The object function of my optimization problem is a*l^3 + p, and l,p are the optiml variables, however, the magnitude of a is 10^(-28). The cvx cannot slove the problem, how should I deal with this problem.
If you need to maintain that relative relation between terms containing the two variables, I don’t think there is any rescaling which is likely to be adequate. Your problem may be beyond the ability of the double precision solvers called by CVX to handle. Your best chance with CVX might be to improve scaling as much as you can, then use Gurobi as solver and include the statement
which will cause at least some of Gurobi’s calculations to be done in quad precision.
Even quad precision might not be enough. Maybe octuple precision would be in order. But that will require something other than CVX.
Or just try to come up with a better formulation.