The Ipopt Solver Backend

Ipopt is an open-source solver for large-scale nonlinear optimization. Use it to solve continuous nonlinear programs (NLP) defined using RelationalAI’s SolverModel API.

Capabilities

Ipopt supports the following problem types:

Problem Type	Supported
Linear Programs (LP)
Mixed-Integer Linear Programs (MILP)
Quadratic Programs (QP)
Quadratically Constrained Programs (QCP)
Nonlinear Programs (NLP)
Constraint Programming (CP)
Discrete Variables
Continuous Variables

Version and Licensing

Ipopt is bundled with RelationalAI and does not require a license key or additional setup.

Detail	Info
Version	`3.14.4`
License Type	Eclipse Public License (Free and Open Source)

Usage

To use Ipopt with RelationalAI, you can define a prescriptive model using the SolverModel API and specify "ipopt" as the solver backend:

import relationalai as rai
from relationalai.std import alias
from relationalai.experimental import solvers

# Create a RAI model.
model = rai.Model("RegionalWaterAllocation")

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# Declare water distribution stage type.
WaterSource = model.Type("WaterSource")
Region = model.Type("Region")
Consumer = model.Type("Consumer")
Consumer.region.declare()  # Region where the consumer is located.
Consumer.demand.declare()  # Water demand of the consumer.

# Declare Pipeline type to connect stages.
Pipeline = model.Type("Pipeline")
Pipeline.source.declare()
Pipeline.destination.declare()
Pipeline.capacity.declare()

# Define water sources.
with model.rule(dynamic=True):
    WaterSource.add(name="Reservoir")

# Define regions.
for region in [
    {"id": 1, "name": "North"},
    {"id": 2, "name": "South"},
    {"id": 3, "name": "East"},
    {"id": 4, "name": "West"},
]:
    with model.rule():
        Region.add(id=region["id"]).set(name=region["name"])

# Define consumers.
for consumer in [
    {"id": 1, "region_id": 1, "name": "Residential", "demand": 150.0},
    {"id": 2, "region_id": 1, "name": "Industrial", "demand": 60.0},
    {"id": 3, "region_id": 1, "name": "Farms", "demand": 40.0},
    {"id": 4, "region_id": 2, "name": "Residential", "demand": 80.0},
    {"id": 5, "region_id": 2, "name": "Industrial", "demand": 140.0},
    {"id": 6, "region_id": 2, "name": "Farms", "demand": 50.0},
    {"id": 7, "region_id": 3, "name": "Residential", "demand": 90.0},
    {"id": 8, "region_id": 3, "name": "Industrial", "demand": 180.0},
    {"id": 9, "region_id": 4, "name": "Residential", "demand": 40.0},
    {"id": 10, "region_id": 4, "name": "Industrial", "demand": 30.0},
    {"id": 11, "region_id": 4, "name": "Farms", "demand": 200.0},
]:
    with model.rule():
        region = Region(id=consumer["region_id"])
        Consumer.add(id=consumer["id"]).set(
                name=consumer["name"],
                region=region,
                demand=consumer["demand"]
        )

# Define pipelines from the reservoir to each region.
for region_id, capacity in [(1, 260.0), (2, 300.0), (3, 200.0), (4, 220.0)]:
    with model.rule():
        source = WaterSource(name="Reservoir")
        region = Region(id=region_id)
        Pipeline.add(source=source, destination=region).set(capacity=capacity)

# Define pipelines between consumers and their regions.
for consumer_id, capacity in [
    (1, 150.0), (2, 60.0), (3, 40.0),   # North
    (4, 80.0), (5, 140.0), (6, 50.0),   # South
    (7, 90.0), (8, 180.0),              # East
    (9, 40.0), (10, 30.0), (11, 200.0)  # West
]:
    with model.rule():
        consumer = Consumer(id=consumer_id)
        region = consumer.region
        Pipeline.add(source=region, destination=consumer).set(capacity=capacity)

