Constraints

A guide for constraints in InfiniteOpt. See the respective technical manual for more details.

Overview

Constraints are a key part of infinite dimensional problems and serve as a fundamental utility of InfiniteOpt. In particular, InfiniteOpt supports finite constraints that entail finite variables and/or measures that fully remove any infinite parameter dependencies (e.g., first stage constraints), infinite constraints that are enforced over the entire domain of its infinite parameter dependencies (e.g., path constraints), and restricted constraints which are enforced over some specified sub-domain of its infinite parameter dependencies (e.g., boundary conditions). This page will highlight how to implement these types of constraints in InfiniteOpt.

Basic Usage

Principally, the @constraint macro is used to define constraints. First, let's set up an infinite model with variables that we can add constraints to:

julia> model = InfiniteModel();

julia> @infinite_parameter(model, t in [0, 10]);

julia> @infinite_parameter(model, x[1:2] in [-2, 2]);

julia> @variable(model, ya, Infinite(t, x));

julia> @variable(model, yb, Infinite(t));

julia> @variable(model, z[1:2]);

Scalar Constraints

Scalar constraints use scalar functions of variables. For example, let's define the constraint $||z||^2 + 2y_a(t, x) \leq 0, \ \forall t \in [0, 10], x \in [-2, 2]^2$ using @constraint:

julia> @constraint(model, c1, sum(z[i]^2 for i = 1:2) + 2ya <= 0)
c1 : z[1]² + z[2]² + 2 ya(t, x) ≤ 0, ∀ t ∈ [0, 10], x[1] ∈ [-2, 2], x[2] ∈ [-2, 2]

Thus, we added an infinite constraint (which infinite with respect to t and x) to model and stored the corresponding constraint reference to c1. Note that this is enforced over the full infinite domains of the infinite parameters t and x which are implicitly used by c1. For scalar constraints like this one, the allowed constraint operators are ==, <=, ≤, >=, and ≥.

Similarly, we can define an array of constraints with varied indexes by including an additional argument before the constraint expression. For example, let's define $3z_i - 14 = 0, \ \forall i \in \{1,2\}$:

julia> @constraint(model, c2[i = 1:2], 3z[i] - 14 == 0)
2-element Vector{InfOptConstraintRef}:
 c2[1] : 3 z[1] = 14
 c2[2] : 3 z[2] = 14

Thus, we added two constraints to model and stored a vector of the corresponding constraint references to the Julia variable c2. To learn more about building containers of constraints please see JuMP's constraint container documentation.

Multi-Dimensional Constraints

Building upon JuMP we support a variety of multi-dimensional constraint types. For example, we can define the vector constraint:

julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
 1  2
 3  4

julia> b = [5, 6]
2-element Vector{Int64}:
 5
 6

julia> @constraint(model, A * z - b in MOI.Nonnegatives(2))
[z[1] + 2 z[2] - 5, 3 z[1] + 4 z[2] - 6] ∈ MathOptInterface.Nonnegatives(2)

See JuMP's constraint documentation for a thorough tutorial on the accepted syntax and constraint types.

Restricted Constraints

Restricted constraints entail an infinite domain (determined by the infinite parameters they explicitly/implicitly depend on) that is restricted to a certain sub-domain. Such constraints are common for enforcing initial/boundary conditions and for enforcing path constraints over a certain sub-domain.

These types of constraints are defined adding DomainRestrictions. For example, let's add the initial condition $y_b(0) = 0$:

julia> @constraint(model, initial, yb == 0, DomainRestrictions(t => 0))
initial : yb(t) = 0, ∀ t = 0

Thus, we have added a constraint to model defined over the sub-domain $t = 0$ in accordance with the initial condition.

julia> @constraint(model, yb(0) == 0)
yb(0) = 0

More complex sub-domains can be specified by simply adding more restrictions. To illustrate this, let's define the constraint $2y_b^2(t, x) + z_1 \geq 3, \ \forall t = 0, \ x \in [-1, 1]^2$:

julia> @constraint(model, 2ya^2 + z[1] >= 3, DomainRestrictions(t => 0, x => [-1, 1]))
2 ya(t, x)² + z[1] ≥ 3, ∀ t = 0, x[1] ∈ [-1, 1], x[2] ∈ [-1, 1]

Now we have added constraints to our model, and it is ready to be solved!

