Constrained optimization

Summary

• $f$ is a real-valued continuous function of two variables on domain $A⊆R^2$ .
• The problem min/max $f(x,y)$ subject to $g(x,y)=c$ is called a two-dimensional constrained optimization problem .
• For a two-dimensional constrained optimization problem the Lagrangian $L$ is defined as the function

$$L(x,y)=f(x,y)-λ(g(x,y)-c)$$

• $λ$ is called the Lagrangian multiplier .
• First order conditions: Suppose that $(x_0,y_0)$ is a local interior extreme point of $f(x,y)$ subject to $g(x,y)=c$ . Then $(x_0,y_0,λ_0)$ must satisfy the following three equations:
• $L'_x\left( x_0,y_0 \right)=0$
• $L'_y\left( x_0,y_0 \right)=0$
• $g(x_0,y_0)=c$
• If $x^*\left( c \right)$ and $y^*\left( c \right)$ solve the constrained optimization problem for a given $c$ then the composite function

$$f^*\left( c \right)=f\left( x^*\left( c \right),y^*\left( c \right) \right)$$

• is called the value function or the indirect objective function .
• $f^*$ is differentiable then

$$\frac{df^*\left( c \right)}{dc}=λ^*\left( c \right)$$

• where $λ^*\left( c \right)$ is the solution for $λ$ for a given $c$ .
• We use this result to interpret $λ_0$ as the approximate increase in the objective function $f$ as $c$ increases by one unit.