# 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.