What is wrong about Abs[x]-a? Is it a bug?

Question

Why is the Out[74] not {1 - a, {x -> -1}}?

 Maximize[{Sqrt[x^2] - a, -1 <= x <= 1}, x]

{1 - a, {x -> -1}}

Answer 1

I suspect the reason this fails is because Mathematica does not know the derivative of Abs[x]. The reason behind this is Abs[x] is nowhere differentiable in the complex plane and Derivative and D take complex derivatives.

In Wolfram Alpha, we had to handle Abs ourselves assuming real variables since most people would want that. If you run WolframAlpha["derivative of Abs[x]"], you'll see the step-by-step solution specifies x is a real variable.

Since Derivative is not a protected symbol, we can add custom SubValues:

Derivative[1][Abs] = Sign;
Derivative[2][Abs] = Derivative[1][Sign] = DiracDelta;

Now Mathematica knows the (real) derivatives of these functions:

D[Abs[x], x]
(* Sign[x] *)

D[Abs[x], {x, 2}]
(* DiracDelta[x] *)

D[Sign[x], x]
(* DiracDelta[x] *)

Unfortunately this does not fix your Maximize issue:

Maximize[{Abs[x] - a, -1 <= x <= 1}, x]
(* Maximize[{-a + Abs[x], -1 <= x <= 1}, x] *)