MOI.Utilities.Model{T}() where {T}

An implementation of ModelLike that supports all functions and sets defined in MOI. It is parameterized by the coefficient type.

Example

julia> import MathOptInterface as MOI

julia> model = MOI.Utilities.Model{Float64}()
MOIU.Model{Float64}
├ ObjectiveSense: FEASIBILITY_SENSE
├ ObjectiveFunctionType: MOI.ScalarAffineFunction{Float64}
├ NumberOfVariables: 0
└ NumberOfConstraints: 0

UniversalFallback

The UniversalFallback can be applied on a MOI.ModelLike model to create the model UniversalFallback(model) supporting any constraint and attribute. This allows to have a specialized implementation in model for performance critical constraints and attributes while still supporting other attributes with a small performance penalty. Note that model is unaware of constraints and attributes stored by UniversalFallback so this is not appropriate if model is an optimizer (for this reason, MOI.optimize! has not been implemented). In that case, optimizer bridges should be used instead.

macro model(
    model_name,
    scalar_sets,
    typed_scalar_sets,
    vector_sets,
    typed_vector_sets,
    scalar_functions,
    typed_scalar_functions,
    vector_functions,
    typed_vector_functions,
    is_optimizer = false
)

Creates a type model_name implementing the MOI model interface and supporting all combinations of the provided functions and sets.

Each typed_ scalar/vector sets/functions argument is a tuple of types. A type is "typed" if it has a coefficient {T} as the first type parameter.

Tuple syntax

To give no set/function, write (). To give one set or function X, write (X,).

is_optimizer

If is_optimizer = true, the resulting struct is a of GenericOptimizer, which is a subtype of MOI.AbstractOptimizer, otherwise, it is a GenericModel, which is a subtype of MOI.ModelLike.

VariableIndex

• The function MOI.VariableIndex must not be given in scalar_functions.
• The model supports MOI.VariableIndex-in-S constraints where S is MOI.EqualTo, MOI.GreaterThan, MOI.LessThan, MOI.Interval, MOI.Integer, MOI.ZeroOne, MOI.Semicontinuous or MOI.Semiinteger.
• The sets supported with MOI.VariableIndex cannot be controlled from the macro; use UniversalFallback to support more sets.

Example

The model describing a linear program would be:

@model(
    LPModel,                                          # model_name
    (),                                               # untyped scalar sets
    (MOI.EqualTo, MOI.GreaterThan, MOI.LessThan, MOI.Interval), #   typed scalar sets
    (MOI.Zeros, MOI.Nonnegatives, MOI.Nonpositives),  # untyped vector sets
    (),                                               #   typed vector sets
    (),                                               # untyped scalar functions
    (MOI.ScalarAffineFunction,),                      #   typed scalar functions
    (MOI.VectorOfVariables,),                         # untyped vector functions
    (MOI.VectorAffineFunction,),                      #   typed vector functions
    false,                                            # is_optimizer
)

mutable struct GenericModel{T,O,V,C} <: AbstractModelLike{T}

Implements a model supporting coefficients of type T and:

• An objective function stored in .objective::O
• Variables and VariableIndex constraints stored in .variable_bounds::V
• F-in-S constraints (excluding VariableIndex constraints) stored in .constraints::C

All interactions take place via the MOI interface, so the types O, V, and C must implement the API as needed for their functionality.

mutable struct GenericOptimizer{T,O,V,C} <: AbstractOptimizer{T}

Implements a model supporting coefficients of type T and:

• An objective function stored in .objective::O
• Variables and VariableIndex constraints stored in .variable_bounds::V
• F-in-S constraints (excluding VariableIndex constraints) stored in .constraints::C

All interactions take place via the MOI interface, so the types O, V, and C must implement the API as needed for their functionality.

ObjectiveContainer{T}

A helper struct to simplify the handling of objective functions in Utilities.Model.

struct VariablesContainer{T} <: AbstractVectorBounds
    set_mask::Vector{UInt16}
    lower::Vector{T}
    upper::Vector{T}
end

A struct for storing variables and VariableIndex-related constraints. Used in MOI.Utilities.Model by default.

mutable struct FreeVariables <: MOI.ModelLike
    n::Int64
    FreeVariables() = new(0)
end

A struct for storing free variables that can be used as the variables field of GenericModel or GenericModel. It represents a model that does not support any constraint nor objective function.

Example

The following model type represents a conic model in geometric form. As opposed to VariablesContainer, FreeVariables does not support constraint bounds so they are bridged into an affine constraint in the MOI.Nonnegatives cone as expected for the geometric conic form.

julia> MOI.Utilities.@product_of_sets(
    Cones,
    MOI.Zeros,
    MOI.Nonnegatives,
    MOI.SecondOrderCone,
    MOI.PositiveSemidefiniteConeTriangle,
);

julia> const ConicModel{T} = MOI.Utilities.GenericOptimizer{
    T,
    MOI.Utilities.ObjectiveContainer{T},
    MOI.Utilities.FreeVariables,
    MOI.Utilities.MatrixOfConstraints{
        T,
        MOI.Utilities.MutableSparseMatrixCSC{
            T,
            Int,
            MOI.Utilities.OneBasedIndexing,
        },
        Vector{T},
        Cones{T},
    },
};

julia> model = MOI.instantiate(ConicModel{Float64}, with_bridge_type=Float64);

julia> x = MOI.add_variable(model)
MathOptInterface.VariableIndex(1)

julia> c = MOI.add_constraint(model, x, MOI.GreaterThan(1.0))
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)

julia> MOI.Bridges.is_bridged(model, c)
true

julia> bridge = MOI.Bridges.bridge(model, c)
MathOptInterface.Bridges.Constraint.VectorizeBridge{Float64, MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives, MathOptInterface.VariableIndex}(MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1), 1.0)

julia> bridge.vector_constraint
MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1)

julia> MOI.Bridges.is_bridged(model, bridge.vector_constraint)
false

mutable struct VectorOfConstraints{
    F<:MOI.AbstractFunction,
    S<:MOI.AbstractSet,
} <: MOI.ModelLike
    constraints::CleverDicts.CleverDict{
        MOI.ConstraintIndex{F,S},
        Tuple{F,S},
        typeof(CleverDicts.key_to_index),
        typeof(CleverDicts.index_to_key),
    }
end

A struct storing F-in-S constraints as a mapping between the constraint indices to the corresponding tuple of function and set.

abstract type StructOfConstraints <: MOI.ModelLike end

A struct storing a subfields other structs storing constraints of different types.

See Utilities.@struct_of_constraints_by_function_types and Utilities.@struct_of_constraints_by_set_types.

