Outer-approximation (OA)

Author: Yousef Aloufi (CHEME 6800 Fall 2021)

Example

Minimize

$$f(x)=y_{1}+y_{2}+{\big (}x_{1}{\big )}^{2}+{\big (}x_{2}{\big )}^{2}$$

Subject to

$${\big (}x_{1}-2{\big )}^{2}-x_{2}\leq 0$$

$$x_{1}-2y_{1}\geq 0$$

$$x_{1}-x_{2}-3{\big (}1-y_{1}{\big )}\geq 0$$

$$x_{1}+y_{1}-1\geq 0$$

$$x_{2}-y_{2}\geq 0$$

$$x_{1}+x_{2}\geq 3y_{1}$$

$$y_{1}+y_{2}\geq 1$$

$$0\leq x_{1}\leq 4$$

$$0\leq x_{2}\leq 4$$

$$y_{1},y_{2}\in {\big \{}0,1{\big \}}$$

Solution
Step 1a: Start from ${\textstyle y_{1}=y_{2}=1}$ and solve the NLP below:
Minimize

$$f=2+{\big (}x_{1}{\big )}^{2}+{\big (}x_{2}{\big )}^{2}$$

Subject to

$${\big (}x_{1}-2{\big )}^{2}-x_{2}\leq 0$$

$$x_{1}-2\geq 0$$

$$x_{1}-x_{2}\geq 0$$

$$x_{1}\geq 0$$

$$x_{2}-1\geq 0$$

$$x_{1}+x_{2}\geq 3$$

$$0\leq x_{1}\leq 4$$

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

Step 1a: Solve the MILP master problem with OA for 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^{*} =[2,1] }  :
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 f\big(x\big) =\big( x_{1} \big)^{2} +\big( x_{2} \big)^{2},~~ \bigtriangledown f\big(x\big)=[2x_{1}~~~~2x_{1}]^{T} ~~for~~x^{*} =[2~~~~1]^{T} } 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\big(x^{*}\big)+ \bigtriangledown f\big(x^{*}\big)^{T}\big(x-x^{*}\big)=5+[4~~~~2] \begin{bmatrix}x_{1}-2 \\x_{2}-1 \end{bmatrix}=5+4\big(x_{1}-2\big)+2\big(x_{2}-1\big)} 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 g\big(x\big)=\big(x_{1}-2\big)^{2}-x_{2},~~ \bigtriangledown g\big(x\big)=[2x_{1}-4~~~~-1]^{T}~~for~~x^{*} =[2~~~~1]^{T} }

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 g\big(x^{*}\big)+ \bigtriangledown g\big(x^{*}\big)^{T}\big(x-x^{*}\big)=-1+[0~~~~-1] \begin{bmatrix}x_{1}-2 \\x_{2}-1 \end{bmatrix}=-x_{2}}