# binmodel

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binmodel converts polynomial expressions involving binary and continuous variables to linear expressions by introducing additional variables and constraints.

### Syntax

[plin1,...,plinN,F] = binmodel(p1,...,pN,Domain)


### Examples

The following example solves a quadratic program with binary variables using a mixed integer linear programming solver, by first converting the quadratic function to a linear expression

x = binvar(5,1);
Q = randn(5);

p = x'*Q*x;
[plinear,F] = binmodel(p)
optimize(F,plinear)


Of course, for this to work, you need a mixed integer linear programming solver (or in the worst-case, BNB together with a linear programming solver).

Products between continuous and binary variables are also supported, but for the big-M modelling to work, you have to specify bounds on the continuous variables

x = binvar(5,1);
y = sdpvar(5,1);
Q = randn(5);
p = x'*Q*y;
[plinear,F] = binmodel(p,[-2 <= y <= 2]);
optimize(F,plinear)


### Comment

The derivation of the linear model is based on simple logic and big-M modelling. A product of two binary variables x and y is replaced with a new binary variable z and the constraints

[z <= x, z <= y, z>= x+y-1];


This idea can be generalized to arbitrary polynomials in binary variables. A product between a binary variable x and a continuous variable w, with known lower and upper bounds L and U, is replaced by a new continuous variable v and the constraints

[ L*x <= v <= x*U, L*(1-x) <= w-v <= U*(1-x)]
optimize(F,plinear)


This can be generalized to expressions having arbitrarily polynomial w.r.t the binary variable.