diff --git a/src/Utilities/Utilities.jl b/src/Utilities/Utilities.jl
index de39a929f7..bac6621917 100644
--- a/src/Utilities/Utilities.jl
+++ b/src/Utilities/Utilities.jl
@@ -30,6 +30,7 @@ include("functions.jl")
 include("mutable_arithmetics.jl")
 include("sets.jl")
 include("constraints.jl")
+include("dense_dict.jl")
 include("copy.jl")
 include("results.jl")
 include("variables.jl")
diff --git a/src/Utilities/copy.jl b/src/Utilities/copy.jl
index b03cc9fbf4..4d1e3a0407 100644
--- a/src/Utilities/copy.jl
+++ b/src/Utilities/copy.jl
@@ -63,12 +63,31 @@ error in case `copy_to` is called with `copy_names` equal to `true`.
 """
 supports_default_copy_to(model::MOI.ModelLike, copy_names::Bool) = false
 
+const DenseVariableDict{V} = DenseDict{MOI.VariableIndex, V, typeof(MOI.index_value), typeof(MOI.VariableIndex)}
+function dense_variable_dict(::Type{V}, n) where V
+    return DenseDict{MOI.VariableIndex, V}(MOI.index_value, MOI.VariableIndex, n)
+end
+
 struct IndexMap <: AbstractDict{MOI.Index, MOI.Index}
-    varmap::Dict{MOI.VariableIndex, MOI.VariableIndex}
+    varmap::Union{DenseVariableDict{MOI.VariableIndex},
+                  Dict{MOI.VariableIndex, MOI.VariableIndex}}
     conmap::Dict{MOI.ConstraintIndex, MOI.ConstraintIndex}
 end
 IndexMap() = IndexMap(Dict{MOI.VariableIndex, MOI.VariableIndex}(),
                       Dict{MOI.ConstraintIndex, MOI.ConstraintIndex}())
+function IndexMap(n)
+    IndexMap(dense_variable_dict(MOI.VariableIndex, n),
+             Dict{MOI.ConstraintIndex, MOI.ConstraintIndex}())
+end
+
+function index_map_for_variable_indices(variables)
+    n = length(variables)
+    if all(i -> variables[i] == MOI.VariableIndex(i), 1:n)
+        return IndexMap(n)
+    else
+        return IndexMap()
+    end
+end
 
 Base.getindex(idxmap::IndexMap, vi::MOI.VariableIndex) = idxmap.varmap[vi]
 function Base.getindex(idxmap::IndexMap, ci::MOI.ConstraintIndex{F, S}) where {F, S}
@@ -298,9 +317,8 @@ the copying a model incrementally.
 function default_copy_to(dest::MOI.ModelLike, src::MOI.ModelLike, copy_names::Bool)
     MOI.empty!(dest)
 
-    idxmap = IndexMap()
-
     vis_src = MOI.get(src, MOI.ListOfVariableIndices())
+    idxmap = index_map_for_variable_indices(vis_src)
     constraint_types = MOI.get(src, MOI.ListOfConstraints())
     single_variable_types = Type{<:MOI.AbstractScalarSet}[]
     vector_of_variables_types = Type{<:MOI.AbstractVectorSet}[]
@@ -659,9 +677,8 @@ the Allocate-Load API.
 function allocate_load(dest::MOI.ModelLike, src::MOI.ModelLike, copy_names::Bool)
     MOI.empty!(dest)
 
