Multivariable optimization, n variables

Summary

• \(f\) is a real-valued continuous function of two variables on domain \(A⊆R^2\) .

Extreme value theorem:

• If \(A\) is closed and bounded then there exists a global maximum point \(\left( x_{max},y_{max} \right)\) in \(A\) and a global minimum point \(\left( x_{min},y_{min} \right)\) in \(A\) .
• If \(z_{max}=f\left( x_{max},y_{max} \right)\) and \(z_{min}=f\left( x_{min},y_{min} \right)\) then \(z_{min}≤f(x,y)≤z_{max}\) for all \(\left( x,y \right)∈A\) .
• The optimization points need not be unique. If \(f(x,y)=2x\) on \(0≤x≤1\) , \(0≤y≤1\) then \(\left( 0,y \right)\) are all minimum points and \(\left( 1,y \right)\) are all maximum points for any \(0≤y≤1\) .
• \(f\) need not be differentiable for the extreme value theorem to hold.

Concave and convex functions.

• If \(f\) is convex function defined on a convex domain and \(\left( x_0,y_0 \right)\) is a stationary point, then \(\left( x_0,y_0 \right)\) is a global minimum point.
• If \(f\) is concave function defined on a convex domain and \(\left( x_0,y_0 \right)\) is a stationary point, then \(\left( x_0,y_0 \right)\) is a global maximum point.
• If \(f\) is strictly convex / concave then \(\left( x_0,y_0 \right)\) is a strict global minimum / maximum point.