# Create a solver model.
solver_model = solvers.SolverModel(model)

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# Specify variables, constraints, and objective.
# Define continuous variables for each pipeline to represent it's flow.
with model.rule():
    pipe = Pipeline()
    solver_model.variable(
        pipe,
        name_args=["pipe", pipe.source.name, pipe.destination.name],
        lower=0,
        upper=pipe.capacity,
    )

# Define a constraint to ensure flow conservation in each region.
# NOTE: `solvers.operators()` overrides infix operators like `==` to build
# solver expressions instead of filter expressions.
with solvers.operators():
    region = Region()
    flow_in = solvers.sum(Pipeline(destination=region), per=[region])
    flow_out = solvers.sum(Pipeline(source=region), per=[region])
    solver_model.constraint(
        flow_in == flow_out,
        name_args=["preserve_flow", region.name]
    )

# Define a constraint satisfy at least half of high-demand consumers' demand.
with model.rule():
    consumer = Consumer()
    consumer.demand >= 100.0  # Filter for high-demand consumers.
    pipe = Pipeline(source=consumer.region, destination=consumer)
    with solvers.operators():
        solver_model.constraint(
            pipe >= .5 * consumer.demand,
            name_args=["satisfy_consumer", consumer.id]
        )

# Define a constraint to limit the total flow from the reservoir.
with solvers.operators():
    total_flow = solvers.sum(Pipeline(source=WaterSource(name="Reservoir")))
    solver_model.constraint(
        total_flow <= 1000.0,  # Max flow from the reservoir
        name_args=["max_reservoir_flow"]
    )

# Define the objective to minimize unmet demand.
with solvers.operators():
    consumer = Consumer()
    pipe = Pipeline(source=consumer.region, destination=consumer)
    unmet_demand_per_consumer = consumer.demand - solvers.sum(pipe, per=[consumer])
    solver_model.min_objective(solvers.sum(unmet_demand_per_consumer))

# Create an Ipopt Solver instance.
solver = solvers.Solver("ipopt")

# Solve the model.
solver_model.solve(solver)

# View the solution's objective value.
with model.query() as select:
    response = select(solver_model.objective_value)
print(f"\n\nUnmet demand (optimized): {response.results.iloc[0, 0]}")

# View the optimized flow for each pipeline.
with model.query() as select:
    pipe = Pipeline()
    response = select(
        alias(pipe.source.name, "source_name"),
        alias(pipe.destination.name, "destination_name"),
        solver_model.value(pipe)
    )
print("\n\nOptimized pipeline flows:")
print(response.results)

Solver Options

In addition to the standard solver options, Ipopt supports other options, like s_max or warm_start_init_point, that can be set when solving a prescriptive model using the SolverModel API.

To set these options, you can pass them as keyword arguments to a SolverModel object’s .solve() method:

import relationalai as rai
from relationalai.std import alias
from relationalai.experimental import solvers

# Create a RAI model.
model = rai.Model("RegionalWaterAllocation")

119 collapsed lines
# Declare water distribution stage type.
WaterSource = model.Type("WaterSource")
Region = model.Type("Region")
Consumer = model.Type("Consumer")
Consumer.region.declare()  # Region where the consumer is located.
Consumer.demand.declare()  # Water demand of the consumer.

# Declare Pipeline type to connect stages.
Pipeline = model.Type("Pipeline")
Pipeline.source.declare()
Pipeline.destination.declare()
Pipeline.capacity.declare()

# Define water sources.
with model.rule(dynamic=True):
    WaterSource.add(name="Reservoir")

# Define regions.
for region in [
    {"id": 1, "name": "North"},
    {"id": 2, "name": "South"},
    {"id": 3, "name": "East"},
    {"id": 4, "name": "West"},
]:
    with model.rule():
        Region.add(id=region["id"]).set(name=region["name"])

