Dear professor:

I have recently been dealing with state constraint with the expression as follows:

**a’x + sqrt(a’Pa) <= b** (1)

where the vector **x** and positive definite matrix **P** are decision variables, and **a**, **b** are constant vectors. In order to convex (1) without direct linearization, I introduce an augmented variable **r**, and transform (1) into following expression:

**a’x + r<= b** (2)

**a’ Pa - r^2 <=0** (3)

**r > 0**(4)

I want to consult whether the expression (3) is convex and whether this transformation is appropriate.

Thank you very much.