Constraint of normal distribution whose parameters are functions of optimization variables

Update: I have solved the problem

I was trying to solve an optimization problem to get the cut-off values for a vector which has a linear dynamics with normal distributed increments.

Suppose I have a portfolio vector x_t, and return vector r_t. In particular r_t follows a multivariate normal distribution with mean vector Mu and covariance matrix Sigma. Then the portfolio value of next period is given by w_{t+1} = r_t’ *x_t. Since this is simply a linear combination of r_t, w_{t+1} should follow a one-dimensional normal distribution with mean = Mu’ * x_t and variance quad_form(x, Sigma)

Now I want to put a loss risk constraint like Pr( w_{t+1} < beta) < alpha. i.e the probability that portfolio value of next period is less than threshold beta cannot be bigger than significance level alpha.

As suggested by this post http://ask.cvxr.com/t/qfunc-and-erfc-in-cvx/32/3 I chose function log_normcdf to express the constraint as

log_normcdf(s) < log(alpha)

where s is the normalized value of beta, i.e s = (beta - Mu’ * x_t/ sqrt( quad_form(x, Sigma))

Now the problem is that “sqrt( quad_form(x, Sigma)” is not compliant with DCP ruleset since it’s a concave composition of a convex expression. I have also tried “norm(quad_form(x, Sigma))” which is not compliant either.

How do I get pass this please? I tried to find post or examples where people use constraint involving normal distribution but I failed. I’m trying to solve a problem similar to Example 4.8 on the book Convex Optimization. Please help!