Utilities.@struct_of_constraints_by_function_types(name, func_types...)

Given a vector of n function types (F1, F2,..., Fn) in func_types, defines a subtype of StructOfConstraints of name name and which type parameters {T, C1, C2, ..., Cn}. It contains n field where the ith field has type Ci and stores the constraints of function type Fi.

The expression Fi can also be a union in which case any constraint for which the function type is in the union is stored in the field with type Ci.

Utilities.@struct_of_constraints_by_set_types(name, func_types...)

Given a vector of n set types (S1, S2,..., Sn) in func_types, defines a subtype of StructOfConstraints of name name and which type parameters {T, C1, C2, ..., Cn}. It contains n field where the ith field has type Ci and stores the constraints of set type Si. The expression Si can also be a union in which case any constraint for which the set type is in the union is stored in the field with type Ci. This can be useful if Ci is a MatrixOfConstraints in order to concatenate the coefficients of constraints of several different set types in the same matrix.

struct_of_constraint_code(struct_name, types, field_types = nothing)

Given a vector of n Union{SymbolFun,_UnionSymbolFS{SymbolFun}} or Union{SymbolSet,_UnionSymbolFS{SymbolSet}} in types, defines a subtype of StructOfConstraints of name name and which type parameters {T, F1, F2, ..., Fn} if field_types is nothing and a {T} otherwise. It contains n field where the ith field has type Ci if field_types is nothing and type field_types[i] otherwise. If types is vector of Union{SymbolFun,_UnionSymbolFS{SymbolFun}} (resp. Union{SymbolSet,_UnionSymbolFS{SymbolSet}}) then the constraints of that function (resp. set) type are stored in the corresponding field.

This function is used by the macros @model, @struct_of_constraints_by_function_types and @struct_of_constraints_by_set_types.

CachingOptimizer

CachingOptimizer is an intermediate layer that stores a cache of the model and links it with an optimizer. It supports incremental model construction and modification even when the optimizer doesn't.

Constructors

    CachingOptimizer(cache::MOI.ModelLike, optimizer::AbstractOptimizer)

Creates a CachingOptimizer in AUTOMATIC mode, with the optimizer optimizer.

The type of the optimizer returned is CachingOptimizer{typeof(optimizer), typeof(cache)} so it does not support the function reset_optimizer(::CachingOptimizer, new_optimizer) if the type of new_optimizer is different from the type of optimizer.

    CachingOptimizer(cache::MOI.ModelLike, mode::CachingOptimizerMode)

Creates a CachingOptimizer in the NO_OPTIMIZER state and mode mode.

The type of the optimizer returned is CachingOptimizer{MOI.AbstractOptimizer,typeof(cache)} so it does support the function reset_optimizer(::CachingOptimizer, new_optimizer) if the type of new_optimizer is different from the type of optimizer.

About the type

States

A CachingOptimizer may be in one of three possible states (CachingOptimizerState):

• NO_OPTIMIZER: The CachingOptimizer does not have any optimizer.
• EMPTY_OPTIMIZER: The CachingOptimizer has an empty optimizer. The optimizer is not synchronized with the cached model.
• ATTACHED_OPTIMIZER: The CachingOptimizer has an optimizer, and it is synchronized with the cached model.

Modes

A CachingOptimizer has two modes of operation (CachingOptimizerMode):

• MANUAL: The only methods that change the state of the CachingOptimizer are Utilities.reset_optimizer, Utilities.drop_optimizer, and Utilities.attach_optimizer. Attempting to perform an operation in the incorrect state results in an error.
• AUTOMATIC: The CachingOptimizer changes its state when necessary. For example, optimize! will automatically call attach_optimizer (an optimizer must have been previously set). Attempting to add a constraint or perform a modification not supported by the optimizer results in a drop to EMPTY_OPTIMIZER mode.

attach_optimizer(model::CachingOptimizer)

Attaches the optimizer to model, copying all model data into it. Can be called only from the EMPTY_OPTIMIZER state. If the copy succeeds, the CachingOptimizer will be in state ATTACHED_OPTIMIZER after the call, otherwise an error is thrown; see MOI.copy_to for more details on which errors can be thrown.

reset_optimizer(m::CachingOptimizer, optimizer::MOI.AbstractOptimizer)

Sets or resets m to have the given empty optimizer optimizer.

Can be called from any state. An assertion error will be thrown if optimizer is not empty.

The CachingOptimizer m will be in state EMPTY_OPTIMIZER after the call.

reset_optimizer(m::CachingOptimizer)

Detaches and empties the current optimizer. Can be called from ATTACHED_OPTIMIZER or EMPTY_OPTIMIZER state. The CachingOptimizer will be in state EMPTY_OPTIMIZER after the call.

drop_optimizer(m::CachingOptimizer)

Drops the optimizer, if one is present. Can be called from any state. The CachingOptimizer will be in state NO_OPTIMIZER after the call.

state(m::CachingOptimizer)::CachingOptimizerState

Returns the state of the CachingOptimizer m. See Utilities.CachingOptimizer.

mode(m::CachingOptimizer)::CachingOptimizerMode

Returns the operating mode of the CachingOptimizer m. See Utilities.CachingOptimizer.

MockOptimizer

MockOptimizer is a fake optimizer especially useful for testing. Its main feature is that it can store the values that should be returned for each attribute.

latex_formulation(model::MOI.ModelLike; kwargs...)

Wrap model in a type so that it can be pretty-printed as text/latex in a notebook like IJulia, or in Documenter.

To render the model, end the cell with latex_formulation(model), or call display(latex_formulation(model)) in to force the display of the model from inside a function.

Possible keyword arguments are:

• simplify_coefficients : Simplify coefficients if possible by omitting them or removing trailing zeros.
• default_name : The name given to variables with an empty name.
• print_types : Print the MOI type of each function and set for clarity.

default_copy_to(dest::MOI.ModelLike, src::MOI.ModelLike)

A default implementation of MOI.copy_to(dest, src) for models that implement the incremental interface, that is, MOI.supports_incremental_interface returns true.

IndexMap()

The dictionary-like object returned by MOI.copy_to.

identity_index_map(model::MOI.ModelLike)

Return an IndexMap that maps all variable and constraint indices of model to themselves.

ModelFilter(filter::Function, model::MOI.ModelLike)

A layer to filter out various components of model.

The filter function takes a single argument, which is each element from the list returned by the attributes below. It returns true if the element should be visible in the filtered model and false otherwise.

