Command line and JSON format

Any language can use moment-sos through a JSON problem file and the command line tool.

JSON problem format

{"n": 2, "sense": "max", "name": "demo",
 "box": {"lo": -1, "hi": 1},
 "boolean": false,
 "objective": [{"c": 1.0, "m": [[0, 2], [1, 1]]}, {"c": 1.0, "m": [[0, 1], [1, 2]]},
               {"c": -1.0, "m": [[0, 1]]}, {"c": -1.0, "m": [[1, 1]]}],
 "constraints": [{"kind": "ineq", "terms": [{"c": 1.0, "m": []}, {"c": -1.0, "m": [[0, 2]]}, {"c": -1.0, "m": [[1, 2]]}]}]}

  • A term {"c": c, "m": [[i, p], ...]} is \(c\prod x_i^{p}\) with 0-based variable indices; "m": [] is a constant.

  • kind is "ineq" (poly \(\ge0\)) or "eq" (poly \(=0\)).

  • box holds scalars or length-\(n\) lists and must contain the feasible set; omit it for Boolean problems.

  • boolean is true, false or a length-\(n\) list of booleans.

From Python:

import moment_sos as po

x, y = po.variables(2)
problem = po.Problem(x**2*y + x*y**2 - x - y, [x**2 + y**2 <= 1], box=(-1, 1))
d = po.problem_to_dict(problem)            # or po.save_problem(problem, "problem.json")
problem2 = po.problem_from_dict(d)
print(problem2)

Command line

moment-sos solve problem.json --order 2 --out result.json
moment-sos solve problem.json --method alm --cliques auto --time-limit 600 --rigorous
moment-sos version

option

meaning

--order r

relaxation order (default: smallest admissible)

--method smooth | alm

smoothing + L-BFGS (default) or dual augmented Lagrangian

--cliques dense | auto | JSON

e.g. '[[0,1,2],[2,3]]'

--time-limit s

seconds

--rigorous

also compute the verified exact-arithmetic bound

--no-round

skip primal recovery

--out path

result JSON (- = stdout, the default)

The result file holds bound, bound_kind, order, blocks, largest_block, time, evaluations, history ([time, evaluations, best bound] triples in the original units), x and value (the best feasible point and its value, null if none) and, with --rigorous, a rigorous object.