diff --git a/src/optimizer_interface.jl b/src/optimizer_interface.jl
index 79941426f8f..1068113d7d4 100644
--- a/src/optimizer_interface.jl
+++ b/src/optimizer_interface.jl
@@ -985,6 +985,114 @@ function optimizer_index(x::Union{VariableRef,ConstraintRef{Model}})
     return _moi_optimizer_index(backend(model), index(x))
 end
 
+struct ObjectiveFunctionAttribute{A,F}
+    attr::A
+end
+
+"""
+    struct ObjectiveDualStart <: MOI.AbstractModelAttribute end
+
+If the objective function had a dual, it would be `-1` for the Lagrangian
+function to be the same.
+When the `MOI.Bridges.Objective.SlackBridge` is used, it creates a constraint.
+The dual of this constraint is therefore `-1` as well.
+When setting this attribute, it allows to set the constraint dual of this
+constraint.
+"""
+struct ObjectiveDualStart <: MOI.AbstractModelAttribute end
+# Defining it for `MOI.set` leads to ambiguity
+function MOI.throw_set_error_fallback(
+    ::MOI.ModelLike,
+    ::ObjectiveDualStart,
+    value,
+)
+    return nothing
+end
+
+"""
+    struct ObjectiveSlackGapPrimalStart <: MOI.AbstractModelAttribute end
+
+If the objective function had a dual, it would be `-1` for the Lagrangian
+function to be the same.
+When the `MOI.Bridges.Objective.SlackBridge` is used, it creates a constraint.
+The dual of this constraint is therefore `-1` as well.
+When setting this attribute, it allows to set the constraint dual of this
+constraint.
+"""
+struct ObjectiveSlackGapPrimalStart <: MOI.AbstractModelAttribute end
+function MOI.throw_set_error_fallback(
+    ::MOI.ModelLike,
+    ::ObjectiveSlackGapPrimalStart,
+    value,
+)
+    return nothing
+end
+
+function MOI.set(
+    b::MOI.Bridges.AbstractBridgeOptimizer,
+    attr::ObjectiveFunctionAttribute{A,F},
+    value,
+) where {A,F}
+    obj_attr = MOI.ObjectiveFunction{F}()
+    if MOI.Bridges.is_bridged(b, obj_attr)
+        return MOI.set(
+            MOI.Bridges.recursive_model(b),
+            attr,
+            MOI.Bridges.bridge(b, obj_attr),
+            value,
+        )
+    else
+        return MOI.set(b.model, attr.attr, value)
+    end
+end
+
+function MOI.set(
+    b::MOI.Bridges.AbstractBridgeOptimizer,
+    attr::Union{ObjectiveDualStart,ObjectiveSlackGapPrimalStart},
+    value,
+)
+    if MOI.Bridges.is_objective_bridged(b)
+        F = MOI.Bridges.Objective.function_type(
+            MOI.Bridges.Objective.bridges(b),
+        )
+        return MOI.set(
+            b,
+            ObjectiveFunctionAttribute{typeof(attr),F}(attr),
+            value,
+        )
+    else
+        return MOI.set(b.model, attr, value)
+    end
+end
+
+function MOI.set(
+    model::MOI.ModelLike,
+    ::ObjectiveFunctionAttribute{ObjectiveDualStart},
+    b::MOI.Bridges.Objective.SlackBridge,
+    value,
+)
+    return MOI.set(model, MOI.ConstraintDualStart(), b.constraint, value)
+end
+
+function MOI.set(
+    model::MOI.ModelLike,
+    ::ObjectiveFunctionAttribute{ObjectiveSlackGapPrimalStart},
+    b::MOI.Bridges.Objective.SlackBridge{T},
+    value,
+) where {T}
+    # `f(x) - slack = value` so `slack = f(x) - value`
+    fun = MOI.get(model, MOI.ConstraintFunction(), b.constraint)
+    set = MOI.get(model, MOI.ConstraintSet(), b.constraint)
+    MOI.Utilities.operate!(-, T, fun, MOI.constant(set))
+    # `fun = f - slack` so we remove the term `-slack` to get `f`
+    f = MOI.Utilities.remove_variable(fun, b.slack)
+    f_val = MOI.Utilities.eval_variables(f) do v
+        return MOI.get(model, MOI.VariablePrimalStart(), v)
+    end
+    MOI.set(model, MOI.VariablePrimalStart(), b.slack, f_val - value)
+    return MOI.set(model, MOI.ConstraintPrimalStart(), b.constraint, value)
+end
+
 """
     set_start_values(
         model::Model;
@@ -1090,6 +1198,8 @@ function set_start_values(
     for (ci, dual_start) in constraint_dual
         set_dual_start_value(ci, dual_start)
     end
+    MOI.set(model, ObjectiveDualStart(), -1.0)
+    MOI.set(model, ObjectiveSlackGapPrimalStart(), 0.0)
     return
 end