The components that are filtered are:

• Entire constraint types via:
  • MOI.ListOfConstraintTypesPresent
• Individual constraints via:
  • MOI.ListOfConstraintIndices{F,S}
• Specific attributes via:
  • MOI.ListOfModelAttributesSet
  • MOI.ListOfConstraintAttributesSet
  • MOI.ListOfVariableAttributesSet

See below for examples of how this works.

Example: copy model excluding integer constraints

Use the do syntax to provide a single function.

filtered_src = MOI.Utilities.ModelFilter(src) do item
    return item != (MOI.VariableIndex, MOI.Integer)
end
MOI.copy_to(dest, filtered_src)

Example: copy model excluding names

Use type dispatch to simplify the implementation:

my_filter(::Any) = true  # Note the generic fallback
my_filter(::MOI.VariableName) = false
my_filter(::MOI.ConstraintName) = false
filtered_src = MOI.Utilities.ModelFilter(my_filter, src)
MOI.copy_to(dest, filtered_src)

Example: copy irreducible infeasible subsystem

my_filter(::Any) = true  # Note the generic fallback
function my_filter(ci::MOI.ConstraintIndex)
    status = MOI.get(dest, MOI.ConstraintConflictStatus(), ci)
    return status != MOI.NOT_IN_CONFLICT
end
filtered_src = MOI.Utilities.ModelFilter(my_filter, src)
MOI.copy_to(dest, filtered_src)

loadfromstring!(model, s)

A utility function to aid writing tests.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> MOI.Utilities.loadfromstring!(model, """
       variables: x, y, z
       constrainedvariable: [a, b, c] in Nonnegatives(3)
       minobjective::Float64: 2x + 3y
       con1: x + y <= 1.0
       con2: [x, y] in Nonnegatives(2)
       x >= 0.0
       """)

Notes

Special labels are:

• variables
• minobjective
• maxobjectives

Everything else denotes a constraint with a name.

Append ::T to use an element type of T when parsing the function.

Do not name VariableIndex constraints.

Exceptions

• x - y does NOT currently parse. Instead, write x + -1.0 * y.
• x^2 does NOT currently parse. Instead, write x * x.

PenaltyRelaxation(
    penalties = Dict{MOI.ConstraintIndex,Float64}();
    default::Union{Nothing,T} = 1.0,
)

A problem modifier that, when passed to MOI.modify, destructively modifies the model in-place to create a penalized relaxation of the constraints.

Reformulation

See Utilities.ScalarPenaltyRelaxation for details of the reformulation.

For each constraint ci, the penalty passed to Utilities.ScalarPenaltyRelaxation is get(penalties, ci, default). If the value is nothing, because ci does not exist in penalties and default = nothing, then the constraint is skipped.

Return value

MOI.modify(model, PenaltyRelaxation()) returns a Dict{MOI.ConstraintIndex,MOI.ScalarAffineFunction} that maps each constraint index to the corresponding y + z as a MOI.ScalarAffineFunction. In an optimal solution, query the value of these functions to compute the violation of each constraint.

Relax a subset of constraints

To relax a subset of constraints, pass a penalties dictionary and set default = nothing.

Supported constraint types

The penalty relaxation is currently limited to modifying MOI.ScalarAffineFunction and MOI.ScalarQuadraticFunction constraints in the linear sets MOI.LessThan, MOI.GreaterThan, MOI.EqualTo and MOI.Interval.

It does not include variable bound or integrality constraints, because these cannot be modified in-place.

To modify variable bounds, rewrite them as linear constraints.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model);

julia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));

julia> map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(default = 2.0));

julia> print(model)
Minimize ScalarAffineFunction{Float64}:
 0.0 + 2.0 v[2]

Subject to:

ScalarAffineFunction{Float64}-in-LessThan{Float64}
 0.0 + 1.0 v[1] - 1.0 v[2] <= 2.0

VariableIndex-in-GreaterThan{Float64}
 v[2] >= 0.0

julia> map[c] isa MOI.ScalarAffineFunction{Float64}
true

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model);

julia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));

julia> map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(Dict(c => 3.0)));

julia> print(model)
Minimize ScalarAffineFunction{Float64}:
 0.0 + 3.0 v[2]

Subject to:

ScalarAffineFunction{Float64}-in-LessThan{Float64}
 0.0 + 1.0 v[1] - 1.0 v[2] <= 2.0

VariableIndex-in-GreaterThan{Float64}
 v[2] >= 0.0

julia> map[c] isa MOI.ScalarAffineFunction{Float64}
true

ScalarPenaltyRelaxation(penalty::T) where {T}

A problem modifier that, when passed to MOI.modify, destructively modifies the constraint in-place to create a penalized relaxation of the constraint.

Reformulation

The penalty relaxation modifies constraints of the form $f(x) \in S$ into $f(x) + y - z \in S$, where $y, z \ge 0$, and then it introduces a penalty term into the objective of $a \times (y + z)$ (if minimizing, else $-a$), where $a$ is penalty

When S is MOI.LessThan or MOI.GreaterThan, we omit y or z respectively as a performance optimization.

Return value

MOI.modify(model, ci, ScalarPenaltyRelaxation(penalty)) returns y + z as a MOI.ScalarAffineFunction. In an optimal solution, query the value of this function to compute the violation of the constraint.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model);

julia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));

julia> f = MOI.modify(model, c, MOI.Utilities.ScalarPenaltyRelaxation(2.0));

julia> print(model)
Minimize ScalarAffineFunction{Float64}:
 0.0 + 2.0 v[2]

Subject to:

ScalarAffineFunction{Float64}-in-LessThan{Float64}
 0.0 + 1.0 v[1] - 1.0 v[2] <= 2.0

VariableIndex-in-GreaterThan{Float64}
 v[2] >= 0.0

julia> f isa MOI.ScalarAffineFunction{Float64}
true

mutable struct MatrixOfConstraints{T,AT,BT,ST} <: MOI.ModelLike
    coefficients::AT
    constants::BT
    sets::ST
    caches::Vector{Any}
    are_indices_mapped::Vector{BitSet}
    final_touch::Bool
end

Represent ScalarAffineFunction and VectorAffinefunction constraints in a matrix form where the linear coefficients of the functions are stored in the coefficients field, the constants of the functions or sets are stored in the constants field. Additional information about the sets are stored in the sets field.

This model can only be used as the constraints field of a MOI.Utilities.AbstractModel.

When the constraints are added, they are stored in the caches field. They are only loaded in the coefficients and constants fields once MOI.Utilities.final_touch is called. For this reason, MatrixOfConstraints should not be used by an incremental interface. Use MOI.copy_to instead.

