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Consider a $K\times 1$ vector $x$ and define $$ \mathcal{X}\equiv\{x\in \mathbb{R}^K: A*x\leq b\} $$ where $A$ is a $J\times K$ matrix containing known real scalars and $b$ is a $J\times 1$ vector containing known real scalars.

Is it correct to say that $$ \mathcal{X}\equiv [\min_{x\in \mathcal{X}} x_1, \max_{x\in \mathcal{X}} x_1]\times [\min_{x\in \mathcal{X}} x_2, \max_{x\in \mathcal{X}} x_2]\times ...\times [\min_{x\in \mathcal{X}} x_K, \max_{x\in \mathcal{X}} x_K] $$ where "$\times$" denotes Cartesian product?

Star
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No, you are suggesting the the feasible region must be rectangular but this is not true.

For example, we have have $x \ge 0, y \ge 0, x+y \le 1$. The feasible region is triangular.

It is not equal to $[0,1] \times [0,1]$.

Siong Thye Goh
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  • Thanks. Is it correct to say that by computing $[\min_{x\in \mathcal{X}} x_k, \max_{x\in \mathcal{X}} x_k]$ I only get the projection of the feasible set on the $k$-th dimension? – Star May 28 '20 at 14:35
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    yes, that is the projection. – Siong Thye Goh May 28 '20 at 15:09