Probability & Statistics/Constraints in LP

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Question
Hello Sir,
I have an assignment due to firday, I'm supposed to find the optimal solution of the objective function, however, I'm still facing a problem. I have two constraints, but on the grapth they never intersect (in the upper right field since x1, x2 >= 0) is it possible for two constraints not to intersect? if it is, how can I find the optimal solution then?!

Thank you in advance

Answer
A linear program might not have the constraints intersect, for example:

x≥0
y≥0
y-3x≤0
2x-y≤5

The feasible region is not bounded, which means the objective function may be unbounded on that region and the max/min of the objective function may not exist. For example, P(x,y)=x+y.

However, it may still be possible. For example, if your region is:

x≥0
y≥0
x≤3

and your objective function is:

P(x,y) = x-y,

then the optimum is at (3,0). The rationale "if a maximum occurs, it occurs at a corner" is still valid, but if the feasible region is unbounded, this maximum might not occur at all. (Same for a minimum.)


Further reading:
https://mathprelims.wordpress.com/2009/06/05/lp-part-3-the-feasible-region/

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Clyde Oliver

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I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

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