The constraints can be added in two different ways:

1. With add_constraint, in which case a canonicalized copy of the function is stored in caches.
2. With pass_nonvariable_constraints, in which case the functions and sets are stored themselves in caches without mapping the variable indices. The corresponding index in caches is added in are_indices_mapped. This avoids doing a copy of the function in case the getter of CanonicalConstraintFunction does not make a copy for the source model, for example, this is the case of VectorOfConstraints.

We illustrate this with an example. Suppose a model is copied from a src::MOI.Utilities.Model to a bridged model with a MatrixOfConstraints. For all the types that are not bridged, the constraints will be copied with pass_nonvariable_constraints. Hence the functions stored in caches are exactly the same as the ones stored in src. This is ok since this is only during the copy_to operation during which src cannot be modified. On the other hand, for the types that are bridged, the functions added may contain duplicates even if the functions did not contain duplicates in src so duplicates are removed with MOI.Utilities.canonical.

Interface

The .coefficients::AT type must implement:

• AT()
• MOI.empty(::AT)!
• MOI.Utilities.add_column
• MOI.Utilities.set_number_of_rows
• MOI.Utilities.allocate_terms
• MOI.Utilities.load_terms
• MOI.Utilities.final_touch

The .constants::BT type must implement:

• BT()
• Base.empty!(::BT)
• Base.resize(::BT)
• MOI.Utilities.load_constants
• MOI.Utilities.function_constants
• MOI.Utilities.set_from_constants

The .sets::ST type must implement:

• ST()
• MOI.is_empty(::ST)
• MOI.empty(::ST)
• MOI.dimension(::ST)
• MOI.is_valid(::ST, ::MOI.ConstraintIndex)
• MOI.get(::ST, ::MOI.ListOfConstraintTypesPresent)
• MOI.get(::ST, ::MOI.NumberOfConstraints)
• MOI.get(::ST, ::MOI.ListOfConstraintIndices)
• MOI.Utilities.set_types
• MOI.Utilities.set_index
• MOI.Utilities.add_set
• MOI.Utilities.rows
• MOI.Utilities.final_touch

add_column(coefficients)::Nothing

Tell coefficients to pre-allocate datastructures as needed to store one column.

allocate_terms(coefficients, index_map, func)::Nothing

Tell coefficients that the terms of the function func where the variable indices are mapped with index_map will be loaded with load_terms.

The function func must be canonicalized before calling allocate_terms. See is_canonical.

set_number_of_rows(coefficients, n)::Nothing

Tell coefficients to pre-allocate datastructures as needed to store n rows.

load_terms(coefficients, index_map, func, offset)::Nothing

Loads the terms of func to coefficients, mapping the variable indices with index_map.

The ith dimension of func is loaded at the (offset + i)th row of coefficients.

The function must be allocated first with allocate_terms.

The function func must be canonicalized, see is_canonical.

final_touch(coefficients)::Nothing

Informs the coefficients that all functions have been added with load_terms. No more modification is allowed unless MOI.empty! is called.

final_touch(sets)::Nothing

Informs the sets that all functions have been added with add_set. No more modification is allowed unless MOI.empty! is called.

extract_function(coefficients, row::Integer, constant::T) where {T}

Return the MOI.ScalarAffineFunction{T} function corresponding to row row in coefficients.

extract_function(
    coefficients,
    rows::UnitRange,
    constants::Vector{T},
) where{T}

Return the MOI.VectorAffineFunction{T} function corresponding to rows rows in coefficients.

mutable struct MutableSparseMatrixCSC{Tv,Ti<:Integer,I<:AbstractIndexing}
    indexing::I
    m::Int
    n::Int
    colptr::Vector{Ti}
    rowval::Vector{Ti}
    nzval::Vector{Tv}
    nz_added::Vector{Ti}
end

Matrix type loading sparse matrices in the Compressed Sparse Column format. The indexing used is indexing, see AbstractIndexing. The other fields have the same meaning than for SparseArrays.SparseMatrixCSC except that the indexing is different unless indexing is OneBasedIndexing. In addition, nz_added is used to cache the number of non-zero terms that have been added to each column due to the incremental nature of load_terms.

The matrix is loaded in 5 steps:

1. MOI.empty! is called.
2. MOI.Utilities.add_column and MOI.Utilities.allocate_terms are called in any order.
3. MOI.Utilities.set_number_of_rows is called.
4. MOI.Utilities.load_terms is called for each affine function.
5. MOI.Utilities.final_touch is called.

abstract type AbstractIndexing end

Indexing to be used for storing the row and column indices of MutableSparseMatrixCSC. See ZeroBasedIndexing and OneBasedIndexing.

struct ZeroBasedIndexing <: AbstractIndexing end

Zero-based indexing: the ith row or column has index i - 1. This is useful when the vectors of row and column indices need to be communicated to a library using zero-based indexing such as C libraries.

struct ZeroBasedIndexing <: AbstractIndexing end

One-based indexing: the ith row or column has index i. This enables an allocation-free conversion of MutableSparseMatrixCSC to SparseArrays.SparseMatrixCSC.

load_constants(constants, offset, func_or_set)::Nothing

This function loads the constants of func_or_set in constants at an offset of offset. Where offset is the sum of the dimensions of the constraints already loaded. The storage should be preallocated with resize! before calling this function.

This function should be implemented to be usable as storage of constants for MatrixOfConstraints.

The constants are loaded in three steps:

1. Base.empty! is called.
2. Base.resize! is called with the sum of the dimensions of all constraints.
3. MOI.Utilities.load_constants is called for each function for vector constraint or set for scalar constraint.

function_constants(constants, rows)

This function returns the function constants that were loaded with load_constants at the rows rows.

This function should be implemented to be usable as storage of constants for MatrixOfConstraints.

set_from_constants(constants, S::Type, rows)::S

This function returns an instance of the set S for which the constants where loaded with load_constants at the rows rows.

This function should be implemented to be usable as storage of constants for MatrixOfConstraints.

modify_constants(constants, row::Integer, new_constant::T) where {T}
modify_constants(
    constants,
    rows::AbstractVector{<:Integer},
    new_constants::AbstractVector{T},
) where {T}

Modify constants in-place to store new_constant in the row row, or rows rows.

This function must be implemented to enable MOI.ScalarConstantChange and MOI.VectorConstantChange for MatrixOfConstraints.

struct Hyperrectangle{T} <: AbstractVectorBounds
    lower::Vector{T}
    upper::Vector{T}
end

A struct for the .constants field in MatrixOfConstraints.

set_index(sets, ::Type{S})::Union{Int,Nothing} where {S<:MOI.AbstractSet}

Return an integer corresponding to the index of the set type in the list given by set_types.

