I am solving 2-variable system of first order conditions for an optimization problem with NLsolve. I know that the optimal values of the optimization problem satisfy some conditions. For example, the optimal values should satisfy T_1 = H_1^{-1}(K_1, x_1) < T_2 \leq H_2^{-1}(K_2/\delta, x_2). How can I supply that information on the inequalities to nlsolve? I want to enforce that T_1 < T_2 and also ideally speed up the computation of T_2 by providing bounds on it.

I’m not an expert on this!
As far as I know, NLsolve doesn’t seem to support nonlinear constraints. (Ok, as a hack, you could define your own Jacobian and maybe the solvers will then never go into the infeasible regions. But, I think that comes with the risk of messing up the methods as they are not supposed to be used in that way.)

I am afraid that solving it through JuMP may be slower. Remember that I will have to solve the same problem many times. I guess I can try it at the worst.

Actually, the solution for t1 is independent of the solution for t2, so I can use the (faster?) univariate methods in Roots.jl. Then I can find t2 with the same univariate approach.