# Define consumers.
for consumer in [
    {"id": 1, "region_id": 1, "name": "Residential", "demand": 150.0},
    {"id": 2, "region_id": 1, "name": "Industrial", "demand": 60.0},
    {"id": 3, "region_id": 1, "name": "Farms", "demand": 40.0},
    {"id": 4, "region_id": 2, "name": "Residential", "demand": 80.0},
    {"id": 5, "region_id": 2, "name": "Industrial", "demand": 140.0},
    {"id": 6, "region_id": 2, "name": "Farms", "demand": 50.0},
    {"id": 7, "region_id": 3, "name": "Residential", "demand": 90.0},
    {"id": 8, "region_id": 3, "name": "Industrial", "demand": 180.0},
    {"id": 9, "region_id": 4, "name": "Residential", "demand": 40.0},
    {"id": 10, "region_id": 4, "name": "Industrial", "demand": 30.0},
    {"id": 11, "region_id": 4, "name": "Farms", "demand": 200.0},
]:
    with model.rule():
        region = Region(id=consumer["region_id"])
        Consumer.add(id=consumer["id"]).set(
                name=consumer["name"],
                region=region,
                demand=consumer["demand"]
        )

# Define pipelines from the reservoir to each region.
for region_id, capacity in [(1, 260.0), (2, 300.0), (3, 200.0), (4, 220.0)]:
    with model.rule():
        source = WaterSource(name="Reservoir")
        region = Region(id=region_id)
        Pipeline.add(source=source, destination=region).set(capacity=capacity)

# Define pipelines between consumers and their regions.
for consumer_id, capacity in [
    (1, 150.0), (2, 60.0), (3, 40.0),   # North
    (4, 80.0), (5, 140.0), (6, 50.0),   # South
    (7, 90.0), (8, 180.0),              # East
    (9, 40.0), (10, 30.0), (11, 200.0)  # West
]:
    with model.rule():
        consumer = Consumer(id=consumer_id)
        region = consumer.region
        Pipeline.add(source=region, destination=consumer).set(capacity=capacity)

# Create a solver model.
solver_model = solvers.SolverModel(model)

# Specify variables, constraints, and objective.
# Define continuous variables for each pipeline to represent it's flow.
with model.rule():
    pipe = Pipeline()
    solver_model.variable(
        pipe,
        name_args=["pipe", pipe.source.name, pipe.destination.name],
        lower=0,
        upper=pipe.capacity,
    )

# Define a constraint to ensure flow conservation in each region.
# NOTE: `solvers.operators()` overrides infix operators like `==` to build
# solver expressions instead of filter expressions.
with solvers.operators():
    region = Region()
    flow_in = solvers.sum(Pipeline(destination=region), per=[region])
    flow_out = solvers.sum(Pipeline(source=region), per=[region])
    solver_model.constraint(
        flow_in == flow_out,
        name_args=["preserve_flow", region.name]
    )

# Define a constraint satisfy at least half of high-demand consumers' demand.
with model.rule():
    consumer = Consumer()
    consumer.demand >= 100.0  # Filter for high-demand consumers.
    pipe = Pipeline(source=consumer.region, destination=consumer)
    with solvers.operators():
        solver_model.constraint(
            pipe >= .5 * consumer.demand,
            name_args=["satisfy_consumer", consumer.id]
        )

# Define a constraint to limit the total flow from the reservoir.
with solvers.operators():
    total_flow = solvers.sum(Pipeline(source=WaterSource(name="Reservoir")))
    solver_model.constraint(
        total_flow <= 1000.0,  # Max flow from the reservoir
        name_args=["max_reservoir_flow"]
    )

# Define the objective to minimize unmet demand.
with solvers.operators():
    consumer = Consumer()
    pipe = Pipeline(source=consumer.region, destination=consumer)
    unmet_demand_per_consumer = consumer.demand - solvers.sum(pipe, per=[consumer])
    solver_model.min_objective(solvers.sum(unmet_demand_per_consumer))

# Create a HiGHS Solver instance.
solver = solvers.Solver("ipopt")

# Solve the model.
solver_model.solve(solver, s_max=1000, warm_start_init_point="yes")

Limitations

Some RAI operators, like solvers.if_then_else(), aren’t supported.
Ipopt requires a single, scalar objective, even if multi-objective support is added to RAI in the future.