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I am not able to understand the part "without solving show that it has an optimal solution"

Please help.

Thanks

P.S: Question in attachment enter image description here]1

AJ_
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Hint:

Show that the feasible set is non-empty and show that the feasible set is compact.

Think of what are the values that $x_i$ can take? Try to deduce a possible range from let say, the first and third constraint. Show that it is bounded.

Siong Thye Goh
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  • The question is supposed to be answered in simple concepts, so if I show Z is bounded by showing that each xi is less than a particular value and more than 0 then I get Z to be bounded. Then it means Z HAS to have an optimal solution right? – AJ_ Oct 26 '17 at 07:13
  • more than zero? nothing wrong with equal to zero. but yup, show that is is bounded and closed (hopefully proven in your class), optimizing a continuous function over a compact set has an optimal solution. – Siong Thye Goh Oct 26 '17 at 07:27
  • Thanks for above, also how do I prove it's closed. Nowhere in my class notes unfortunately. – AJ_ Oct 26 '17 at 10:46
  • construct a sequence of feasible points that converges. show that the limit is feasible as well. – Siong Thye Goh Oct 26 '17 at 17:28
  • the answer to the first question tells you that the feasible set is not empty (for example $x=(0,2,0,6)$ is a basic feasible solution). So, for the second question you have to prove that the objective function is bounded from above all over the feasible set. One way is to prove that all the variables are bounded from above. Have a look to the first constraint....... – Marcello Sammarra Oct 26 '17 at 18:13