If S is not part of the list, return nothing.

set_types(sets)::Vector{Type}

Return the list of the types of the sets allowed in sets.

add_set(sets, i)::Int64

Add a scalar set of type index i.

add_set(sets, i, dim)::Int64

Add a vector set of type index i and dimension dim.

Both methods return a unique Int64 of the set that can be used to reference this set.

rows(sets, ci::MOI.ConstraintIndex)::Union{Int,UnitRange{Int}}

Return the rows in 1:MOI.dimension(sets) corresponding to the set of id ci.value.

For scalar sets, this returns an Int. For vector sets, this returns an UnitRange{Int}.

num_rows(sets::OrderedProductOfSets, ::Type{S}) where {S}

Return the number of rows corresponding to a set of type S. That is, it is the sum of the dimensions of the sets of type S.

set_with_dimension(::Type{S}, dim) where {S<:MOI.AbstractVectorSet}

Returns the instance of S of MOI.dimension dim. This needs to be implemented for sets of type S to be useable with MatrixOfConstraints.

abstract type ProductOfSets{T} end

Represents a cartesian product of sets of given types.

abstract type MixOfScalarSets{T} <: ProductOfSets{T} end

Product of scalar sets in the order the constraints are added, mixing the constraints of different types.

Use @mix_of_scalar_sets to generate a new subtype.

@mix_of_scalar_sets(name, set_types...)

Generate a new MixOfScalarSets subtype.

Example

julia> import MathOptInterface as MOI

julia> MOI.Utilities.@mix_of_scalar_sets(
           MixedIntegerLinearProgramSets,
           MOI.GreaterThan{T},
           MOI.LessThan{T},
           MOI.EqualTo{T},
           MOI.Integer,
       )

abstract type OrderedProductOfSets{T} <: ProductOfSets{T} end

Product of sets in the order the constraints are added, grouping the constraints of the same types contiguously.

Use @product_of_sets to generate new subtypes.

@product_of_sets(name, set_types...)

Generate a new OrderedProductOfSets subtype.

Example

julia> import MathOptInterface as MOI

julia> MOI.Utilities.@product_of_sets(
           LinearOrthants,
           MOI.Zeros,
           MOI.Nonnegatives,
           MOI.Nonpositives,
           MOI.ZeroOne,
       )

get_fallback(model::MOI.ModelLike, ::MOI.ObjectiveValue)

Compute the objective function value using the VariablePrimal results and the ObjectiveFunction value.

get_fallback(
    model::MOI.ModelLike,
    ::MOI.DualObjectiveValue,
    ::Type{T},
)::T where {T}

Compute the dual objective value of type T using the ConstraintDual results and the ConstraintFunction and ConstraintSet values.

Note that the nonlinear part of the model is ignored.

get_fallback(
    model::MOI.ModelLike,
    ::MOI.ConstraintPrimal,
    constraint_index::MOI.ConstraintIndex,
)

Compute the value of the function of the constraint of index constraint_index using the VariablePrimal results and the ConstraintFunction values.

get_fallback(
    model::MOI.ModelLike,
    attr::MOI.ConstraintDual,
    ci::MOI.ConstraintIndex{Union{MOI.VariableIndex,MOI.VectorOfVariables}},
    ::Type{T} = Float64,
) where {T}

Compute the dual of the constraint of index ci using the ConstraintDual of other constraints and the ConstraintFunction values.

Throws an error if some constraints are quadratic or if there is one another MOI.VariableIndex-in-S or MOI.VectorOfVariables-in-S constraint with one of the variables in the function of the constraint ci.

eval_variables(value_fn::Function, f::MOI.AbstractFunction)

Returns the value of function f if each variable index vi is evaluated as value_fn(vi).

Note that value_fn must return a Number. See substitute_variables for a similar function where value_fn returns an MOI.AbstractScalarFunction.

map_indices(index_map::Function, attr::MOI.AnyAttribute, x::X)::X where {X}

Substitute any MOI.VariableIndex (resp. MOI.ConstraintIndex) in x by the MOI.VariableIndex (resp. MOI.ConstraintIndex) of the same type given by index_map(x).

When to implement this method for new types X

This function is used by implementations of MOI.copy_to on constraint functions, attribute values and submittable values. If you define a new attribute whose values x::X contain variable or constraint indices, you must also implement this function.

map_indices(
    variable_map::AbstractDict{T,T},
    x::X,
)::X where {T<:MOI.Index,X}

Shortcut for map_indices(vi -> variable_map[vi], x).

substitute_variables(variable_map::Function, x)

Substitute any MOI.VariableIndex in x by variable_map(x). The variable_map function returns either MOI.VariableIndex or MOI.ScalarAffineFunction, see eval_variables for a similar function where variable_map returns a number.

This function is used by bridge optimizers on constraint functions, attribute values and submittable values when at least one variable bridge is used hence it needs to be implemented for custom types that are meant to be used as attribute or submittable value.

filter_variables(keep::Function, f::AbstractFunction)

Return a new function f with the variable vi such that !keep(vi) removed.

WARNING: Don't define filter_variables(::Function, ...) because Julia will not specialize on this. Define instead filter_variables(::F, ...) where {F<:Function}.

remove_variable(f::AbstractFunction, vi::VariableIndex)

Return a new function f with the variable vi removed.

remove_variable(
    f::MOI.AbstractFunction,
    s::MOI.AbstractSet,
    vi::MOI.VariableIndex,
)

Return a tuple (g, t) representing the constraint f-in-s with the variable vi removed. That is, the terms containing the variable vi in the function f are removed and the dimension of the set s is updated if needed (for example, when f is a VectorOfVariables with vi being one of the variables).

all_coefficients(p::Function, f::MOI.AbstractFunction)

Determine whether predicate p returns true for all coefficients of f, returning false as soon as the first coefficient of f for which p returns false is encountered (short-circuiting). Similar to all.

unsafe_add(t1::MOI.ScalarAffineTerm, t2::MOI.ScalarAffineTerm)

Sums the coefficients of t1 and t2 and returns an output MOI.ScalarAffineTerm. It is unsafe because it uses the variable of t1 as the variable of the output without checking that it is equal to that of t2.

unsafe_add(t1::MOI.ScalarQuadraticTerm, t2::MOI.ScalarQuadraticTerm)

Sums the coefficients of t1 and t2 and returns an output MOI.ScalarQuadraticTerm. It is unsafe because it uses the variable's of t1 as the variable's of the output without checking that they are the same (up to permutation) to those of t2.

unsafe_add(t1::MOI.VectorAffineTerm, t2::MOI.VectorAffineTerm)

Sums the coefficients of t1 and t2 and returns an output MOI.VectorAffineTerm. It is unsafe because it uses the output_index and variable of t1 as the output_index and variable of the output term without checking that they are equal to those of t2.

isapprox_zero(f::MOI.AbstractFunction, tol)

Return a Bool indicating whether the function f is approximately zero using tol as a tolerance.

