Modelling on/off behaviour leads to poor performance

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An extremely common mistake beginners make in the development of models is that they have binary variables representing some type of on/off behaviour, and then multiply other variables with this binary to represent things being turned off.

Typical model could for instance be something along the lines of

x = sdpvar(N,1);
u = sdpvar(N,1);
d = binvar(N,1);
...
Model = [Model, x(k) == x(k-1) + u(k)*d(k)]
...

The product here completely kills this model, as it introduces a nonconvex bilinear equality constraint, thus landing us in a very hard nonconvex nonlinear integer program. A significantly better model is to introduce a new variable to represent the product, and then model the product using simple linear logic. This will keep us in the comfortable world of mixed-integer linear programming (considering these constraints only of course)

x = sdpvar(N,1);
u = sdpvar(N,1);
d = binvar(N,1);
w = sdpvar(N,1);
...
Model = [Model, x(k) == x(k-1) + w(k), implies(d(k)==1, w(k)==u(k), implies(d(k)==0, w(k)==0]
...

The implication will be converted to linear equalities using the big-M modelling framework, but you can of course do it manually if you are familiar with the method

Model = [Model, x(k) == x(k-1) + w(k), -M*d(k) <= w(k) <= M*d(k), -M*(1-d(k)) <= w(k)-u(k) <= M*(1-d(k))]

Alternatively, there is a command binmodel to use in more complex scenarios if you are too lazy to detect and untangle all the nonlinear products.

Some more on-off modelling is discussed n the unit commitment example.

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