I'm dealing with the following question:
d1=-1/b1*log((p1*t-a1)/c1)
d2=-1/b2*log((p2*t-a2)/c2)
d3=-1/b3*log((p3*t-a3)/c3)
where a1,a2,a3,b1,b2,b3,c1,c2,c3,p1,p2,p3 are known, and theoretically d1=d2=d3>0, so I try to find the proper t to minimize (d1-d2)^2+(d2-d3)^2+(d3-d1)^2. I try to use lmfit to do it, but don't know to impose the restrictions like d>0 and the thing in log is greater than 0, namely (p*t-a)/c>0. There are simple examples in lmfit's document, in my case I don't know how to do. Can anybody helps? Thank you very much.
Assuming you know the values of b1, b2, b3, c1, c2, c3, you can place bounds on the value of t so that d1>0. If b1 is positive,
d1>0 means that log((p1*t-a1)/c1) > 0, so that the argument to the log function is greater than 1 (not only "greater than 0"). That means p1*t-a1 > c1 or t > (c1 + a1)/p1, and so on. So, you can create a Parameter for t and set its minimum value to be the largest of [(c1+a1)/p1, (c2+a2)/p2, (c3+a3)/p3] to satisfy d1,d2,d3>0.