Sum of exponential growth and decay Suppose we have the equation:
$$y(t_i) = C_0 + C_1 e^{-\lambda_1 t_i} + C_2 e^{\lambda_2 t_i}$$
where $C_0$, $C_1$, $\lambda_1$, $C_2$, and $\lambda_2 \ge 0$ 
This is equivalent to the summation of an exponential decay curve with that of exponential growth.  

Thus, one can see that 


*

*$C_0$ is equivalent to the sum of the asymptotic value for the decay part as $t \rightarrow \infty$ and the asymptotic values for the gowth part as $t \rightarrow -\infty$  

*$y(0) = C_0 + C_1 + C_2$

If I have a sample of n observations, what are some good starting approximations for the estimates of $C_1$, $\lambda_1$, $C_2$, and $\lambda_2$ which I can then pass into a non-linear optimization algorithm.  I've explored ODE solutions to these types of equations, but cannot find a good "least squares" type function of the observed data.
I am considering partitioning the observations close to the position of the inflection point in the data, and estimating the $\lambda_1, \lambda_2$ from the least squares solution to the log transform of the data points to the left and the right, and then solving for $C_0, C_1,$ and $C_2$ after substituting $\hat{\lambda_1}$ and $\hat{\lambda_2}$ into the above equation.
Any alternative solutions would be greatly appreciated.

UPDATE:
@JJacquelin gave an excellent answer.  I worked out the solution myself to confirm that the equations hold.  
Here is an illustration:


However, there is at least one instances where the approach breaks down.


*

*The expression for $p \ or \ q = \frac{B \pm \sqrt{B^2 + 4A}}{2}$ can't be calculated when $B^2 + 4A < 0$.





In addition :  With your data publish latter, the numerical calculus and result is :

