To find the minimum of $(h_1 + h_2)(w_1 + w_2 + w_3)$ under the constraints:

$h_1 \geq 0, \quad h_2 \geq 0, \quad w_1 \geq 0, \quad w_2 \geq 0, \quad w_3 \geq 0$

$h_1 w_2 = 2, \quad h_2 w_1 = 4, \quad h_2 w_3 = 4,$

we need to analyze the problem carefully to determine if it is a convex optimization problem and to find the minimum value.

Step-by-Step Solution

1. Express $h_1$ and $h_2$ in terms of $w_2, w_1,$ and $w_3$:

From the constraints: $h_1 = \frac{2}{w_2}, \quad h_2 = \frac{4}{w_1}, \quad h_2 = \frac{4}{w_3}.$

Since $h_2$ is expressed twice, set the two expressions for $h_2$ equal to each other:

$\frac{4}{w_1} = \frac{4}{w_3}.$

Thus, $w_1 = w_3$.

2. Rewrite the objective function:

$(h_1 + h_2)(w_1 + w_2 + w_3) = \left( \frac{2}{w_2} + \frac{4}{w_1} \right)(w_1 + w_2 + w_1).$

Since $w_1 = w_3$, we can simplify:

$= \left( \frac{2}{w_2} + \frac{4}{w_1} \right)(2w_1 + w_2).$

3. Substitute $w_1 = w_3$ into the objective function:

$= \frac{2(2w_1 + w_2)}{w_2} + \frac{4(2w_1 + w_2)}{w_1}.$

Simplify further:

$= \frac{4w_1 + 2w_2}{w_2} + \frac{8w_1 + 4w_2}{w_1}.$

$= \frac{4w_1}{w_2} + 2 + \frac{8w_1}{w_1} + \frac{4w_2}{w_1}.$

$= \frac{4w_1}{w_2} + 2 + 8 + \frac{4w_2}{w_1}.$

$= \frac{4w_1}{w_2} + 10 + \frac{4w_2}{w_1}.$

4. Minimize the function:

Consider the terms $\frac{4w_1}{w_2}$ and $\frac{4w_2}{w_1}$. By the AM-GM inequality:

$\frac{4w_1}{w_2} + \frac{4w_2}{w_1} \geq 2 \sqrt{\frac{4w_1}{w_2} \cdot \frac{4w_2}{w_1}} = 2 \times 4 = 8.$

Thus, the objective function is:

$\frac{4w_1}{w_2} + 10 + \frac{4w_2}{w_1} \geq 8 + 10 = 18.$

Conclusion:

The minimum value of $(h_1 + h_2)(w_1 + w_2 + w_3)$ is 18.

This minimum is achieved when $\frac{4w_1}{w_2} = \frac{4w_2}{w_1} \Rightarrow w_1 = w_2$. Thus, $w_1 = w_2 = w_3$, and consequently $w_1 = w_3 = 2$ and $w_2 = 1$.

Convexity:

To determine if this is a convex optimization problem, we observe that the objective function is not a quadratic function and involves products and reciprocals of the variables, which are not convex in general. However, after using the AM-GM inequality, the function suggests that it behaves well under optimization techniques, but strictly speaking, it is not convex in the traditional sense. Thus, it is not a standard convex optimization problem.

Follow-Up Questions:

1. What are other methods to solve non-convex optimization problems?
2. Can we use numerical optimization techniques for such problems?
3. What is the significance of convexity in optimization problems?
4. How does the AM-GM inequality help in optimization problems?
5. What are some real-life applications of such optimization problems?

Tip:

Always try to simplify the constraints and objective function when solving optimization problems; this often reveals the underlying structure and makes it easier to find a solution.