Norm-2 equation constraint of vector

Hello everyone,

Recently I have met a optimization problem with norm-2 equation constraint, but I can’t find a good method to solve it.

For example, \boldsymbol{x}=(x_1,\cdots,x_n,\cdots,x_N) is the optimization variable, with element x_n between 0 and 1. Now a norm-2 equation constraint \Vert \boldsymbol{x} \Vert_2=1 is employed. How to solve this constraint, especially in CVX?

One method I think maybe add two inequality constraints to replace it.

\Vert \boldsymbol{x} \Vert_2 \leq 1

\Vert \boldsymbol{x} \Vert_2 \geq 1

Then utilizing successive convex approximation or majorization-minimization to approximate the second inequation (with \geq). It works in CVX with MOSEK, but converges too slowly. How can I solve it?

That may be about all you can do in CVX.

You can look at

for some tips some years back by Stephen Boyd.

Thank you for your response. Despite this reply being 13 years old, it remains highly relevant to my queation.
The patterns of history indeed tend to recur.