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.