6800 Fall 2021)
Introduction
Theory
Example
Minimize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)= y_{1} +y_{2} + \big(x_{1}\big)^{2} +\big(x_{2}\big)^{2} }
Subject to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \big(x_{1}-2\big)^{2}-x_{2} \leq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}-2y_{1} \geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}-x_{2}-3 \big(1-y_{1}\big) \geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}+y_{1}-1\geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{2}-y_{2}\geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}+x_{2}\geq 3y_{1}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{1}+y_{2}\geq 1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq x_{1}\leq 4}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq x_{2}\leq 4}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{1},y_{2} \in \big\{0,1\big\} }
Solution
Step 1a: Start from
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y_{1}=y_{2}=1}
and solve the NLP below:
Minimize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f= 2+ \big(x_{1}\big)^{2} +\big(x_{2}\big)^{2} }
Subject to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \big(x_{1}-2\big)^{2}-x_{2} \leq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}-2 \geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}-x_{2} \geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1} \geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{2}-1 \geq 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}+x_{2} \geq 3}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq x_{1}\leq 4}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq x_{2}\leq 4}
Solution: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x_{1}=2, x_{2}=1}
, Upper Bound = 7