In order for a company to accuse an organization of willful infringement, or alleging that they are actively and knowingly infringing on a patent, must the patent holder first send the allegedly infringing company notice?

If so, how long after notice is sent until the alleged infringement can be called willful if it continues?

Or, can willful infringement apply to a company that simply failed to search?

In order for a company to accuse an organization of willful infringement, or allege that they are actively and knowingly infringing on a patent, must the patent holder first send the allegedly infringing company notice?

If so, how long after notice is sent until the alleged infringement can be called willful if it continues?

Or, can willful infringement apply to a company that simply failed to search?