Question

What the best supplements claims to promote relaxation and relieve anxiety.

Answer 1

Supplements that 'claim' to promote relaxation and less anxiety include

Gamma-Aminobutyric Acid
L-Theanine
Lemon balm
Passionflower
Melatonin  

However, there are more ways to relive or help relieve anxiety and stress like

Meditation
Getting a massage
Taking a mindful shower
Going for a walk
Using your favorite lotion while using your 5 senses 
Playing a game, going out with friends, solve a puzzle or logic problems
Do something artsy
Reading
Just taking a break from the situation
Listen to music
Write a letter, write down your thoughts in a diary/journal, talk to someone supportive

I listed some of the main ones that are commonly used but just do something enjoyable. This wont solve your problems or completely make you feel better but it can give you a break which is something we all need at times.
Other things like antidepressants or things for anxiety may be needed if you are constantly feeling this, there could be a unknown disorder your battling through like, an anxiety disorder, or something of that nature.