Maxima and minima of the function $f(x,y)=4xy$ on $M'=\{ (x,y) \in \mathbb{R}^2 |\max(|x|,|y|)\leq4 \}$

Question

I have shown that there are no extreme points in the inner part of $M'$.

So what is left is to find the maxima and minima of the function $f(x,y)=4xy$ on $M=\{ (x,y) \in \mathbb{R}^2 \mid \max(|x|,|y|)=4 \}$.

Intuitively I know that $|f(x,y)|\leq 4^3$ on $M$ and the extrema are attained, since $M$ is compact, and the points $(4,4)$ and $(-4,-4)$ are the maxima on $M$ and $(4,-4)$ and $(-4,4)$ are the minima. But the function is very "friendly" and one can see it immediately. I wonder how one could solve this with "harder" functions.

Usually I solve such problems with the method of Lagrange multipliers, however it does not seem to me that $M$ is a differentiable manifold, since $g(x,y)=\max(|x|,|y|)-4$ is not continuously differentiable. Is there a method to solve for extreme points under such side conditions?

Answer 1

Hint.

Examine the contour level plot for $f(x,y)$ in black, with the feasible region in light blue.

Answer 2

The approach I would suggest for more complicated functions / boundaries.

Check for critical points in the interior of the region.

$\frac {\partial F}{\partial x} = 0$ and $\frac {\partial F}{\partial y} = 0$

Test if these are max/min or saddles.

$4(F_{xx})(F_{yy})>(F_{xy})^2$

Check the boundary. i.e Find a parameterization of the boundary.

$x = g(t), y = h(t)\\ \frac {d}{dt} F(g(t),h(t)) = 0$

If the boundary is not smooth, check the corners.

Answer 3

Here is a rather full context that i read for my university course:

https://www.amazon.com/Introduction-Optimization-Edwin-K-Chong/dp/1118279018

To summarize it in here, first define the problem formulation: we seek for $$\min_{x\in\Omega}f(x)$$where $f(x)$ is our target function to be minimized and $\Omega$ as our feasible region created by intersection of all constraints. A feasible direction in a point of feasible region ,roughly say, is a direction whose to the point along the direction is possible only if we remain in feasible region. In the interior points, the feasible direction is all the vectors in the point for example for the feasible region $$\Omega=\{(x,y)|x\ge 0\}$$ the feasible direction for points $(x>0,y)$ is all the vectors emanating from the point and for points $(0,y)$ the feasible direction is a vector $(d_1,d_2)$ where $$d_1\ge 0$$.

First Order Necessary Condition (FONC): if $x_0$ is a local minimizer of $f(x)$ then$$d^T\cdot\nabla f(x_0)\ge 0\quad\,\quad\text{for all feasible }d$$

Second Order Necessary Condition (SONC): if $x_0$ is a local minimizer of $f(x)$ and $d^T\cdot\nabla f(x_0)= 0\quad\,\quad\text{for all feasible }d$ then$$d^T\cdot H_f(x_0)\cdot d\ge 0\quad\,\quad\text{for all feasible }d$$where $H_f$ denotes Hessian.

Second Order Sufficient Condition (SOSC): if for some point $x_0$ $$d^T\cdot\nabla f(x_0)\ge 0\\d^T\cdot H_f(x_0)\cdot d>0\quad\,\quad\text{for all feasible }d$$then $x_0$ is a local minimizer