# Minimize $f(x)=\max\{|x_1|,|x_2|\}$ subject to $g(x)=a_1x_1+a_2x_2+b\leq 0$

$$\begin{array}{ll} \text{minimize} & f(x) := \max \{ |x_1|,|x_2| \}\\\\ \text{subject to} & g(x) := a_1 x_1 + a_2 x_2 + b \leq 0\end{array}$$

where $f, g$ : $\mathbb{R}^{2}\rightarrow \mathbb{R}$, and $a_{1}, a_{2}, b > 0$.

Since the global minimum of $f$, $(0,0)$, does not satisfy the constraint, and since $f$ is convex, the minimum is attained on the boundary of $C:=\{x \mid g(x)\leq 0\}$. Hence, the minimizer is a point of the form

$$\left(\frac{-b-a_2x_2}{a_1},x_2\right)$$

Intuitively, I can see that at the minimum, we must have $x_1=x_2$, giving us $\left(\frac{-b}{a_1+a_2}, \frac{-b}{a_1+a_2}\right)$ as the minimizer. How can I show this rigorously?

Generalizing and rephrasing slightly, we have the following optimization problem in $\mathrm x \in \mathbb R^n$

$$\begin{array}{ll} \text{minimize} & \| \mathrm x \|_{\infty}\\ \text{subject to} & \mathrm a^\top \mathrm x \leq b\end{array}$$

where vector $\mathrm a \in \mathbb R^n$ has no zero entries. There are two cases to consider:

• If the origin is in the feasible region, then the minimum is attained at the origin.

• If the origin is not in the feasible region, then the minimum is attained at its boundary.

Hence, in the latter case, we have a least-norm problem in the $\infty$-norm

$$\begin{array}{ll} \text{minimize} & \| \mathrm x \|_{\infty}\\ \text{subject to} & \mathrm a^\top \mathrm x = b\end{array}$$

In $\mathbb R^2$, the feasible region of this least-norm problem is a line that is neither vertical nor horizontal (due to the restrictions on vector $\mathrm a \in \mathbb R^2$). We would like to find the point on this line that is closest to the origin in the $\infty$-norm. Since the level sets of $\| \mathrm x \|_{\infty}$ are squares centered at the origin, then the minimizer is a square's vertex and, thus, its entries have the same absolute value.

[ image source ]

Let $(x_1,x_2):=(y_1,y_2)$ be a minimizing pair of $f(x_1,x_2)$ under $g(x_1,x_2)\leq 0$. Clearly, $y_1\leq 0$ or $y_2\leq0$. If $y_1<y_2$, then $$(x_1,x_2)=\left(y_1+\epsilon,y_2-\frac{a_1}{a_2}\epsilon\right)$$ is a solution for all real numbers $\epsilon$. What happens if $\epsilon$ is a very small positive real number, say $$0<\epsilon\ll\frac{y_2-y_1}{1+\frac{a_1}{a_2}}\,?$$ Do something similar when $y_1>y_2$.

As we can observe in the attached figures for

$$\min\left(\max\left(|x|,|y|\right)\right)\;\;\mbox{s. t.}\;\; ax+by + c\le 0$$

with $a = -2,b = 1, c = 6$ first a spatial representation and second as the level surfaces and the feasible region in light blue,

if the origin is at the feasible region interior the the answer is $0$ otherwise the answer is at one of the solutions

$$ax+by+c=0\\ x+y = 0$$

and

$$ax+by+c=0\\ x-y = 0$$

or at

$$\min\left(\left|\frac{c}{a-b}\right|,\left|\frac{c}{a+b}\right|\right)$$