A naive explanation of infeasibility and unboundedness

Specific Domains Optimization (Mathematical)
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1

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”.

2

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.