Hi, you may refer to the following discussion:

https://www.researchgate.net/post/How-binary-variables-can-be-modeled-using-continuous-variables

Best

I found this RG link, kindly check, it may be useful:

https://www.researchgate.net/post/Can-binary-variables-be-transformed-to-continous-variables-in-MILP

thank you

Two simple examples of mapping continuous variables into two valued ones.

Suppose a point moves along straight line under continually acting displacement vector. The result of measurement is fraction part of point position. If the observation result is defined as 0 when the fraction is less than half, and as 1 when it is greater than half we get exactly (0,1) binary result.

More complicated example of a tossed coin that is closer to quantum mechanics. The wave function in that case is R=exp(1/2*I*omega*t). It acts on the bivector of the coin plane O as operator of rotation inv(R)*O*R. The result of observation is binary valued variable: which side of the coin is seen from given direction.

Respoding to Dr Alexandre.

The method of rounding down or up if continuous output value is less or more than 0.5 is inaccurate and would directly lead to infeasable solutions.

To transform liquidity values of the continuous variables to crystalize correctly to the integer binary value (0 or 1), we need to define a tiny tolerance value delta. if the difference between 0 and continuous solution value or between 1 and the associated continuous solution value is less than tolerance delta, then you fix (assign) 0 or 1 to this variable while checking if any constraints are violated and while observing improvement of the lower bounds in general branch-and-bound algorithm. If any constraints are violated, check which variables can be swapped and which ones causing infeasibility and recrystalize.

To answer Mr Anas

Yes you relax the integrality constraints on the binary variables then you solve the continous problem with simlpex or interior point and cast this relaxation in a branch-and-bound algorithm.

At each iteration of the B&B you devise what i call a crystalization methodology whereby continuous values change phase from liquid-continuous to crystal solid integers 0 or 1 while verifying feasibility (no constraint violations). Make your tolerance dynamic so the algorithm can vary its tiny value depending on lower bound progress at every B&B iteration. That is it.

large-scale problems involving millions of vars and constraints in airline optimized planning and scheduling, in petrochemical industries, in automotive industries and construction etc... have been solved to optimality and solution time depends on how we parallelize our COMPUTERS.

Binary