Here s are the optimization variables and h and w are vectors of compatible sizes.
Let us consider, the parameters \alpha, \beta, \phi, h and w are known.

You can let CVX invoke its successive approximation method (with appropriate warnings and caveats) for exp in both the objective and constraints and write the CVX code virtually identically to your formulation above, presuming that phi > 0 , which is necessary for the constraints to be convex.

% h and w of compatible dimensions with code below
cvx_begin
variable s(T)
minimize(exp(alpha'*s))
for t=1:T
1/2*((exp(alpha(t)*s(t))-1)/phi(t)+phi(t)*beta(t)^2) <= h(:,t)'*w(:,t)
end
cvx_end

Any vectorization is left as an exercise for the OP

@Mark L. Stone: Thanks for your reply. Please consider my answer to my own question. Please let me about the computational complexities or the solver running timeā¦