-    idxmap = IndexMap()
-
     vis_src = MOI.get(src, MOI.ListOfVariableIndices())
+    idxmap = index_map_for_variable_indices(vis_src)
     constraint_types = MOI.get(src, MOI.ListOfConstraints())
     single_variable_types = [S for (F, S) in constraint_types
                              if F == MOI.SingleVariable]
diff --git a/src/Utilities/dense_dict.jl b/src/Utilities/dense_dict.jl
new file mode 100644
index 0000000000..a7ac300baf
--- /dev/null
+++ b/src/Utilities/dense_dict.jl
@@ -0,0 +1,41 @@
+"""
+    struct DenseDict{K, V, F, I} <: AbstractDict{K, V}
+        hash::F
+        inverse_hash::I
+        set::BitSet
+        map::Vector{V}
+    end
+
+Same as `Dict{K, V}` but `hash(key)` is assumed to belong to `eachindex(map)`.
+"""
+struct DenseDict{K, V, F, I} <: AbstractDict{K, V}
+    hash::F
+    inverse_hash::I
+    set::BitSet
+    map::Vector{V}
+    function DenseDict{K, V}(hash, inverse_hash, n) where {K, V}
+        set = BitSet()
+        sizehint!(set, n)
+        return new{K, V, typeof(hash), typeof(inverse_hash)}(hash, inverse_hash, set, Vector{K}(undef, n))
+    end
+end
+
+# Implementation of the `AbstractDict` API.
+# Base.empty(::DenseDict, ::Type{K}, ::Type{V}) not implemented
+function Base.iterate(d::DenseDict, args...)
+    itr = iterate(d.set, args...)
+    if itr === nothing
+        return nothing
+    else
+        el, i = itr
+        return d.inverse_hash(el) => d.map[el], i
+    end
+end
+Base.length(d::DenseDict) = length(d.set)
+Base.haskey(dict::DenseDict, key) = dict.hash(key) in dict.set
+Base.getindex(dict::DenseDict, key) = dict.map[dict.hash(key)]
+function Base.setindex!(dict::DenseDict, value, key)
+    h = dict.hash(key)
+    push!(dict.set, h)
+    dict.map[h] = value
+end
diff --git a/src/indextypes.jl b/src/indextypes.jl
index bd7ad7dfa4..82afa66f40 100644
--- a/src/indextypes.jl
+++ b/src/indextypes.jl
@@ -19,6 +19,7 @@ index.value == MOI.get(model, MOI.ConstraintFunction(), index).variable.value
 struct ConstraintIndex{F, S}
     value::Int64
 end
+index_value(ci::ConstraintIndex) = ci.value
 
 """
     VariableIndex
@@ -29,6 +30,7 @@ To allow for deletion, indices need not be consecutive.
 struct VariableIndex
     value::Int64
 end
+index_value(vi::VariableIndex) = vi.value
 
 # The default hash is slow. It's important for the performance of dictionaries
 # of VariableIndices to define our own.
diff --git a/test/Utilities/Utilities.jl b/test/Utilities/Utilities.jl
index fc4db83c80..a42ba9e132 100644
--- a/test/Utilities/Utilities.jl
+++ b/test/Utilities/Utilities.jl
@@ -30,6 +30,9 @@ end
 @testset "Caching Optimizer" begin
     include("cachingoptimizer.jl")
 end
+@testset "DenseDict" begin
+    include("dense_dict.jl")
+end
 @testset "Copy" begin
     include("copy.jl")
 end
diff --git a/test/Utilities/copy.jl b/test/Utilities/copy.jl
index 56f8bd7ff2..142c8622f6 100644
--- a/test/Utilities/copy.jl
+++ b/test/Utilities/copy.jl
@@ -4,6 +4,7 @@ const MOI = MathOptInterface
 const MOIT = MOI.Test
 const MOIU = MOI.Utilities
 
+
 include("../dummy.jl")
 
 remove_moi(x::String) = replace(x, "MathOptInterface." => "")
@@ -170,7 +171,7 @@ mutable struct ReverseOrderConstrainedVariablesModel <: AbstractConstrainedVaria
     inner                   ::MOIU.Model{Float64}
     ReverseOrderConstrainedVariablesModel() = new(MOI.ConstraintIndex[], MOIU.Model{Float64}())
 end
-        
+
 
 
 MOI.add_variables(model::AbstractConstrainedVariablesModel, n) = MOI.add_variables(model.inner, n)
diff --git a/test/Utilities/dense_dict.jl b/test/Utilities/dense_dict.jl
new file mode 100644
index 0000000000..14edf1affe
--- /dev/null
+++ b/test/Utilities/dense_dict.jl
@@ -0,0 +1,34 @@
+using Test
+import MathOptInterface
+const MOI = MathOptInterface
+
+mul2(x) = x * 2
+div2(x) = div(x, 2)
+d = MOI.Utilities.DenseDict{Int, Float64}(div2, mul2, 3)
+
+d[4] = 0.25
+@test !haskey(d, 2)
+@test haskey(d, 4)
+@test !haskey(d, 6)
+@test length(d) == 1
+@test collect(d) == [4 => 0.25]
+@test d[4] == 0.25
+
+d[2] = 1.5
+@test haskey(d, 2)
+@test haskey(d, 4)
+@test !haskey(d, 6)
+@test length(d) == 2
+@test collect(d) == [2 => 1.5, 4 => 0.25]
+@test d[2] == 1.5
+@test d[4] == 0.25
+
+d[6] = 0.75
+@test haskey(d, 2)
+@test haskey(d, 4)
+@test haskey(d, 6)
+@test length(d) == 3
+@test collect(d) == [2 => 1.5, 4 => 0.25, 6 => 0.75]
+@test d[2] == 1.5
+@test d[4] == 0.25
+@test d[6] == 0.75