Important note

This function assumes that f does not contain any duplicate terms, you might want to first call canonical if that is not guaranteed.

Example

julia> import MathOptInterface as MOI

julia> x = MOI.VariableIndex(1)
MOI.VariableIndex(1)

julia> f = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.([1, -1], [x, x]), 0)
(0) + (1) MOI.VariableIndex(1) - (1) MOI.VariableIndex(1)

julia> MOI.Utilities.isapprox_zero(f, 1e-8)
false

julia> MOI.Utilities.isapprox_zero(MOI.Utilities.canonical(f), 1e-8)
true

modify_function(f::AbstractFunction, change::AbstractFunctionModification)

Return a copy of the function f, modified according to change.

zero_with_output_dimension(::Type{T}, output_dimension::Integer) where {T}

Create an instance of type T with the output dimension output_dimension.

This is mostly useful in Bridges, when code needs to be agnostic to the type of vector-valued function that is passed in.

is_canonical(f::Union{ScalarAffineFunction, VectorAffineFunction})

Returns a Bool indicating whether the function is in canonical form. See canonical.

is_canonical(f::Union{ScalarQuadraticFunction, VectorQuadraticFunction})

Returns a Bool indicating whether the function is in canonical form. See canonical.

canonical(f::MOI.AbstractFunction)

Returns the function in a canonical form, that is,

• A term appear only once.
• The coefficients are nonzero.
• The terms appear in increasing order of variable where there the order of the variables is the order of their value.
• For a AbstractVectorFunction, the terms are sorted in ascending order of output index.

The output of canonical can be assumed to be a copy of f, even for VectorOfVariables.

Example

julia> import MathOptInterface as MOI

julia> x, y, z = MOI.VariableIndex.(1:3);

julia> f = MOI.ScalarAffineFunction(
           MOI.ScalarAffineTerm.(Float64[2, 1, 3, -2, -3], [y, x, z, x, z]),
           5.0,
       )
5.0 + 2.0 MOI.VariableIndex(2) + 1.0 MOI.VariableIndex(1) + 3.0 MOI.VariableIndex(3) - 2.0 MOI.VariableIndex(1) - 3.0 MOI.VariableIndex(3)

julia> MOI.Utilities.canonical(f)
5.0 - 1.0 MOI.VariableIndex(1) + 2.0 MOI.VariableIndex(2)

canonicalize!(f::Union{ScalarAffineFunction, VectorAffineFunction})

Convert a function to canonical form in-place, without allocating a copy to hold the result. See canonical.

canonicalize!(f::Union{ScalarQuadraticFunction, VectorQuadraticFunction})

Convert a function to canonical form in-place, without allocating a copy to hold the result. See canonical.

scalar_type(F::Type{<:MOI.AbstractVectorFunction})

Type of functions obtained by indexing objects obtained by calling eachscalar on functions of type F.

scalarize(func::MOI.VectorOfVariables, ignore_constants::Bool = false)

Returns a vector of scalar functions making up the vector function in the form of a Vector{MOI.SingleVariable}.

See also eachscalar.

scalarize(func::MOI.VectorAffineFunction{T}, ignore_constants::Bool = false)

Returns a vector of scalar functions making up the vector function in the form of a Vector{MOI.ScalarAffineFunction{T}}.

See also eachscalar.

scalarize(func::MOI.VectorQuadraticFunction{T}, ignore_constants::Bool = false)

Returns a vector of scalar functions making up the vector function in the form of a Vector{MOI.ScalarQuadraticFunction{T}}.

See also eachscalar.

eachscalar(f::MOI.AbstractVectorFunction)

Returns an iterator for the scalar components of the vector function.

See also scalarize.

eachscalar(f::MOI.AbstractVector)

Returns an iterator for the scalar components of the vector.

promote_operation(
    op::Function,
    ::Type{T},
    ArgsTypes::Type{<:Union{T,AbstractVector{T},MOI.AbstractFunction}}...,
) where {T<:Number}

Compute the return type of the call operate(op, T, args...), where the types of the arguments args are ArgsTypes.

One assumption is that the element type T is invariant under each operation. That is, op(::T, ::T)::T where op is a +, -, *, and /.

There are six methods for which we implement Utilities.promote_operation:

1. + a. promote_operation(::typeof(+), ::Type{T}, ::Type{F1}, ::Type{F2})
2. - a. promote_operation(::typeof(-), ::Type{T}, ::Type{F}) b. promote_operation(::typeof(-), ::Type{T}, ::Type{F1}, ::Type{F2})
3. * a. promote_operation(::typeof(*), ::Type{T}, ::Type{T}, ::Type{F}) b. promote_operation(::typeof(*), ::Type{T}, ::Type{F}, ::Type{T}) c. promote_operation(::typeof(*), ::Type{T}, ::Type{F1}, ::Type{F2}) where F1 and F2 are VariableIndex or ScalarAffineFunction d. promote_operation(::typeof(*), ::Type{T}, ::Type{<:Diagonal{T}}, ::Type{F}
4. / a. promote_operation(::typeof(/), ::Type{T}, ::Type{F}, ::Type{T})
5. vcat a. promote_operation(::typeof(vcat), ::Type{T}, ::Type{F}...)
6. imag a. promote_operation(::typeof(imag), ::Type{T}, ::Type{F}) where F is VariableIndex or VectorOfVariables

In each case, F (or F1 and F2) is one of the ten supported types, with a restriction that the mathematical operation makes sense, for example, we don't define promote_operation(-, T, F1, F2) where F1 is a scalar-valued function and F2 is a vector-valued function. The ten supported types are:

1. ::T
2. ::VariableIndex
3. ::ScalarAffineFunction{T}
4. ::ScalarQuadraticFunction{T}
5. ::ScalarNonlinearFunction
6. ::AbstractVector{T}
7. ::VectorOfVariables
8. ::VectorAffineFunction{T}
9. ::VectorQuadraticFunction{T}
10. ::VectorNonlinearFunction

operate(
    op::Function,
    ::Type{T},
    args::Union{T,MOI.AbstractFunction}...,
)::MOI.AbstractFunction where {T<:Number}

Returns an MOI.AbstractFunction representing the function resulting from the operation op(args...) on functions of coefficient type T.

No argument can be modified.

