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Suppose we have a standard optimization problem. $A'$ is an optimal solution to the problem. If we add a constraint to our original optimization problem, and $A'$ satisfies the new constraint, then is $A'$ still optimal for the new optimization problem? If so, prove it.

My solution: Yes, $A'$ is still optimal for the new optimization problem. By contradiction, suppose it wasn't optimal for the new problem.

WLOG, assume the original problem was a minimization one, we have for the original problem for all $x$ in the original feasible region, $O$, $c' A'<c'x$. Since $A'$ is not optimal for the new problem, there exists some $x_1$ in the new feasible region, $N$, such that $c'x_1<c'A'$. However, since we just added a constraint and kept all the previous constraints, the new feasible region must be a subset of the original feasible region. Thus $x_1 \in$ O. But, this means that $A'$ is not optimal for the original problem, since $x_1$ is in the original feasible region, $O$ but $c'x_1<c'A'$. Thus, we have reached a contradiction.

Is this proof correct? Or is it incorrect or perhaps not rigorous enough? For reference the knowledge we have at our disposal is Bertsimas: Linear Optimization Chapter 1. Thanks.

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Yes, you have gotten the main idea if the standard optimization problem is a linear one.

Minor comment if the problem is restricted to linear programming:

If $A'$ is optimal, then we can write $$c'A' \color{blue}\le c'x.$$

Suppose not, let $f$ be the objective function and rather than $c'x$, write $f(x)$.

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  • $\begingroup$ thanks. I forgot that I should put $\leq$ and not $<$. The equals part slightly confuses me. What happens, in the case of equality if $c'A'=c'x$ then no contradiction is reached? $\endgroup$ – Boy Wonder Sep 9 '19 at 3:13
  • $\begingroup$ actually nevermind so in the first inequality I should write $c'A' \leq c'x$, but the second inequality in the proof is correct with $c'x_1<c'A'$? thanks $\endgroup$ – Boy Wonder Sep 9 '19 at 3:14
  • $\begingroup$ including the equality because it is possibel that $x=A'$ and also there can be multiple solution. Yes, suppose it is not optimal, then we can find $x_1$ that performs strictly better. $\endgroup$ – Siong Thye Goh Sep 9 '19 at 3:23
  • $\begingroup$ this might not be a proper question, but what happens if the new optimization problem has no optimal solutions, then supposing A' is not optimal, how can i be sure that i can find $x_1$ that performs better. now again im slightly confused. thanks $\endgroup$ – Boy Wonder Sep 9 '19 at 3:33
  • $\begingroup$ when we say that $A'$ is feasible but not optimal, it means there is a better solution, the optimal solution need not exists for the argument to work. In layman term, if you are a participant in a contest but not the best, means someone beat you, that person who beat you need not be the champion. $\endgroup$ – Siong Thye Goh Sep 9 '19 at 3:41

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