Complex objective functions

[FedeClaudi]

Hi,

I'm trying to implement an @objective function which requires calling argmin on a variable. The full story behind is a bit long, but basically I'm using a Bezier function to estimate the penalty cost for a term Fu and I need argmin to go from Fu to the parameter for the Bezier curve.

    # bezier curve
    t = 0:.001:1

    P0_x = -750
    P1_x = -350
    P2_x = -350
    P3_x = 3000
    
    P0_y = 1
    P1_y = 0
    P2_y = 0
    P3_y = 1

    Px(t) = P0_x * (1 - t)^3 + P1_x * 3(1 - t)^2 * t + P2_x * 3(1 - t) * t^2 + P3_x * t^3
    Py(t) = P0_y * (1 - t)^3 + P1_y * 3(1 - t)^2 * t + P2_y * 3(1 - t) * t^2 + P3_y * t^3
    X = Px.(t)

    cost(Fu) = Py(
        argmin(abs.(X .- Fu))
    )

    @objective(model, Min, ∫(SF + α*cost(Fu), s))
    optimize!(model)

However I then get this error:

MethodError: no method matching isless(::InfiniteOpt.NLPExpr, ::InfiniteOpt.NLPExpr)

I've tried changing it to value(Fu) but then I get an error saying that Fu(s) is not in the transcription model.

[pulsipher]

The problem is that argmin is not one of the accepted nonlinear functions. You can always check with all_registered_functions to see what functions can be used to build nonlinear expressions. Like JuMP, we can register new nonlinear functions to use. A current limitation of this interface is that each function must be scalar valued (not vector inputs/outputs). We hope to remove this limitation in the relatively near future (#229).

In this case, we cannot register directly since argmin is vector valued. However, if Fu is a scalar (or we can easily enumerate it with scalar components) then we can register the cost function:

@register(model, cost(Fu))

And then use it when defining the objective.

[pulsipher]

For more info, see https://infiniteopt.github.io/InfiniteOpt.jl/stable/guide/expression/#Function-Registration

[FedeClaudi]

it worked perfectly, thank you!