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I want to find the maximal $\psi_1$ for the following linear programming problem: \begin{align} \max \frac{2}{3025}(525+121\psi_3 + 1089 \psi_4), \text{ s.t.} \\ \end{align} \begin{array} \text{0} \leq 1450 - 3267 \psi_3 - 5203 \psi_4 \\ \text{0} \leq \psi_3\\ \text{0} \leq \psi_4 \end{array}

Intuitively we want to make $\psi_3$ and $\psi_4$ as large as possible. My textbook states that:

Since the constraints are linear, it follows that either all the weight should be put on $\psi_3$ or $\psi_4$.

Why is that the case?

I have drawn the constraints, I see that the possible values should be inside a triangle. But why the optimal values should be one of the two corners of the triangle?

Victor
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    In a linear program, it's always the case that if there is an optimal solution (the LP isn't unbounded), then there is an optimal solution that lies at a corner of the feasible region. There may be additional optimal solutions (with the same optimal objective value) along edges or higher dimensional faces of the feasible region. – Brian Borchers Apr 06 '19 at 20:43
  • @BrianBorchers Do you have a proof or some kind of intuition of why this is the case? – Victor Apr 06 '19 at 20:48
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    This is a standard theorem in linear programming that is usually proved in courses on that subject. Chvatal's Linear Programming has a nice constructive proof based on the simplex method. – Brian Borchers Apr 06 '19 at 21:08
  • This topic (called the fundamental theorem of linear programming), is discussed here with some follow-up here. – David M. Apr 07 '19 at 18:26

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