Solving continuous Minimax Optimization problem

Question

I want to solve a linear programming minimax problem here mathematically without using software:

$$\begin{align*} \text{min}\ \text{max} \quad & \{x_1,x_2,x_3\} \\ \text{s.t.} \quad & x_1 + x_2 + x_3 = 15 \end{align*}$$

Or it can be written

$$\begin{align*} \text{min} \quad & Z \\ \text{s.t.} \quad & x_1 + x_2 + x_3 = 15 \\ & Z \ge x_1 \\ & Z \ge x_2 \\ & Z \ge x_3 \\ \end{align*}$$

I was wondering if someone could help me or provide me with good lecture notes having an explanation with examples?

Edited: The above problem seems solvable by inspection method, but if we consider the following problem we can't get the optimal solution by inspection (optimal solution I have obtained using the software is $x_1=0$, $x_2=0.39216$, $x_3=0.29412$, $x_4=0.31373$ and $z=-1.1765$, but I don't know how to solve it manually/mathematically, as well): $$\begin{align*} \text{min} \quad & Z \\ \text{s.t.} \quad & x_1 + x_2 + x_3+x_4 = 1 \\ & Z \ge x_1-3x_2 \\ & Z \ge x_1-4x_3 \\ & Z \ge x_1-7x_4 \\ & Z \ge x_2-5x_4 \\ & Z \ge x_3-5x_4 \\ & x_1,x_2,x_3,x_4\ge 0 \\ \end{align*}$$

Answer 1

Although this is a linear programming problem, it can really be solved by inspection. Think about how you'd solve the problem if there were only two variables, i.e.:

$$\begin{align*} \text{min}\ \text{max} \quad & \{x_1,x_2\} \\ \text{s.t.} \quad & x_1 + x_2 = 15 \end{align*}$$

Now can you extend your approach to handle 3 variables?

Answer 2

The optimal solution is 15/3 for the three variables. Any other assignment is such that at least one of the variables takes a value larger than 15/3.

Answer 3

For a proof that the solution $x=(5,5,5)$ is optimal, use a dual multiplier $1/3$ for each constraint: \begin{align} \frac{1}{3}(x_1 + x_2 + x_3) &= \frac{1}{3}\cdot 15 \\ \frac{1}{3}Z &\ge \frac{1}{3}x_1 \\ \frac{1}{3}Z &\ge \frac{1}{3}x_2 \\ \frac{1}{3}Z &\ge \frac{1}{3}x_3 \\ \end{align} Adding these up yields $Z \ge 5$.