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Consider $$\text{max} \ 5x_1+3x_2$$ $$s.t.\ 2x_1+x_2\le 8$$ $$3x_1+2x_2\ge 6$$ $$x_1,x_2\ge0$$

Change the objective function by another function such that the resulting program has infinite optimal solutions.


Any hint please?

How will I solve this?

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Make the objective function a constant and every feasible point is an optimal solution.

Remark: You still have to prove that the feasible set has infinitely many points. You might like to use convexity to prove this.

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  • $\begingroup$ How do I make the function a constant? $\endgroup$ – user441848 Feb 4 '18 at 8:01
  • $\begingroup$ Let the objective function be a constant, say $0$, $1$, or even $\pi$. That is every point is equally optimal. $\endgroup$ – Siong Thye Goh Feb 4 '18 at 8:02
  • $\begingroup$ $ \max 0x_1+0x_2$. $\endgroup$ – Siong Thye Goh Feb 4 '18 at 18:04
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You can visualize this very nicely, see https://www.desmos.com/calculator/jiukwzxdxt

If you change the objective function so that its contour lines are perpendicular to one of the edges of the feasible region, you also get infinitely many solutions.

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