i have a problem
Give h1, h2, w1, w2, w3 >= 0
h1*w2 = 2
h2*w1 = 4
h2*w3 = 4
find minimum of (h1+h2)*(w1+w2+w3)
Is this convex optimization ?

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.

Explore a convex optimization problem where the goal is to minimize (h1 + h2)(w1 + w2 + w3). Learn the step-by-step solution and the application of the AM-GM inequality in constraint optimization. Suitable for advanced mathematics enthusiasts.

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