mdls

mdls is a small, decorator-based implementation of multi-directional local search (MDLS) for multi-objective optimisation. You supply an evaluate function and one move per objective; the library owns the non-dominated archive, selection, budgets and statistics.

The method follows Tricoire (2012). This package is a generalised, reusable version of the MDLS engine used in ORBITAL.

Install

pip install pymdls

The distribution is named pymdls; the import package is mdls, so the code below always uses from mdls import ....

Python 3.10 or newer is required. NumPy and Numba are installed automatically.

For the example's plotting command, install the extra:

pip install "pymdls[examples]"

Quick start

import numpy as np

from mdls import MDLS, move


def evaluate(x, inst):
    """MDLS minimises; g <= 0 means feasible."""
    return -(inst["obj"] @ x), [inst["weight"] @ x - inst["cap"]]


def _toggle(x, inst, rng, k, n_toggle):
    """One attempt: flip items, keep the copy if objective k improves."""
    y = x.copy()
    flip = rng.choice(x.size, size=n_toggle, replace=False)
    y[flip] = ~y[flip]
    if inst["weight"] @ y > inst["cap"]:
        return None
    return y if inst["obj"][k] @ y > inst["obj"][k] @ x else None


@move(improves=0, attempts=8)
def toggle_0(x, inst, rng, n_toggle=2):
    return _toggle(x, inst, rng, 0, n_toggle)


@move(improves=1, attempts=8)
def toggle_1(x, inst, rng, n_toggle=2):
    return _toggle(x, inst, rng, 1, n_toggle)


rng = np.random.default_rng(0)
n = 20
inst = {"obj": rng.random((2, n)), "weight": rng.random(n), "cap": 0.3 * n}
moves = [toggle_0(n_toggle=2), toggle_1(n_toggle=2)]
search = MDLS(evaluate, moves=moves, instance=inst)

res = search.run([np.zeros(n, dtype=bool)], max_eval=500, seed=0)
print(res.summary())
print(res.objectives)

What the library provides

• Archive of feasible, non-dominated solutions with a relative tolerance, duplicate rejection and dominated-point eviction.
• Move contract: a move is one improving attempt, optionally retried up to attempts times by the @move decorator.
• Balanced search: every iteration applies one sampled move per objective from the same archived base solution.
• Budgets: stop on iterations (max_iter), evaluations (max_eval, including initial solutions) or wall-clock time (time_limit).
• Statistics: per-move counts of calls, no-moves, non-descents, infeasible results, dominated results, accepted results and time.
• Extension points: is_feasible, selection, copy_solution, and the select, offer, copy and on_iteration hooks.

Moves are ordinary functions and stay readable; Numba is optional but is used inside the library's dominance scan and can be used inside your own moves.

Examples

The repository includes a bi-objective 0/1 knapsack:

python examples/knapsack.py run      # one search, writes front.csv
python examples/knapsack.py plot     # fronts for up to four seeds, writes front.png

The full tutorial, including the move contract and every option, is in user_guide.md.

Testing

pip install -e ".[dev,examples]"
pytest

License

MIT. See LICENSE.

Citation

If you use this package in academic work, cite the MDLS paper and optionally the software itself (see CITATION.cff):

Tricoire, F. (2012). Multi-directional local search. Computers & Operations Research, 39(12), 3089–3101. https://doi.org/10.1016/j.cor.2012.03.010