Skip to content

Discrete and Mixed-Integer Variables

By default every variable is continuous. Marking some or all of them as integer-valued takes one field, variables.types, but it changes what the rest of the configuration has to look like: an integer variable cannot be differentiated, so the problem needs a method that searches without gradients, and that method needs bounds.

There are two runnable scripts for this page: examples/discrete.py, where every variable is an integer, and examples/mixed.py, where continuous and integer variables appear in one problem.

Warning

Only differential_evolution handles integer variables. The other SciPy methods silently treat them as continuous — no error, only a fractional answer to a problem you meant to be discrete. Choosing the method is not optional here.

All variables integer

discrete.py maximizes min(3x, y) over two integers, subject to x + y <= 10. The types field takes a single value that applies to every variable:

config: dict[str, Any] = {
    "variables": {
        "variable_count": 2,
        "types": "integer",
        "lower_bounds": [0.0, 0.0],
        "upper_bounds": [10.0, 10.0],
    },
    "optimizer": {
        "max_functions": 5,
    },
    "backend": {
        "method": "differential_evolution",
        "options": {"rng": 4},
        "parallel": False,
    },
}

Three things go together. types marks the variables as integers; the backend section selects differential_evolution, the only method that will respect that; and lower_bounds / upper_bounds are mandatory, because that method searches within a box rather than stepping from a start point.

types takes the strings "real" and "integer". These are the values of VariableType, whose members are strings, so VariableType.INTEGER equals "integer". The scripts use the strings; the members, imported from ropt.enums, are accepted in their place.

The types field, and every other field of the variables section, is described under variables.

The objective is an ordinary evaluation function. It receives the variables as floats that happen to hold integral values:

def function(
    variables: NDArray[np.float64],
    _context: EvaluationFunctionContext,
) -> float | list[float]:
    x, y = variables
    objective = -min(3.0 * x, y)
    if linear:
        return float(objective)
    return [float(objective), float(x + y)]

The script imposes x + y <= 10 as a nonlinear constraint by default, and as a linear one with --linear. Both forms work unchanged with integer variables; see Constraints.

Mixing continuous and integer variables

To make only some variables discrete, give types one entry per variable instead of a single value. mixed.py does this for the first four variables of an ensemble Rosenbrock problem, keeping two continuous and two integer:

CONFIG: dict[str, Any] = {
    "variables": {
        "variable_count": DIM,
        "perturbation_magnitudes": 1e-6,
        "lower_bounds": 0.0,
        "upper_bounds": 10.0,
        "types": ["real", "real", "integer", "integer"],
    },
    "backend": {
        "method": "differential_evolution",
        "options": {"rng": 4},
        "max_iterations": 50,
    },
    "realizations": {
        "weights": [1.0] * REALIZATIONS,
    },
}

Nothing else changes. The realizations, the objective and the call to optimize are the same as in Ensemble-Based Optimization — being partly discrete is a property of the variables, not of the problem around them.

A gradient-free method reaches an answer by evaluating many points rather than by following a gradient, and the cost profile differs accordingly:

  • perturbation_magnitudes is unused. No perturbations are evaluated, because no gradient is estimated.
  • Limit the optimization with max_functions or max_iterations rather than a convergence tolerance.
  • The result is reproducible only if the method's own generator is seeded — hence "options": {"rng": 4} in both scripts.