sum of absolute values constraint in semi definite programming

Question

I want to further my real world semi definite programming optimization problem with a constraint on sum of absolute values. For example:

abs(x1) + abs(x2) + abs(         


        

Answer 1

As an alternative to Warren's solution involving 2^n constraints for a sum of n absolute value terms, one could introduce n extra variables y1, y2, ..., yn and write the following n pairs of inequalites

-y1 <= x1 <= y1
-y2 <= x2 <= y2
...
-yn <= xn <= yn

which, combined with a single equality

y1+y2+...+yn = 10

are equivalent to the original constraint

abs(x1) + abs(x2) + ... + abs(xn) <= 10

Total cost: n new variables and 2n+1 linear constraints.