Lagrangian Optimization

Alg. Lagrangian Optimization.

Let:

• $f(x)$ the target function to optimize
• $g(x)=c$ is the constraint function
• $\lambda$ is the Lagrange multiplier.

1. (when optimizing for budget constraint) Do a Monotonic Transformation on the Utility Function to make the function easier to manipulate
2. The Lagrangian function is constructed to find the maximum or minimum of a target function subject to constraints:
   • The Lagrangian: $\mathcal{L}(\vec{x},\lambda )=f(\vec{x})-\lambda (g(\vec{x})-c)$
   • $\lambda$ is an unknown constant
3. The first-order necessary conditions (also known as KKT conditions) are found by taking the derivative of the Lagrangian with respect to all variables and the Lagrange multipliers, and setting them equal to zero:
   • For all $i$, $\frac{\partial L}{\partial x_i}=0$
   • $\frac{\partial L}{\partial \lambda }=0$
4. Feasibility condition:
   • Is it in the feasible region: $g(x)=c$?
5. Solve for $\vec{x},\lambda$ ← this is the optimal point

Example

Worked Example

![[Pasted image 20230905153050.png|Worked Example|625]] Utility Maximization