one single value target variable I have a set of 20 observations, the target variable which is a binary variable (0 or 1), in this set has one unique value which is 1. 

I want to predict this variable for a large sample, so I want to use a regression algorithm. The problem is that the target variable take one value, there is no observation that can describe the 0 class. 

Is there any solution to handle this situation. 

Thank you in advance.
Welcome to the site! As Media has mentioned it is a classification problem not prediction problem. If you meant Logistic Regression then it makes sense. 

You need to remember one thing if the outcome is only one value then algorithm would also return the same value irrespective of anything. if you don't have negative cases then you cannot use Logistic regression and it doesn't know when to say 0 as the model wasn't trained on it. 

If possible using Business Matter Experts advice populate some dummy data for 0's and balance the data. Train your model and see what the model is going to say.

Else you cannot use any kind of classification algorithm.