Data Structure

Here we detail the data structures used to store constraints in InfiniteOpt. In general, constraints in JuMP are of the form: function in set where function corresponds to a JuMP expression and set corresponds to a MOI set. This leads to the following data structures:

The above combos are then stored in JuMP.ScalarConstraints and `JuMP.VectorConstraints.

Restricted constraints are built upon this data structure where the underlying constraint is created in the same manner. Then the specified DomainRestrictions are added by creating a DomainRestrictedConstraint which stores the JuMP.AbstractConstraint and the restrictions.

These constraint objects are what store constraints in InfiniteModels. And these are pointed to by InfOptConstraintRefs.

Definition

In this section, we describe the ins and outs of defining constraints. Note that this process is analogous to the manner in which variables are defined and added to the model.

Manual Definition

Defining a constraint principally involves the following steps:

• Define the constraint information (i.e., function, set, and domain restrictions)
• Construct a concrete subtype of JuMP.AbstractConstraint to store the constraint information
• Add the AbstractConstraint object to an InfiniteModel and assign a name
• Create an InfOptConstraintRef that points to the constraint object stored in the model.

The constraint objects are specified via JuMP.build_constraint which requires that the user provides a function, set, and optionally include domain restrictions. For example, let's build a scalar constraint $3y_a(t, x) - y_b^2(t) \leq 0, \ \forall t \in [0, 10], x \in [-2, 2]^2$ over its full infinite domain (i.e., have no DomainRestrictions):

julia> constr = build_constraint(error, 3ya - yb^2, MOI.LessThan(0.0));

Now the built constraint object can be added to the infinite model via add_constraint. Let's do so with our example and assign it the name of c3 (note that adding a name is optional):

julia> cref = add_constraint(model, constr, "c3")
c3 : -yb(t)² + 3 ya(t, x) ≤ 0, ∀ t ∈ [0, 10], x[1] ∈ [-2, 2], x[2] ∈ [-2, 2]

Thus, we have made our constraint and added it model and now have a constraint reference cref that we can use to access it.

The @constraint macro automate the above steps.

Macro Definition

As mentioned above in the Basic Usage section, the @constraint macro should be used to define constraints with the syntax: @constraint(model::InfiniteModel, [container/name_expr], constr_expr, [rs::DomainRestrictions]).

The second argument is optional and is used to assign a name and/or define indexing variables to be used in the constraint expression. When a name is provided it is registered and cannot be used again for another constraint or variable name. The indexing expression can be used to produce an array of constraints as shown below (notice this is equivalent to looping over individual @constraint calls):

julia> crefs = @constraint(model, [i = 1:2], 2z[i] - yb == 0)
2-element Vector{InfOptConstraintRef}:
 2 z[1] - yb(t) = 0, ∀ t ∈ [0, 10]
 2 z[2] - yb(t) = 0, ∀ t ∈ [0, 10]

julia> crefs = Vector{InfOptConstraintRef}(undef, 2);

julia> for i = 1:2
           crefs[i] = @constraint(model, 2z[i] - yb == 0)
       end

julia> crefs
2-element Vector{InfOptConstraintRef}:
 2 z[1] - yb(t) = 0, ∀ t ∈ [0, 10]
 2 z[2] - yb(t) = 0, ∀ t ∈ [0, 10]

Please refer to JuMP's constraint container documentation for a thorough tutorial on creating containers of constraints.

