Hello Morteza,

It is nonlinear. I am not sure if you can reformulate it such that the resulting formulation is linear!

If you provide a brief explanation of your problem and the formulation you have, that may help investigating for possible linear counterparts.

Since you refer to MILP and MINLP I assume the variables are in a mathematical program, right?

Indeed, multiplying (decision or utility) variables will make the model nonlinear. Irrespective of whether you multiply binary variables with one another (like quadratic assignment problems) or binary variables with integer variables. Even if you multiply two continuous variables with one another in the model in which there are binary variables the result is MINLP.

As Morteza Davari suggested, one could (possibly) linearize the expression, but that will be expression-specific.

Now: WHY is it nonlinear?

By definition, a "linear" expression of variables' value changes linearly with the change in a variable: change in output is linear for the change in input. When variables are multiplied, the output is a function of not the change in one, bit TWO variables. Straight line versus parabola, in its simplest form.

Hi Morteza

If you multiply variables of any sort the resultant mathematical formulation become nonlinear. (It follows from the definition of linearity)

In general it is computationally less expensive to solve linear formulations than nonlinear formulations. I would thus attempt to reformulate the problem so that it become linear again. Whether it can be done and how to do it, is formulation specific.

nonlinear but with a specific shape. It is or zero or real.

Yes it makes it nonlinear. However you can convert this type of expression to a linear discrete type by the use of new binary variables.

Many thanks for all of your worthwhile answers.

I am going to find an equivalent MILP model for NLP or MINLP problems and I have heard there are some well-known linearization methods in this regard.

Could you please recommend some useful papers or books to me about linearization methods?

Morteza

Your product just tries to express that the continuous variable should be zero if the binary variable is zero.

In the Framework of a MILP you could as well avoid the product by generalized upper bounds on the continuous variable - if it has a natural upper bound -- AND if the variable does not occur elsewhere without that product.

In that case just add the two inequalities

binary * lower_bound

Dear Dr. Gollmer,

I highly appreciate your worthwhile answer. Indeed, it is the exact answer I was looking for. Many thanks for your contribution.

Regards,

Morteza

Hi

In attach you can find a draft about Linearization Techniques

to solve nonlinear problems with MILP models

best regards,

jose barahona da fonseca.

Hello, Morteza.

It is a real variable conditioned to constraints, for example if x2 is real such that

 L

I think its become Integer/Discrete rather than Nonlinear as non-linearity is dependent on binary variable value to be one.

hi

consider A a real variable and flag a binary variable

if the constraint is for example

A*flag + B >= C

then this can be implemented by two constraints in GAMS syntax:

constr1.. B +  flag * M =G= C;

* if flag=0 this constraint is equivalent to B >= C

*flag= 1 this constraint is always true which is equivalent to be removed from the model

constr2.. A + B + (1-flag)*M =G=C;

*when flag=1 this constraint is equivalent to A+B >=C 

*when flag=0 this constraint is always true which is equivalent to be removed from the model

in attach you can find a draft with more linearization techniques

best regards,

jbf.

this way you solve a nonlinear problem, with a multiplication of two variables, with a linear MILP model without the multiplication of variables.

best regards,

jbf.

This will become a non-linear problem. Check the trick here to linearize it:

http://www.leandro-coelho.com/linearization-product-variables/

I think you want to have a continuous variable to become zero as long as the binary variable is zero. i.e. x_j takes values in the union of {0} and the interval [lb,ub]. This is called a semi-continuos variable and is handled by commercial solvers like CPLEX and Gurobi internally.

If there are a lower and an upper bound for x_j you could also model it yourself by the generalized upper bounds

lb*switch

Thanks a million for all your answers.

To find a linear expression for the production of a binary variable and a continuous variable, see the appendix of my recent paper:

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