Recently, I intended to minimize an objective function which involves the square root of binary variable after transformation. Most of the studies use the relaxation method to relax the binary variable into continuous variable between 0 and 1. However, this method cannot be applied to my problem, since the square root of the variable is still concave. Therefore, I want to know can I adopt the following transformation.

Original form:

\sqrt {{a_i}{B_i}}

where {a_i} \in \{ 0,1\} is binary variable and {{B_i}} is a constant.

Since {a_i} is a binary variable, can I take {a_i} out of the square root and relaxed it into \left( {0,1} \right) ?

{a_i}\sqrt {{B_i}} ,{\rm{ }}0 \le {a_i} \le 1

If it is right, the complexity of our optimization problem can be greatly reduced.

# Can I take the binary variable out of the square root?

The square root of binary variable a_i (which takes value 0 or 1) is equal to the variable a_i. So a_i\sqrt{B_i} = \sqrt{a_iB_i} However, there is still a binary variable, even though not inside a square root.

As to the merits of relaxing the binary variable, we can’t determine much without more information. In general, you can’t just relax a binary variable and expect to get the correct answer to the original problem. However, unless you are solving an SDP, binary variables can be declared as such in CVX, if Gurobi or Mosek are specified as the solver. How long it takes the solver to solve the problem is another matter.

Thank you Mark for your quick response, your suggestion is very helpful.