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I was watching the lecture videos on Discrete Optimization on Coursera and I came across something I couldn't understand. A solution is feasible if the following constraints (picture below) are satisfied.

enter image description here

I have been wondering if it possible for the solution to satisfy the constraint yet be unfeasible like in the image below ? Is there a simple proof (pictorial) that shows why the scenario below is impossible ?

I have been thinking it through and I am wondering if the magic lies when you add the slack variables which brings the 'plots' into a higher dimension and when you solve for m variables (assuming m constraints) at that higher dimension, you are guaranteed that the solution (at the original dimension) will always be a feasible solution.

enter image description here

Kong
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  • If it satisfy some constraints, it may be feasible/unfeasible, in can be observed in the picture. If it satisfy all constraints, it must be feasible because that's the very definition of feasibility. It's good to look for visualisations, but they can only offer you a limited vision up to $n=3$, so I prefer the algebraic explanations at the top. – GNUSupporter 8964民主女神 地下教會 Dec 16 '17 at 01:25

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