# Equivalent representation in linear programming

I have a very simple linear problem: $$\begin{split}\min_{x}&\ x^2\\\text{s.t. }&\ a_1x_1+a_2x_2=b\end{split}$$ Suppose I want to write this problem equivalently as in Find the equivalent linear program. Unlike the problem in the link, I have equality. Can I write it equivalently as: $$\begin{split}\min_{x,\alpha,\beta}&\ x^2\\\text{s.t. }&\ a_1x_1=\alpha b,\ a_2x_2=\beta b,\ \alpha+\beta=1.\end{split}$$ The converse is intuitive: Given $$\{x,\alpha,\beta\}$$ feasible for the second problem, adding the first and second constraints gives the constraint of the first problem. But the forward part is not clear, especially because I have never seen an equality constraint written like this. Any help would be highly appreciated.

• What does $x^2$ mean in your objective? Commented Apr 11, 2022 at 17:46
• @RobPratt It does not signify anything other than the problem being convex (but I don't think it's necessary for the doubt ). Commented Apr 12, 2022 at 10:32

You are correct. Suppose $$x$$ is feasible. Then $$a_1x_1 + a_2x_2 = b \iff (a_1x_1 + a_2x_2 = \left(\alpha + \beta)b \text{ and } \alpha + \beta = 1\right)$$. Therefore any solution satisfying the latter conditions is also feasible.
1. $$\begin{split}\min_{x}&\ x\\ \text{s.t. }&\ x \geq 2 \\ & x \in \mathbb{R} \end{split}$$
$$\begin{split}\min_{x}&\ x\\ \text{s.t. }&\ x \geq 2 \\ & x\geq 1 \\ & x \in \mathbb{R}\end{split}$$
$$\begin{split}\min_{x,a}&\ x\\ \text{s.t. }&\ x \geq a \\ & a = 2 \\ & x \in \mathbb{R}\end{split}$$