Indicator Constraints¶

The LXIndicatorTerm represents conditional constraints: “if binary_var equals condition, then constraint holds”.

Overview¶

Indicator constraints express if-then logic:

\[\text{if } b = 1 \text{ then } ax \leq c\]

Common Use Cases:

Conditional capacity constraints
Minimum order quantities
Fixed charges and activation
Logical implications in scheduling
Resource dependencies

Linearization Method (Big-M)¶

For constraint “if b = condition then expr ≤ rhs”, linearize as:

\[\begin{split}\text{expr} \leq \text{rhs} + M \cdot (1 - b) \quad \text{if condition = True} \\ \text{expr} \leq \text{rhs} + M \cdot b \quad \text{if condition = False}\end{split}\]

Where M is a sufficiently large constant (Big-M).

Key Properties:

No auxiliary variables needed
1 constraint per indicator term
Requires computing appropriate M value
M computed from variable bounds

Basic Usage¶

Minimum Order Quantity¶

from lumix import LXVariable, LXLinearExpression
from lumix.nonlinear import LXIndicatorTerm

# Binary: order placed?
order_placed = LXVariable[Product, int]("ordered").binary()

# Continuous: order quantity
quantity = LXVariable[Product, float]("quantity").continuous().bounds(0, 1000)

# If order_placed = 1, then quantity >= 100
min_order = LXIndicatorTerm(
    binary_var=order_placed,
    condition=True,
    linear_expr=LXLinearExpression().add_term(quantity, 1.0),
    sense='>=',
    rhs=100.0
)

Facility Capacity¶

# If facility is open, capacity <= max_capacity
capacity_limit = LXIndicatorTerm(
    binary_var=is_open,
    condition=True,
    linear_expr=LXLinearExpression().add_term(flow, 1.0),
    sense='<=',
    rhs=500.0
)

Complete Examples¶

Example 1: Facility Location with Minimum Throughput¶

from dataclasses import dataclass
from lumix import LXModel, LXVariable, LXLinearExpression
from lumix.nonlinear import LXIndicatorTerm

@dataclass
class Facility:
    id: str
    min_throughput: float
    max_throughput: float

facilities = [...]

is_open = LXVariable[Facility, int]("is_open").binary().from_data(facilities)
throughput = LXVariable[Facility, float]("throughput").continuous().bounds(0).from_data(facilities)

# If open, throughput >= min_throughput
min_throughput_constraints = [
    LXIndicatorTerm(
        binary_var=is_open,
        condition=True,
        linear_expr=LXLinearExpression().add_term(throughput, 1.0),
        sense='>=',
        rhs=f.min_throughput
    )
    for f in facilities
]

See full documentation for more examples.