MathOptSetDistances

Set of functions to compute distances and projections to sets defined in MathOptInterface.jl.

Distance to set

set_distance(d::D, v::V, s::S) defines the distance of a point v to a set s. The distance is always 0 if v ∈ s. S is a MOI.AbstractSet, v is a scalar or vector value and d a type of distance that is a subtype of AbstractDistance.

New sets should implement at least set_distance(::DefaultDistance, v::V, s::MySet).

Projection on set

projection_on_set(d::D, v, s::S) returns the point on S that is closest to v with respect to the distance d. projection_gradient_on_set returns the gradient of this projection, i.e. the transpose of the Jacobian.

Gradients as ChainRules

Gradients projection_gradient_on_set eagerly computes the full derivative matrix. This is often simpler to test and implement, but leads to unnecessary allocations and expensive operations. They are also implemented using ChainRulesCore.jl methods rrule and frule. Both methods should be implemented for each derivative and tested against projection_gradient_on_set and FiniteDifferences.jl.

Special matrix types

When some gradients or projections have structural zeros (sparsity patterns), they can and should return non-standard matrices including FillArrays Zeros, Eyes, Ones, Fill, sparse arrays and LinearAlgebra.Diagonal.