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minkowski_sum for SPZ and zonotopic set #3493

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May 14, 2024
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minkowski_sum for SPZ and zonotopic set #3493

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21 changes: 20 additions & 1 deletion src/ConcreteOperations/minkowski_sum.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -454,7 +454,8 @@ end

# ZeroSet is the neutral element (+ disambiguation)
for T in [:LazySet, :AbstractPolyhedron, :AbstractPolytope, :AbstractZonotope,
:AbstractHyperrectangle, :AbstractSingleton, :DensePolynomialZonotope]
:AbstractHyperrectangle, :AbstractSingleton, :DensePolynomialZonotope,
:SparsePolynomialZonotope]
@eval begin
@commutative minkowski_sum(::ZeroSet, X::$T) = X
end
Expand Down Expand Up @@ -498,6 +499,13 @@ Compute the Minkowski sum of two sparse polyomial zonotopes.
### Output

The Minkowski sum of `P1` and `P2`.

### Algorithm

See Proposition 3.1.19 in [1].

[1] Kochdumper. *Extensions of polynomial zonotopes and their application to
verification of cyber-physical systems.* PhD diss., TU Munich, 2022.
"""
function minkowski_sum(P1::SparsePolynomialZonotope,
P2::SparsePolynomialZonotope)
Expand All @@ -508,6 +516,17 @@ function minkowski_sum(P1::SparsePolynomialZonotope,
return SparsePolynomialZonotope(c, G, GI, E)
end

# See Proposition 3.1.19 in [1].
# [1] Kochdumper. *Extensions of polynomial zonotopes and their application to
# verification of cyber-physical systems.* PhD diss., TU Munich, 2022.
@commutative function minkowski_sum(PZ::SparsePolynomialZonotope, Z::AbstractZonotope)
c = center(PZ) + center(Z)
G = genmat_dep(PZ)
GI = hcat(genmat_indep(PZ), genmat(Z))
E = expmat(PZ)
return SparsePolynomialZonotope(c, G, GI, E)
end

# Given two balls of the same p-norm, their Minkowski sum is again a p-norm ball,
# which can be seen as follows:
#
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9 changes: 9 additions & 0 deletions test/ConcreteOperations/minkowski_sum.jl
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Expand Up @@ -11,6 +11,15 @@ for N in [Float64, Float32, Rational{Int}]
B2 = B(N[4, 5], N(6))
@test minkowski_sum(B1, B2) == B(N[5, 7], N(9))
end

H1 = Hyperrectangle(N[1, 2], N[3, 4])
H2 = Hyperrectangle(N[5, 6], N[7, 8])
@test minkowski_sum(H1, H2) == Hyperrectangle(N[6, 8], N[10, 12])
PZ = minkowski_sum(convert(SparsePolynomialZonotope, H1), H2)
# equality is not required but approximates the equivalence check
@test PZ == SparsePolynomialZonotope(N[6, 8], N[3 0; 0 4], N[7 0; 0 8], [1 0; 0 1], 1:2)
PZ = minkowski_sum(H1, convert(SparsePolynomialZonotope, H2))
@test PZ == SparsePolynomialZonotope(N[6, 8], N[7 0; 0 8], N[3 0; 0 4], [1 0; 0 1], 1:2)
end

for N in [Float64, Float32]
Expand Down
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