First way (quick restart): After finishing any mission you will come to a screen that shows the scores and also it's the screen you are supposed to like the mission. And all you have to do is press (Y, and Right Trigger).

Second way (full restart, change settings, buy weapons and armor) For this restart you will need to fail the mission, any way works, easiest way is for everyone to die. Then when you're at the screen that shows the score click A to continue. Then at the main screen where it shows all of the missions, click restart and there you go!