I am looking to How do you solve for the optimum values with the lowest MSE for the coefficients and dampening constant in exponential smoothing and ARIMA models? What are the equation used?
ARIMA coefficients are estimated using Log Likelihood function. For background, you can consult either the classic text book by Box, Jenkins & Reinsel or much more accessible book by Pankratz. I tried to post an example in this website which also has references to lecture notes. I used R optim function to solve for ARMA parameters.
As far as Exponential smoothing, there are two publications that I know of, both in the Omega - International Journal of Management Science show how to solve for optimal Parameters using spreadsheet. Both the papers give you very clear numerical illustrations with equations (see below for links). If you would want to get indepth on equations I would recommend an excellent text by Spyros G. Makridakis,Steven C. Wheelwright, Rob J Hyndman or this free text book by Hyndman and Athanasopoulos. I like them both.