Constraint on count of variables >0 in JuMP?

Specific Domains Optimization (Mathematical)
1

I have a model with a control variable as follows:

@variable(model, 0 ≤ u[1:(T+1)] ≤ u_max)

I want to constrain the number of elements in u greater than 0 to be less than a threshold, u_nonzero_max. How can I do this in JuMP?

2

Here you go:

T, u_max, u_non_zero_max = 3, 4.0, 2
model = Model()
@variable(model, 0 <= u[1:(T+1)] <= u_max)
@variable(model, z[1:(T+1)], Bin)
@constraint(model, [t in 1:T+1], u[t] <= u_max * z[t])
@constraint(model, sum(z) <= u_non_zero_max)
3

Thanks @odow! Unfortunately, my MINLP problem is way slower to fit than my NLP problem. Is there a workaround to do this to avoid binary variables? My searches to find an alternative haven’t amounted to much.

4

Unfortunately, my MINLP problem is way slower to fit than my NLP problem

This is expected. The M-choose-N-non-zero is quite a nasty constraint.

Is there a workaround to do this to avoid binary variables?

No good ones. Consider the case when T = 1 and u_non_zero_max = 1. There are two decisions variables, and the feasible region is the L formed by the x- and y-axes from [0, u_max]. This is the same as a classical complementarity constraint, and you’ve made it harder by increasing the dimension :smile:

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