pyfiltersqp

Open-source SQP (Sequential Quadratic Programming) solver in Python with L-BFGS Hessian approximation, l₁-merit and filter line searches, and first-class warm-starting. Faster than IPOPT on most CUTEst problems we've benchmarked.

Why

The open-source NLP ecosystem is dominated by interior-point methods (IPOPT, ECOS, Clarabel). For SQP there's no modern, MIT-licensed, Python-friendly option with sparse Jacobians and warm-starting baked in. This fills that gap.

Headline benchmarks

vs cyipopt IPOPT on the classical CUTEst deck (47 problems, n up to 5000):

41 of 45 successfully-solved problems beat IPOPT, by 1.3× – 70×. See scripts/bench_cutest.py for the harness.

Install

pip install pyfiltersqp                 # core
pip install "pyfiltersqp[numba]"        # +Numba JIT (2× faster L-BFGS recursion)
pip install "pyfiltersqp[dev]"          # +pytest for development

Requires Python ≥ 3.11. Core deps: numpy, scipy, osqp.

Quick start

import numpy as np
from pyfiltersqp import NLPProblem, solve

# HS71 — the classic 4-var SQP demo problem.
problem = NLPProblem(
    n=4, m_eq=1, m_ineq=1,
    xl=np.ones(4), xu=5 * np.ones(4),
    objective = lambda x: x[0]*x[3]*(x[0]+x[1]+x[2]) + x[2],
    gradient  = lambda x: np.array([
        x[3]*(2*x[0]+x[1]+x[2]),
        x[0]*x[3],
        x[0]*x[3] + 1,
        x[0]*(x[0]+x[1]+x[2]),
    ]),
    eq_constraints = lambda x: np.array([x[0]**2+x[1]**2+x[2]**2+x[3]**2 - 40]),
    eq_jacobian    = lambda x: np.array([[2*x[0], 2*x[1], 2*x[2], 2*x[3]]]),
    ineq_constraints = lambda x: np.array([25 - x[0]*x[1]*x[2]*x[3]]),
    ineq_jacobian    = lambda x: np.array([[
        -x[1]*x[2]*x[3], -x[0]*x[2]*x[3],
        -x[0]*x[1]*x[3], -x[0]*x[1]*x[2],
    ]]),
)

result = solve(problem, x0=np.array([1., 5., 5., 1.]))
print(result.success, result.obj)        # True, 17.014017...

Algorithm

• Outer: Sequential Quadratic Programming with l₁-merit Armijo line search; filter line search opt-in (Fletcher & Leyffer, 2002) for degenerate problems.
• Hessian: L-BFGS with Powell damping. Optional Numba JIT for the forward recursion. Compact (Byrd-Lu-Nocedal-Zhu) representation under the hood.
• QP backend (auto-selected by problem shape):
  • ImplicitLBFGSQP — primal-dual active-set + Schur complement, applies B⁻¹ via the L-BFGS two-loop recursion in O(n·m) without ever forming the dense n×n Hessian. Best for small-to-medium n with few active constraints.
  • WoodburyADMM — Woodbury-identity ADMM on the compact L-BFGS factors, polished via a Schur solve on the inferred active set. Cost per ADMM step is O(n·m) and independent of the active set. Best for large n or many constraints.
  • OSQP — the original v0.1 path. Retained as a deprecated fallback; every benchmark we've run is faster on the implicit / Woodbury paths.
• Robustness: second-order correction on Maratos-class line search collapse, mid-solve fallback to filter on stagnation, anti-cycling (Bland's rule + one-step tabu) in active-set, watchdog reset of L-BFGS on tiny-α streaks.

See DESIGN.md for the full algorithm specification.

Warm-starting

Designed for iterative MPCC strategies and homotopy methods where the constraint set changes slightly between solves:

result1 = solve(problem_v1, x0)
result2 = solve(problem_v2,
                x0=result1.x,
                lam_eq0=result1.lam_eq,
                lam_ineq0=result1.lam_ineq)

Roadmap

See ROADMAP.md and CHANGELOG.md. The next real-engineering items are PyPI publish (this commit), Woodbury custom KKT solver inside OSQP for the n > 1000 range, and a projected-CG QP path for n > 10 000.

References

• Nocedal & Wright, Numerical Optimization, 2nd ed. (2006) — chapters 18 (SQP), 7 (L-BFGS)
• Fletcher & Leyffer (2002), "Nonlinear programming without a penalty function" — filter line search
• Byrd, Lu, Nocedal & Zhu (1994), "Representations of quasi-Newton matrices and their use in limited memory methods" — compact L-BFGS
• Powell (1978), "A fast algorithm for nonlinearly constrained optimization calculations" — damped BFGS
• Stellato et al. (2020), OSQP

License

MIT — see LICENSE.