Thanks very much, but I am actually looking for Auxiliary Principle Technique of Variational Inequalities @Sudev

Are you referring to the "auxiliary problem principle", first devised by Guy Cohen in the 1980s? Here is one of his major papers:

http://link.springer.com/article/10.1007/BF00934554#page-1

My book "Nonlinear Programming and Variational Inequalities" has tons of stuff on exactly that type of methodology: 

http://link.springer.com/book/10.1007%2F978-1-4757-2991-7

Let me know if this is the sort of thing you are looking for, and I can give you more links.

Actually, the mixed variational inequality is not very special. It seems, all projection methods for the classical VI can be simply adjusted to the mixed VI by changing projection to the proximal operator.

Thank you all for your contributions, I appreciate it.

Dear prof Muhammad, you can explain your problem on variational inequalities in my email

[email protected]

best regards