Methods

1. + a. operate(::typeof(+), ::Type{T}, ::F1) b. operate(::typeof(+), ::Type{T}, ::F1, ::F2) c. operate(::typeof(+), ::Type{T}, ::F1...)
2. - a. operate(::typeof(-), ::Type{T}, ::F) b. operate(::typeof(-), ::Type{T}, ::F1, ::F2)
3. * a. operate(::typeof(*), ::Type{T}, ::T, ::F) b. operate(::typeof(*), ::Type{T}, ::F, ::T) c. operate(::typeof(*), ::Type{T}, ::F1, ::F2) where F1 and F2 are VariableIndex or ScalarAffineFunction d. operate(::typeof(*), ::Type{T}, ::Diagonal{T}, ::F)
4. / a. operate(::typeof(/), ::Type{T}, ::F, ::T)
5. vcat a. operate(::typeof(vcat), ::Type{T}, ::F...)
6. imag a. operate(::typeof(imag), ::Type{T}, ::F) where F is VariableIndex or VectorOfVariables

One assumption is that the element type T is invariant under each operation. That is, op(::T, ::T)::T where op is a +, -, *, and /.

In each case, F (or F1 and F2) is one of the ten supported types, with a restriction that the mathematical operation makes sense, for example, we don't define promote_operation(-, T, F1, F2) where F1 is a scalar-valued function and F2 is a vector-valued function. The ten supported types are:

1. ::T
2. ::VariableIndex
3. ::ScalarAffineFunction{T}
4. ::ScalarQuadraticFunction{T}
5. ::ScalarNonlinearFunction
6. ::AbstractVector{T}
7. ::VectorOfVariables
8. ::VectorAffineFunction{T}
9. ::VectorQuadraticFunction{T}
10. ::VectorNonlinearFunction

operate!(
    op::Function,
    ::Type{T},
    args::Union{T,MOI.AbstractFunction}...,
)::MOI.AbstractFunction where {T<:Number}

Returns an MOI.AbstractFunction representing the function resulting from the operation op(args...) on functions of coefficient type T.

The first argument may be modified, in which case the return value is identical to the first argument. For operations which cannot be implemented in-place, this function returns a new object.

operate_output_index!(
    op::Union{typeof(+),typeof(-)},
    ::Type{T},
    output_index::Integer,
    f::Union{AbstractVector{T},MOI.AbstractVectorFunction}
    g::Union{T,MOI.AbstractScalarFunction}...
) where {T<:Number}

Return an MOI.AbstractVectorFunction in which the scalar function in row output_index is the result of op(f[output_index], g).

The functions at output index different to output_index are the same as the functions at the same output index in func. The first argument may be modified.

Methods

1. + a. operate_output_index!(+, ::Type{T}, ::Int, ::VectorF, ::ScalarF)
2. - a. operate_output_index!(-, ::Type{T}, ::Int, ::VectorF, ::ScalarF)

vectorize(x::AbstractVector{<:Number})

Returns x.

vectorize(x::AbstractVector{MOI.VariableIndex})

Returns the vector of scalar affine functions in the form of a MOI.VectorAffineFunction{T}.

vectorize(funcs::AbstractVector{MOI.ScalarAffineFunction{T}}) where T

Returns the vector of scalar affine functions in the form of a MOI.VectorAffineFunction{T}.

vectorize(funcs::AbstractVector{MOI.ScalarQuadraticFunction{T}}) where T

Returns the vector of scalar quadratic functions in the form of a MOI.VectorQuadraticFunction{T}.

shift_constant(set::MOI.AbstractScalarSet, offset)

Returns a new scalar set new_set such that func-in-set is equivalent to func + offset-in-new_set.

Use supports_shift_constant to check if the set supports shifting:

if MOI.Utilities.supports_shift_constant(typeof(set))
    new_set = MOI.Utilities.shift_constant(set, -func.constant)
    func.constant = 0
    MOI.add_constraint(model, func, new_set)
else
    MOI.add_constraint(model, func, set)
end

Note for developers

Only define this function if it makes sense and you have implemented supports_shift_constant to return true.

Example

julia> import MathOptInterface as MOI

julia> set = MOI.Interval(-2.0, 3.0)
MathOptInterface.Interval{Float64}(-2.0, 3.0)

julia> MOI.Utilities.supports_shift_constant(typeof(set))
true

julia> MOI.Utilities.shift_constant(set, 1.0)
MathOptInterface.Interval{Float64}(-1.0, 4.0)

supports_shift_constant(::Type{S}) where {S<:MOI.AbstractSet}

Return true if shift_constant is defined for set S.

See also shift_constant.

Example

julia> import MathOptInterface as MOI

julia> MOI.Utilities.supports_shift_constant(MOI.Interval{Float64})
true

julia> MOI.Utilities.supports_shift_constant(MOI.ZeroOne)
false

normalize_and_add_constraint(
    model::MOI.ModelLike,
    func::MOI.AbstractScalarFunction,
    set::MOI.AbstractScalarSet;
    allow_modify_function::Bool = false,
)

Adds the scalar constraint obtained by moving the constant term in func to the set in model. If allow_modify_function is true then the function func can be modified.

normalize_constant(
    func::MOI.AbstractScalarFunction,
    set::MOI.AbstractScalarSet;
    allow_modify_function::Bool = false,
)

Return the func-in-set constraint in normalized form. That is, if func is MOI.ScalarQuadraticFunction or MOI.ScalarAffineFunction, the constant is moved to the set. If allow_modify_function is true then the function func can be modified.

get_bounds(model::MOI.ModelLike, ::Type{T}, x::MOI.VariableIndex)

Return a tuple (lb, ub) of type Tuple{T, T}, where lb and ub are lower and upper bounds, respectively, imposed on x in model.

get_bounds(
    model::MOI.ModelLike,
    bounds_cache::Dict{MOI.VariableIndex,NTuple{2,T}},
    f::MOI.ScalarAffineFunction{T},
) where {T} --> Union{Nothing,NTuple{2,T}}

Return the lower and upper bound of f as a tuple. If the domain is not bounded, return nothing.

get_bounds(
    model::MOI.ModelLike,
    bounds_cache::Dict{MOI.VariableIndex,NTuple{2,T}},
    x::MOI.VariableIndex,
) where {T} --> Union{Nothing,NTuple{2,T}}

Return the lower and upper bound of x as a tuple. If the domain is not bounded, return nothing.

Similar to get_bounds(::MOI.ModelLike, ::Type{T}, ::MOI.VariableIndex), except that the second argument is a cache which maps variables to their bounds and avoids repeated lookups.

is_diagonal_vectorized_index(index::Base.Integer)

Return whether index is the index of a diagonal element in a MOI.AbstractSymmetricMatrixSetTriangle set.

side_dimension_for_vectorized_dimension(n::Integer)

Return the dimension d such that MOI.dimension(MOI.PositiveSemidefiniteConeTriangle(d)) is n.

abstract type AbstractDistance end

An abstract type used to enable dispatch of Utilities.distance_to_set.