Any constraint type supported by JuMP can be specified in the constr_expr argument. This includes a wealth of constraint types including:

• Variable constraints
• Scalar constraints
• Vector constraints
• Conic constraints
• Indicator constraints
• Semi-definite constraints

For example, we could define the following semi-definite constraint:

julia> @constraint(model, [yb 2yb; 3yb 4yb] >= ones(2, 2), PSDCone())
[yb(t) - 1    2 yb(t) - 1
 3 yb(t) - 1  4 yb(t) - 1] ∈ PSDCone(), ∀ t ∈ [0, 10]

See JuMP's constraint documentation for a thorough tutorial on the accepted syntax and constraint types.

Finally, restrictions on the inherent infinite domain of a constraint can be specified via DomainRestrictions with the rs argument. The accepted syntax is DomainRestrictions(restricts...) where each argument of restricts can be any of the following forms:

• pref => value
• pref => [lb, ub]
• pref => IntervalDomain(lb, ub)
• prefs => value
• prefs => [lb, ub]
• prefs => IntervalDomain(lb, ub).

Note that pref and prefs must correspond to infinite parameters.

For example, we can define the constraint $y_a^2(t, x) + z_i \leq 1$ and restrict the infinite domain of $x_i$ to be $[0, 1]$:

julia> @constraint(model, [i = 1:2], ya^2 + z[i] <= 1, DomainRestrictions(x[i] => [0, 1]))
2-element Vector{InfOptConstraintRef}:
 ya(t, x)² + z[1] ≤ 1, ∀ t ∈ [0, 10], x[1] ∈ [0, 1], x[2] ∈ [-2, 2]
 ya(t, x)² + z[2] ≤ 1, ∀ t ∈ [0, 10], x[1] ∈ [-2, 2], x[2] ∈ [0, 1]

Queries

In this section, we describe a variety of methods to extract constraint information.

Basic

A number of constraint properties can be extracted via constraint references. Principally, the validity, name, model, index, and constraint object can be queried via is_valid, name, owner_model, index, and constraint_object, respectively. These methods all constitute extensions of JuMP methods and follow exactly the same behavior. Let's try them out with the following example:

julia> is_valid(model, c1) # check if contained in model
true

julia> name(c1) # get the name
"c1"

julia> m = owner_model(c1); # get the model it is added to

julia> index(c1) # get the constraint's index
InfOptConstraintIndex(1)

julia> constr = constraint_object(c1); # get the raw constraint datatype

Also, constraint_by_name can be used to retrieve a constraint reference if only the name is known and its name is unique. For example, let's extract the reference for "c1":

julia> cref = constraint_by_name(model, "c1")
c1 : z[1]² + z[2]² + 2 ya(t, x) ≤ 0, ∀ t ∈ [0, 10], x[1] ∈ [-2, 2], x[2] ∈ [-2, 2]

Domain Restrictions

As explained above, restricted constraints serve as a key capability of InfiniteOpt. Information about domain restrictions can be obtained via has_domain_restrictions and domain_restrictions which indicate if a constraint is restricted and what its DomainRestrictions are, respectively. These are exemplified below:

julia> has_domain_restrictions(c1) # check if constraint is bounded
false

julia> has_domain_restrictions(initial)
true

julia> domain_restrictions(initial)
Subdomain restrictions (1): t = 0

General

Constraints can be defined in a number of ways symbolically that differ from how it is actually stored in the model. This principally occurs since like terms and constants are combined where possible with the variable terms on the left-hand side and the constant on the right-hand side. For instance, the constraint $2y_b(t) + 3y_b(t) - 2 \leq 1 + z_1$ would be normalized $5y_b(t) - z_1 \leq 3$. In accordance with this behavior, normalized_rhs and normalized_coefficient can be used to query the normalized right-hand side and the coefficient of a particular variable reference, respectively. Let's employ the above example to illustrate this:

julia> @constraint(model, constr, 2yb + 3yb - 2 <= 1 + z[1])
constr : 5 yb(t) - z[1] ≤ 3, ∀ t ∈ [0, 10]

