Let's say I have a linear optimization problem with some uncertain parameters
Case 1: Uncertain Parameters are present independently in different constraints
$$maximize\ x_1+x_2$$
$$ax_1+bx_2\leq 6 $$
$$x_1,x_2\geq 0$$
$$\bar{a}=3, \bar{b}=5$$
where a & b are uncertain parameter $a\in [1.5,4.5 ]$($\pm 1.5$ from mean) and $b\in [3,7 ]$ ($\pm 2$ from mean).
I solved this using the robust optimization approach as given in this article.
Robust counterpart of above equation can be written as below.
$$maximize\ x_1+x_2$$
$$3x_1+5x_2+\Gamma z+ p_1 +p_2 \leq 6$$
$$z+p_1\geq 1.5y_1$$
$$z+p_2\geq 2y_2$$
$$y_1\geq x_1$$
$$y_2\geq x_2$$
$$z,x_1,x_2,y_1,y_2,p_1,p_2 \geq 0$$
Here $\Gamma$ is budget of uncertainty, $\Gamma \in [0,2]$ , upper limit of $\Gamma$ is maximum no of uncertain parameter.
Case2: Uncertain Parameter is being shared in multiple constraints
I solved above using simple Linear programming problem. But I am stuck when uncertain parameter is being shared across multiple constraints. For example
$$maximize\ x_1+x_2$$
$$ax_1+bx_2\leq 6 $$
$$x_1 \geq b+1$$
$$x_1,x_2\geq 0$$
$$\bar{a}=3, \bar{b}=5$$
where a & b are uncertain parameter $a\in [1.5,4.5 ]$($\pm 1.5$ from mean) and $b\in [3,7 ]$ ($\pm 2$ from mean).
Case 3: Uncertain Parameters are being multiplied
In another case when uncertain parameters are being multiplied to each other, for example
$$maximize\ x_1+x_2$$
$$ax_1+abx_2\leq 6 $$
$$x_1 \geq b+1$$
$$x_1,x_2\geq 0$$
$$\bar{a}=3, \bar{b}=5$$
where a & b are uncertain parameter $a\in [1.5,4.5 ]$($\pm 1.5$ from mean) and $b\in [3,7 ]$ ($\pm 2$ from mean).
What would be mathematical robust counterpart for case 2 and case 3?
I face the same question (for case 2) now. Could you please tell me whether you have addressed this problem?