ProjectionUpperBoundDistance() <: AbstractDistance

An upper bound on the minimum distance between point and the closest feasible point in set.

Definition of distance

The minimum distance is computed as:

\[d(x, \mathcal{K}) = \min_{y \in \mathcal{K}} || x - y ||\]

where $x$ is point and $\mathcal{K}$ is set. The norm is computed as:

\[||x|| = \sqrt{f(x, x, \mathcal{K})}\]

where $f$ is Utilities.set_dot.

In the default case, where the set does not have a specialized method for Utilities.set_dot, the norm is equivalent to the Euclidean norm $||x|| = \sqrt{\sum x_i^2}$.

Why an upper bound?

In most cases, distance_to_set should return the smallest upper bound, but it may return a larger value if the smallest upper bound is expensive to compute.

For example, given an epigraph from of a conic set, $\{(t, x) | f(x) \le t\}$, it may be simpler to return $\delta$ such that $f(x) \le t + \delta$, rather than computing the nearest projection onto the set.

If the distance is not the smallest upper bound, the docstring of the appropriate distance_to_set method must describe the way that the distance is computed.

distance_to_set(
    [d::AbstractDistance = ProjectionUpperBoundDistance()],]
    point::T,
    set::MOI.AbstractScalarSet,
) where {T}

distance_to_set(
    [d::AbstractDistance = ProjectionUpperBoundDistance(),]
    point::AbstractVector{T},
    set::MOI.AbstractVectorSet,
) where {T}

Compute the distance between point and set using the distance metric d. If point is in the set set, this function must return zero(T).

If d is omitted, the default distance is Utilities.ProjectionUpperBoundDistance.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.RotatedSecondOrderCone)

Let (t, u, y...) = x. Return the 2-norm of the vector d such that in x + d, u is projected to 1 if u <= 0, and t is increased such that x + d belongs to the set.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.ExponentialCone)

Let (u, v, w) = x. If v > 0, return the epigraph distance d such that (u, v, w + d) belongs to the set.

If v <= 0 return the 2-norm of the vector d such that x + d = (u, 1, z) where z satisfies the constraints.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.DualExponentialCone)

Let (u, v, w) = x. If u < 0, return the epigraph distance d such that (u, v, w + d) belongs to the set.

If u >= 0 return the 2-norm of the vector d such that x + d = (u, -1, z) where z satisfies the constraints.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.GeometricMeanCone)

Let (t, y...) = x. If all y are non-negative, return the epigraph distance d such that (t + d, y...) belongs to the set.

If any y are strictly negative, return the 2-norm of the vector d that projects negative y elements to 0 and t to ℝ₋.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.PowerCone)

Let (a, b, c) = x. If a and b are non-negative, return the epigraph distance required to increase c such that the constraint is satisfied.

If a or b is strictly negative, return the 2-norm of the vector d such that in the vector x + d: c, and any negative a and b are projected to 0.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.DualPowerCone)

Let (a, b, c) = x. If a and b are non-negative, return the epigraph distance required to increase c such that the constraint is satisfied.

If a or b is strictly negative, return the 2-norm of the vector d such that in the vector x + d: c, and any negative a and b are projected to 0.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.NormOneCone)

Let (t, y...) = x. Return the epigraph distance d such that (t + d, y...) belongs to the set.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.NormInfinityCone)

Let (t, y...) = x. Return the epigraph distance d such that (t + d, y...) belongs to the set.

distance_to_set(::ProjectionUpperBoundDistance, x, ::MOI.RelativeEntropyCone)

Let (u, v..., w...) = x. If v and w are strictly positive, return the epigraph distance required to increase u such that the constraint is satisfied.

If any elements in v or w are non-positive, return the 2-norm of the vector d such that in the vector x + d: any non-positive elements in v and w are projected to 1, and u is projected such that the epigraph constraint holds.

distance_to_set(::ProjectionUpperBoundDistance, x, set::MOI.NormCone)

Let (t, y...) = x. Return the epigraph distance d such that (t + d, y...) belongs to the set.

set_dot(x::AbstractVector, y::AbstractVector, set::AbstractVectorSet)

Return the scalar product between a vector x of the set set and a vector y of the dual of the set s.

set_dot(x, y, set::AbstractScalarSet)

Return the scalar product between a number x of the set set and a number y of the dual of the set s.

DoubleDict{V}

An optimized dictionary to map MOI.ConstraintIndex to values of type V.

Works as a AbstractDict{MOI.ConstraintIndex,V} with minimal differences.

If V is also a MOI.ConstraintIndex, use IndexDoubleDict.

Note that MOI.ConstraintIndex is not a concrete type, opposed to MOI.ConstraintIndex{MOI.VariableIndex, MOI.Integers}, which is a concrete type.

When looping through multiple keys of the same Function-in-Set type, use

inner = dict[F, S]

to return a type-stable DoubleDictInner.

DoubleDictInner{F,S,V}

A type stable inner dictionary of DoubleDict.

IndexDoubleDict

A specialized version of [DoubleDict] in which the values are of type MOI.ConstraintIndex

When looping through multiple keys of the same Function-in-Set type, use

inner = dict[F, S]

to return a type-stable IndexDoubleDictInner.

IndexDoubleDictInner{F,S}

A type stable inner dictionary of IndexDoubleDict.

outer_keys(d::AbstractDoubleDict)

Return an iterator over the outer keys of the AbstractDoubleDict d. Each outer key is a Tuple{Type,Type} so that a double loop can be easily used:

for (F, S) in DoubleDicts.outer_keys(dict)
    for (k, v) in dict[F, S]
        # ...
    end
end

For performance, it is recommended that the inner loop lies in a separate function to guarantee type-stability. Some outer keys (F, S) might lead to an empty dict[F, S]. If you want only nonempty dict[F, S], use nonempty_outer_keys.

nonempty_outer_keys(d::AbstractDoubleDict)

Return a vector of outer keys of the AbstractDoubleDict d.

Only outer keys that have a nonempty set of inner keys will be returned.

Each outer key is a Tuple{Type,Type} so that a double loop can be easily used

for (F, S) in DoubleDicts.nonempty_outer_keys(dict)
    for (k, v) in dict[F, S]
        # ...
    end
end
For performance, it is recommended that the inner loop lies in a separate
function to guarantee type-stability.

If you want an iterator of all current outer keys, use [`outer_keys`](@ref).