To understand how the solution that maximizes the cost function is found, let's analyze the provided solution step-by-step:

Problem Statement

The problem involves solving a constrained optimization problem: $\min J(u_1, u_2) = e^{-(x_1^2 + x_2^2)}$ subject to: $f(x_1, x_2) = (x_1 - 1)^2 + 2x_2 = 3$

Hamiltonian Equation

To solve this problem, the Hamiltonian $H$ is defined as: $H = J + \lambda f$ where $\lambda$ is the Lagrange multiplier.

Solution Process

1. Define the Hamiltonian: $H = e^{-(x_1^2 + x_2^2)} + \lambda ((x_1 - 1)^2 + 2x_2 - 3)$

2. First-order necessary conditions: To find the critical points, we take the partial derivatives of $H$ with respect to $x_1, x_2,$ and $\lambda$, and set them to zero.

$\frac{\partial H}{\partial x_1} = -2x_1 e^{-(x_1^2 + x_2^2)} + \lambda \cdot 2(x_1 - 1) = 0$ $\frac{\partial H}{\partial x_2} = -2x_2 e^{-(x_1^2 + x_2^2)} + \lambda \cdot 2 = 0$ $\frac{\partial H}{\partial \lambda} = (x_1 - 1)^2 + 2x_2 - 3 = 0$

3. Solve the system of equations: The equations above form a system of nonlinear equations. Solving them step-by-step:

• From $\frac{\partial H}{\partial x_2} = 0$: $-2x_2 e^{-(x_1^2 + x_2^2)} + 2\lambda = 0$ $\lambda = x_2 e^{-(x_1^2 + x_2^2)}$

• Substitute $\lambda$ in $\frac{\partial H}{\partial x_1} = 0$: $-2x_1 e^{-(x_1^2 + x_2^2)} + (x_2 e^{-(x_1^2 + x_2^2)}) \cdot 2(x_1 - 1) = 0$ $-2x_1 + 2x_2(x_1 - 1) = 0$ $-2x_1 + 2x_1 x_2 - 2x_2 = 0$ $2x_1 x_2 - 2x_1 = 2x_2$ $x_1 (x_2 - 1) = x_2$ $x_1 = \frac{x_2}{x_2 - 1}$

• Using the constraint $(x_1 - 1)^2 + 2x_2 = 3$: Substitute $x_1$: $\left( \frac{x_2}{x_2 - 1} - 1 \right)^2 + 2x_2 = 3$ $\left( \frac{x_2 - (x_2 - 1)}{x_2 - 1} \right)^2 + 2x_2 = 3$ $\left( \frac{1}{x_2 - 1} \right)^2 + 2x_2 = 3$ $\frac{1}{(x_2 - 1)^2} + 2x_2 = 3$

This equation can be solved to find the values of $x_2$, and consequently $x_1$.

Maximizing the Cost Function

Once the critical points are found, the cost function $J$ needs to be evaluated at these points to determine the maximum. The solution involves verifying if the cost function $J = e^{-(x_1^2 + x_2^2)}$ has a maximum value at these critical points.

1. Calculate $J$ at critical points: Evaluate the cost function $J$ at the obtained critical points.

2. Compare values: The point at which $J$ attains its maximum value will be the solution that maximizes the cost function.

Conclusion

The final solution $(x_1, x_2)$ that maximizes $J$ is obtained after solving the above nonlinear system and verifying the cost function values at these points.

If you need more details on solving the nonlinear equation, please let me know!

Questions and Tip

1. What is the general form of the Hamiltonian in constrained optimization?
2. How do Lagrange multipliers help in solving constrained optimization problems?
3. What are the necessary conditions for optimality in constrained optimization?
4. How do you interpret the partial derivatives of the Hamiltonian?
5. Can you solve a simpler example of constrained optimization using the Hamiltonian approach?
6. How does the exponential function impact the optimization process in this problem?
7. What are the common methods for solving nonlinear equations like those in this problem?
8. What are potential pitfalls in finding critical points for constrained optimization problems?

Tip: When dealing with constrained optimization problems, always ensure to check both necessary and sufficient conditions for optimality to confirm whether the solution is indeed a maximum or minimum.