 # Describe S = <mo fence="false" stretchy="false">{ ( u , t ) &#x2208;<!- Ayanna Trujillo 2022-07-01 Answered
Describe

using finitely many linear inequalities and equations.
It's easy to show S is convex. We can describe S as
$S=\left\{\left(u,t\right)\in {\mathbb{R}}^{n}×\mathbb{R}:\underset{x\in {\mathbb{R}}^{n},Ax\le b}{sup}{u}^{\prime }x\le t\le \underset{x\in {\mathbb{R}}^{n},Cx\le d}{inf}{u}^{\prime }x\right\}.$
Taking the supremum and infimum over infinitely many expressions is not allowed in the finitely many linear inequalities and equations. The LHS and RHS are solutions to linear programming problems and the hint is to use strong duality, but I couldn't find a way to make the number of conditions finite.
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Step 1
This is done with duality. Let me show one inequality:
$\underset{x}{max}\left\{{u}^{T}x:Ax\le b\right\}\le t$
$\underset{y}{min}\left\{{b}^{T}y:{A}^{T}y=u,y\ge 0\right\}\le t$
$\mathrm{\exists }y\ge 0:{b}^{T}y\le t,{A}^{T}y=u$

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