Logistic Regression is a nonlinear regression problem? I come to a statement that logistic regression is a non linear problem. How can one show this?
Is it possible to treat logistic discrimination in terms of equivalent linear regression problem?
 A: Recall that the Logistic regression model is a non linear transformation of $\beta^Tx$


*

*Probability of $(Y = 1)$: $p = \frac{e^{\alpha + \beta_1x_1 + \beta_2 
x_2}}{1 + e^{ \alpha + \beta_1x_1 + \beta_2 x_2}}$

*Odds of $(Y = 1)$: $ \left( \frac{p}{1-p}\right) = e^{\alpha + \beta_1x_1 + \beta_2 
x_2}$

*Log Odds of $(Y = 1)$: $ \log \left( \frac{p}{1-p}\right) = \alpha + \beta_1x_1 + \beta_2 
x_2$


So to answer your question, Logistic regression is indeed non linear in terms of Odds and Probability, however it is linear in terms of Log Odds.

A simple example
Fitting a logistic regression model on the following toy example gives the coefficients $\alpha = -5.05$ and $\beta = 1.3$ 
Plotting the probability $P(Y=1)$ as a function of $X$ clearly shows the non linear relationship 

The Odds of $Y$ being 1 given $X$ is also non linear

Finally the log odds of $Y$ being 1 is a linear relationship

See here for some more details: Calculating confidence intervals for a logistic regression 
A: For the first statement: logistic regression is used when a variable is dichotomous. Since the variable can assume only value 1 or 0, fitting a line assumes a linear relationship which cannot hold for dichotomous outcomes. It can be proved that the linear probability model will not be efficient and, furthermore, nothing ensures that the estimated dependent variable will be bounded between 0 and 1. The logit can solve these problem.
Please clarify your second statement. 