julia> normalized_rhs(constr)
3.0

julia> normalized_coefficient(constr, yb)
5.0

There also exist a number of methods for querying an infinite model about what constraints it contains. list_of_constraint_types can be used query what types of constraints have been added to a model. This is provided as a list of tuples where the first element is the expression type and the second element is the set type (recall that constraints are stored in the form func-in-set). Thus, for our current model we obtain:

julia> list_of_constraint_types(model)
4-element Vector{Tuple{DataType,DataType}}:
 (GenericQuadExpr{Float64, GeneralVariableRef}, MathOptInterface.LessThan{Float64})
 (GenericQuadExpr{Float64, GeneralVariableRef}, MathOptInterface.GreaterThan{Float64})
 (GenericAffExpr{Float64, GeneralVariableRef}, MathOptInterface.LessThan{Float64})
 (GenericAffExpr{Float64, GeneralVariableRef}, MathOptInterface.EqualTo{Float64})

This information is useful when used in combination with the num_constraints and all_constraints methods which can take the expression type and/or the set type as inputs. Here num_constraints provides the number of constraints that match a certain type, and all_constraints returns a list of constraint references matching the criteria provided. These have been extended beyond JuMP functionality such additional methods have been provided for the cases in which one wants to query solely off of set or off expression type. Let's illustrate this with num_constraints:

julia> num_constraints(model) # total number of constraints
16

julia> num_constraints(model, GenericQuadExpr{Float64, GeneralVariableRef})
5

julia> num_constraints(model, MOI.LessThan{Float64})
5

julia> num_constraints(model, GenericQuadExpr{Float64, GeneralVariableRef},
                       MOI.LessThan{Float64})
4                   

Modification

In this section, we highlight a number of methods that can be used to modify existing constraints.

Deletion

All constraints in InfiniteOpt can be removed in like manner to typical JuMP constraints with the appropriate extension of delete. This will remove the corresponding constraint object from the model. However, please note any registered names will remain registered in the infinite model. This means that a constraint with a registered name cannot be repeatedly added and removed using the same name. To exemplify this, let's delete the constraint c1:

julia> delete(model, c1)

General

There also are a number of ways to modify information and characteristics of constraints. First, set_name can be used to specify a new name for a particular constraint. For instance, let's update the name of initial to "init_cond":

julia> set_name(initial, "init_cond")

julia> initial
init_cond : yb(t) = 0, ∀ t = 0

We can also update the normalized right hand side constant value or normalized left hand side variable coefficient value using set_normalized_rhs and set_normalized_coefficient, respectively. Let's again consider the constraint $5y_b(t) - z_1 \leq 3$ as an example. Let's change the constant term to -1 and the y_b(t) coefficient to 2.5:

julia> set_normalized_rhs(constr, -1)

julia> set_normalized_coefficient(constr, yb, 2.5)

julia> constr
constr : 2.5 yb(t) - z[1] ≤ -1, ∀ t ∈ [0, 10]

Domain Restrictions

Domain Restrictions can be added to, modified, or removed from any constraint in InfiniteOpt. Principally, this is accomplished via add_domain_restrictions, set_domain_restrictions, and delete_domain_restrictions.

First, domain restrictions can be added to a constraint via add_domain_restrictions. For example, let's add the bound $t \in [0, 1]$ to constr:

julia> add_domain_restrictions(constr, DomainRestrictions(t => [0, 1]))

julia> constr
constr : 2.5 yb(t) - z[1] ≤ -1, ∀ t ∈ [0, 1]

In similar manner, set_domain_restrictions can be employed to specify what restrictions a constraint has (overwriting any existing ones if forced). It follows the same syntax, so let's use it to change the bounds on t to $t = 0$:

julia> set_domain_restrictions(constr, DomainRestrictions(t => 0), force = true)

julia> constr
constr : 2.5 yb(t) - z[1] ≤ -1, ∀ t = 0

Finally, constraint restrictions can be deleted via delete_domain_restrictions. Now let's delete the domain restrictions associated with our example:

julia> delete_domain_restrictions(constr)

julia> constr
constr : 2.5 yb(t) - z[1] ≤ -1, ∀ t ∈ [0, 10]