Don’t start off worrying about vectorization, worry about whether the optimization problem is convex. It doesn’t look convex to me.
What you call constraints 1 through 3 are better thought of as being expression assignments (they woould be a total no-go as constraints because nonlinear equality constraints are not allowed by CVX or CVXR).
CVX does have a concave approximation for the log of the standard normal distribution function,log_normcdf. Note that you would use normcdf, not nomrpdf, in MATLAB because \Phi is the cdf, not pdf of a standard Normal. However, \sigma_j is a concave function of x_{ij} and I don’t believe that (C_j - \mu)/\sigma_j would be a concave argument needed to use log_normcdf. So even though you could take the log of the objective function, I believe you are out of luck with \Phi((C_j - \mu_j)/\sigma_j) in CVX and I suspect all other DCP type modeling toolds such as CVXR.
If you can come up with a convex formulation of your problem, or at least a good enough convex approximation, then perhaps you could use CVX. Otherwise, you may need to use a non-convex optimizer.
help log_normcdf
log_normcdf Logarithm of the cumulative normal distribution.
Y = log_normcdf(X) is the logarithm of the CDF of the normal
distribution at the point X.1 / x log_normcdf(X) = LOG( ------- | exp(-t^2/2) dt ) sqrt(2) / -Inf For numeric X, log_normcdf(X) is computed using the equivalent expression LOG(0.5*ERFC(-X*SQRT(0.5))). When X is a CVX variable, a a piecewise quadratic *approximation* is employed instead. This approximation gives good results when -4 <= x <= 4, and will be improved in future releases of CVX. For array values of X, the log_normcdf returns an array of identical size with the calculation applied independently to each element. X must be real. Disciplined convex programming information: log_normcdf is concave and nondecreasing in X. Therefore, when used in CVX specifications, X must be concave.
help cvx.max
Disciplined convex/geometric programming information: max is convex, log-log-convex, and nondecreasing in its first two arguments. Thus when used in disciplined convex programs, both arguments must be convex (or affine). In disciplined geometric programs, both arguments must be log-convex/affine.