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.