# A naive explanation of infeasibility and unboundedness

**URL:** https://discourse.julialang.org/t/a-naive-explanation-of-infeasibility-and-unboundedness/127343
**Category:** Optimization (Mathematical)
**Tags:** tutorials, duality
**Created:** [March 25, 2025, 12:00pm UTC](https://discourse.julialang.org/t/a-naive-explanation-of-infeasibility-and-unboundedness/127343 "2025-03-25T12:00:44Z")
**Posts on this page:** 2
**Page:** 1

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### Author: ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)
#### Post date: [March 25, 2025, 12:00pm UTC](https://discourse.julialang.org/t/a-naive-explanation-of-infeasibility-and-unboundedness/127343/1 "2025-03-25T12:00:45Z")

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I demo the process “From unboundedness to an infeasibility system”

Say, let’s employ y \in \mathbb{R} as our **decision variable**. We know  
 +\infty = \sup \{y | y \ge 0\}   
Given any y \ge 0, we know  
0 = \inf \{yx |x \ge 0 \}  
Based on these 2 facts, we have  
+\infty = \sup\_{y \ge 0} \inf\_{x \ge 0} \{y + yx\} \le \inf\_{x \ge 0} \sup\_{y \ge 0} \{y(1 + x)\} =: \mathrm{RHS}  
Inspecting the inner layer of \mathrm{RHS}, we conclude that 1 + x \le 0 should hold, meanwhile y is bound to take 0. Therefore we have  
\mathrm{RHS} = \inf\_x \{0 | 0 \le x \le -1\}  
Thus far, we have proved why _it is a convention that_ “minimize an infeasibility system ends up with the value +\infty”.

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### Author: ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)
#### Post date: [March 25, 2025, 12:37pm UTC](https://discourse.julialang.org/t/a-naive-explanation-of-infeasibility-and-unboundedness/127343/2 "2025-03-25T12:37:48Z")

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Now let’s embark on the reverse direction.  
Suppose we have an **infeasibility system**.  
Whatever it is, we can reduce it to the following system  
\{x : 0 \le x \le -1\}  
Then we promote it to an inf-program, i.e., the \mathrm{RHS} in the last post.  
Then along the reverse direction we end up with an ( **unbounded** ) **dual program**  
\sup\{y | y \ge 0\}

To conclude, the strong LP duality holds (i.e. +\infty = +\infty), under the abnormal case (i.e. the **infeasible** and **unbounded** pair), as long as we adopt the convention that the OBJSENSE of the unbounded problem is `sup`, while the OBJSENSE of the infeasible system is `inf`.