How to deal with $\ell_1$ norm equality constraint?

Hi, is it possible to change the equality constraint ||x||_1=1 to affine equalities? If so, how? Thanks!

FAQ: Why doesn’t CVX accept my problem? [READ THIS FIRST]

Yes, it’s not a convex problem. But is there some good way to deal with norm equality constraint (Besides gradient descent, exact penalty method and sequential convex programming )?

Yes, the easy way is to replace something like ||x||_1 = 1 with ||x||_1 <= 1. Odds are that at the solution, the constraint will be satisfied exactly, so it will have been equivalent. If you need to satisfy it, and this doesn’t work, then there is not an easy way. The ideas of “gadient descent, exact penalty method, sequential convex programming” will not in general help you find the global solution to a nonconvex problem.

Ok, may be that’s the thing I should try. Thanks!