Minimize $\sum^n_{i=1} c_i x_i$ subject to $\sum^n_{i=1} a_i x_i = b$ (a single constraint), $x_i \ge 0$.

  1. Derive a simple test for feasibility of this problem

  2. Assuming the optimal cost is finite, develop a simple method for obtaining an optimal solution directly.

For the first part, I have $\sum^n_{i=1} a_i x_i = b \iff \text {sgn}(b)=\text{sgn}(a_i)$ for some $i \in \{1,..,n\}$ or $b = 0$.

I have trouble obtaining this simple method for part two. Could someone help me out here?

I know $ -\infty < \sum^n_{i=1} c_i x_i$ and $\sum^n_{i=1} a_i x_i = b$.

  • 3
    $\begingroup$ What have tou tried and where did you get stuck? $\endgroup$ – Raphael Sep 23 '14 at 11:07

Hint: Since there are $n+1$ constraints on $n$ variables, there is a solution in which all but one constraint are tight. You can calculate all these solutions explicitly ("on paper"), and choose the best one.

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