How to Rewrite max(t, 0) Using a Minimum

For a real number $t$, the expression $\max\{t,0\}$ is the positive part of $t$: it equals $t$ when $t$ is nonnegative and equals $0$ when $t$ is negative. To write it using a minimum, use the general identity that a maximum is the negative of a minimum after both inputs are negated:

$$\max\{t,0\}=-\min\{-t,0\}.$$

The minus sign outside the minimum is part of the identity. Simply changing max to min does not preserve the value.

Why the identity works

For any real numbers $a$ and $b$,

$$\max\{a,b\}=-\min\{-a,-b\}.$$

Negating reverses the order: if $a\ge b$, then $-a\le -b$. Thus the smaller of $-a$ and $-b$ is $-a$; negating that minimum gives $a$, the original maximum. Taking $a=t$ and $b=0$ gives the desired formula.

You can also verify it by cases:

• If $t\ge 0$, then $-t\le 0$, so $\min\{-t,0\}=-t$. Therefore $-\min\{-t,0\}=t$.
• If $t<0$, then $-t>0$, so $\min\{-t,0\}=0$. Therefore $-\min\{-t,0\}=0$.

These are exactly the two values of $\max\{t,0\}$.

Worked examples

Take $t=4$. Then

$$-\min\{-4,0\}=-(-4)=4=\max\{4,0\}.$$

For $t=-3$, the same expression gives

$$-\min\{3,0\}=-(0)=0=\max\{-3,0\}.$$

The formula also works at the boundary $t=0$: both sides are zero.

Another equivalent form

A related identity is

$$\max\{t,0\}=t-\min\{t,0\}.$$

If $t\ge0$, the minimum is zero and the right side is $t$. If $t<0$, the minimum is $t$ and the right side is $t-t=0$. This form can be useful when an expression already contains $t$ and $\min\{t,0\}$.

Using the rewrite in optimization

The rewrite is an exact algebraic equality, so you can substitute $-\min\{-t,0\}$ wherever $\max\{t,0\}$ appears. For example, for $t=2x-1$,

$$\max\{2x-1,0\}=-\min\{1-2x,0\}.$$

At $x=0$, both expressions equal zero; at $x=1$, both equal one.

Be careful about what the optimization model requires. The graph $y=\max\{t,0\}$ is often represented by the inequalities $y\ge t$ and $y\ge0$. These describe the epigraph: they ensure $y$ is at least the maximum, but do not by themselves force equality. Equality follows in some models when the objective makes $y$ as small as possible; otherwise, additional modeling constraints may be needed. Replacing the maximum with its minimum identity changes none of these modeling requirements.

Common mistakes

• Dropping the outside minus: $\min\{-t,0\}$ equals $-t$ for positive $t$, not $t$.
• Negating only one input: The general identity requires negating both inputs before taking the minimum.
• Assuming a minimum of affine functions is interchangeable with a maximum: A minimum of affine functions is concave, while $\max\{t,0\}$ is convex. In general, you cannot express this positive-part function as a minimum of affine functions without the outside negative sign.