Add the following  constraint to your model 

U(U-1) =e=0 

U is real and 0

I agree with my colleague!

What about using a comparison with e.g. 0.5 ?

...that IP is a special case of NLP can be illustrated by the fact that the constraint xj ε{0,1} can be written as the nonlinear constraint xj(1-xj)=0, please see the book https://www.amazon.com/Introduction-Continuous-Optimization-Foundations-Fundamental/dp/0486802876, i hope to help you

sir, direct conversion from binary to continuous is not feasible. but it can be interpreted if the variable is multiplied by some probabilistic/ continuous variable....or for case o stage1 for true case1 stage 2 and so on! decision making algorithm!

with regards!

Thank you

Is there any Linear model?

In a system such as (AX

Dear Hossein,

Could you please give a simple example?

The well-known cutting plane method has been proposed by DFJ54 (see the reference).  Please find the attached GAMS file to see how to solve the TSP. In this method, in each iteration, a cut is added to the LP problem (in TSP the relaxed LP is Assignment Problem). I hope this example be useful.

It should be noted that as i know, there is not a general method to change binary variables with continuous variables plus additional constraints.

As another clear example, see http://homepages.rpi.edu/~mitchj/handouts/bc_eg/

Or see http://slideplayer.com/slide/8756298/

Dantzig, G., Fulkerson, R., & Johnson, S. (1954). Solution of a large-scale traveling-salesman problem. Journal of the operations research society of America, 2(4), 393-410.

Some remarks to Hossein Karimi's answers.

Linear optimization problems with totally unimodular matrix of constraint parameters are not really integer/binary problems. They have still continuous variables. They just have "guaranteed" integer/binary corner-point solutions because all the corner point (vertices) of the feasible set (the polytope of the system) have all their coordinates integer/binary. But if the problem with TUM constraints has multiple solutions (the same minimum/maximum for two or more corner points) then there are also infinitely many non-corner-point solutions which are not integer/binary (at least for some variables) but they are still valid optimal solutions with respect to the formulation of the  problem. 

Here is a example - the assignment problem Hossein mentioned about

https://en.wikipedia.org/wiki/Assignment_problem

Hi

To this purpose, in “Optimal day-ahead operational planning of microgrids” which is an article in Energy Conversion and Management , 2016, we proposed a general, simple and new mechanism. I believe it can be useful for you too.

With regards

Article Optimal day-ahead operational planning of microgrids

Yes, x is binary if and only if x - x^2 = 0.  So the binary condition can be replaced by a quadratic equation.

(This is a folklore result, noted by, e.g., Shor; Koerner; Poljak, Rendl, & Wolkowicz; Lovasz & Schrijver; Lemarechal & Oustry, etc.)

Unfortunately, the quadratic equation is non-convex, so the resulting continuous quadratic optimisation problem is just as difficult as the original 0-1 optimisation problem.  But this is to be expected, since the binary condition is inherently non-convex.

You should may e explain more clearly what you need.

A binary variable on a discrete space is always continuous ! (In any discrete topology any function is continuous).

Maybe what you need is related to fuzzy sets.

you could use a simple formula X=INT( {f}+0.5 ), where {f} is a fractional part of a positive continuous variable F, and INT is a function for  the nearest integer not exceeding the given value. If {f}

Alexander, you're right - binary variables can be expressed in the way proposed by you. But does it make calculations with binary variables any easier? 

Adam Letchford noticed that using the constraint "x - x^2 = 0" instead of "x binary" doesn't make calculations easier than dealing with the original constraint.

Przemysław, the original question was not how to make the optimization solution easier. The question did not provide the optimization problem details. The question was simply "...Is it possible to mathematically model a binary variable using continuous variables in the optimization problems? ...". I answered exactly the question how it was stated, and I did not guess what it was for, and what the author actually meant.... Thanks for your comment, anyway....

Alexander, you're right - you answered the original question correctly. It was just my supposition what the reason of posing the question was.  The only guess I had was to avoid using the binary/integer optimization by replacing the original formulation with the one which is computationally more efficient.

By the way, the replacement of the binary condition with the quadratic equation can be useful for constructing strong "semidefinite programming" relaxations of various combinatorial optimization problems.  See, e.g., this paper:

http://link.springer.com/article/10.1007/BF01100205