In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics.

Absolute value equations (AVEs), characterised by the inclusion of absolute value functions in the variables, present inherently non‐smooth and computationally challenging problems that are ...

The second part shows that the matrix absolute-value function, the matrix semidefinite-projection function, and the matrix projective residual function are strongly semismooth.