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Consider the following problem

$$ \begin{equation} \begin{aligned} \min && x_2(5 + x_1) \\ s.t. && x_1x_2 &\geq 5 \qquad & (c_1)\\ && x_1^2 + x_2^2 &\leq 20 \qquad &(c_2) \end{aligned} \end{equation} $$

I introduce slack variables $s_1, s_2 \geq 0$ to transform the inequality constraints to equality constraints, and the problem becomes

$$ \begin{equation} \begin{aligned} \min && x_2(5 + x_1) \\ s.t. && -5 + x_1 x_2 - s_1 &= 0 \qquad & (e_1)\\ && 20 - x_1^2 - x_2^2 - s_2 &= 0 \qquad & (e_2)\\ && s_1, s_2 \geq 0 \end{aligned} \end{equation} $$

How does introducing $s_1$ and $s_2$ to my original constraints change the KKT system? Is the gradient of my first constraint

  1. $$ \nabla_x e_1 = \begin{bmatrix} x_2 \\ x_1 \\ -1 \\ 0 \end{bmatrix} $$
  2. $$ \nabla_x e_1 = \begin{bmatrix} x_2 \\ x_1 \end{bmatrix} $$

and if it's (2), how do I deal with the slack variables $s_1, s_2 \geq 0$?

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