Norm-2 equation constraint of vector

[WLTJYHK]

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?

[Mark_L_Stone]

That may be about all you can do in CVX.

You can look at

How to handle nonlinear equality constraints?

mcg’s answer is correct; the problem is not convex, and cannot be expressed in DCP format. more specifically, a constraint like \|Aw\|_2 \leq P (with A a constant matrix and w a variable) is OK (convex, and DCP), but the converse constraint, \|Aw\|_2 \geq P, is not convex, and (therefore) not DCP. i’d like to add that there are heuristic methods for “handling” such problems, using the so-called convex-concave procedure (see EE364b, lecture on sequential convex programming). this is a heuris…

for some tips some years back by Stephen Boyd.

[WLTJYHK]

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.