QTQP

The cutie QP solver is a primal-dual interior point method for solving convex quadratic programs (QPs), implemented in pure python. It solves the primal QP:

    min. (1/2) x.T @ p @ x + c.T @ x
    s.t. a @ x + s = b
         s[:z] == 0
         s[z:] >= 0

With dual:

    max. -(1/2) x.T @ p @ x - b.T @ y
    s.t. p @ x + a.T @ y = -c
         y[z:] >= 0

with data a, b, c, p, z and variables x, y, s. It returns a primal-dual solution when one exists, or a certificate of primal or dual infeasibility otherwise.

Installation

QTQP is available via pip:

python -m pip install qtqp

On supported platforms this also installs the recommended sparse CPU backend automatically:

• Linux / Windows x86_64: py-mkl-pardiso
• macOS arm64: macldlt

To install from source, first clone the repository:

git clone https://github.com/google-deepmind/qtqp.git
cd qtqp

Then, assuming conda is installed, create a new conda environment:

conda create -n tmp python=3.12
conda activate tmp

Finally, install the package:

python -m pip install .

To run the tests, inside the qtqp directory:

python -m pytest .

Tests for optional linear solvers are skipped when the corresponding dependencies are not installed.

Quick start

Here is an example usage (taken from here):

import qtqp
import scipy
import numpy as np

# Set up the problem data
p = scipy.sparse.csc_matrix([[3.0, -1.0], [-1.0, 2.0]])
a = scipy.sparse.csc_matrix([[-1.0, 1.0], [1.0, 0.0], [0.0, 1.0]])
b = np.array([-1, 0.3, -0.5])
c = np.array([-1.0, -1.0])

# Initialize solver
solver = qtqp.QTQP(p=p, a=a, b=b, c=c, z=1)
# Solve!
sol = solver.solve()
print(f'{sol.x=}')
print(f'{sol.y=}')
print(f'{sol.s=}')

You should see output similar to

| QTQP v0.0.7: m=3, n=2, z=1, nnz(A)=4, nnz(P)=4, linear_solver=ACCELERATE, equilibration=RUIZ
|------|------------|------------|----------|----------|----------|----------|----------|----------|----------|
| iter |      pcost |      dcost |     pres |     dres |      gap |   infeas |       mu |  q, p, c |     time |
|------|------------|------------|----------|----------|----------|----------|----------|----------|----------|
|    0 |  1.340e+00 |  1.131e+00 | 1.28e-01 | 1.67e-01 | 2.09e-01 | 1.05e+00 | 1.78e-01 |  1, 1, 1 | 2.99e-03 |
|    1 |  1.221e+00 |  1.226e+00 | 1.60e-02 | 7.08e-03 | 4.23e-03 | 8.21e-01 | 3.93e-03 |  1, 1, 1 | 3.32e-03 |
|    2 |  1.235e+00 |  1.235e+00 | 1.65e-04 | 7.28e-05 | 3.80e-05 | 8.09e-01 | 4.73e-05 |  1, 1, 1 | 3.68e-03 |
|    3 |  1.235e+00 |  1.235e+00 | 4.14e-08 | 1.83e-08 | 9.53e-09 | 8.09e-01 | 1.26e-08 |  1, 1, 1 | 3.98e-03 |
|    4 |  1.235e+00 |  1.235e+00 | 5.08e-12 | 2.38e-12 | 8.62e-13 | 8.09e-01 | 1.26e-12 |  1, 2, 2 | 4.29e-03 |
|------|------------|------------|----------|----------|----------|----------|----------|----------|----------|
| Solved
| Completed IPM steps: 5
sol.x=array([ 0.3, -0.7])
sol.y=array([2.70000000e+00, 2.10000000e+00, 1.47264885e-11])
sol.s=array([0.00000000e+00, 2.72772187e-13, 2.00000000e-01])

API reference

Once installed QTQP is imported using

import qtqp

This exposes the main solver class qtqp.QTQP with constructor:

QTQP(
    *,
    a: scipy.sparse.csc_matrix,
    b: np.ndarray,
    c: np.ndarray,
    z: int,
    p: scipy.sparse.csc_matrix | None = None,
)

Arguments:

• a: (m×n) Constraint matrix.
• b: (m) RHS vector. For inequality rows, values at or above 1e20 * (1 - 1e-9) are reserved: they are treated as +inf (the row is unbounded and removed by presolve, with its dual fixed to 0). Finite data in that range is therefore discarded by design — rescale any genuine constraint whose RHS approaches 1e20 before calling the solver.
• c: (n) Cost vector.
• z: Number of equality constraints (size of the zero cone). Must satisfy 0 ≤ z ≤ m; an all-equality problem (z == m) is solved by the initialization itself, which solves that KKT system exactly, and is graded by the standard criteria at iteration 0 (a singular system is reported as FAILED, or ALMOST_SOLVED if the reduced tolerances hold). Problems with no variables, or with no constraints left after presolve, are rejected.
• p: (n×n) QP matrix. If None, treated as the zero matrix (i.e., LP).

This class has a single API method solve:

solve(
    *,
    tol_feas: float = 1e-8,
    tol_gap_abs: float = 1e-8,
    tol_gap_rel: float = 1e-8,
    tol_infeas_abs: float = 1e-8,
    tol_infeas_rel: float = 1e-8,
    certificate_ktratio: float = 1e9,
    max_iter: int = 100,
    step_size_scale: float = 0.99,
    min_static_regularization: float = 1e-8,
    max_iterative_refinement_steps: int = 20,
    linear_solver_atol: float = 1e-12,
    linear_solver_rtol: float = 1e-12,
    linear_solver: qtqp.LinearSolver = qtqp.LinearSolver.AUTO,
    verbose: bool = True,
    equilibration_strategy: qtqp.EquilibrationStrategy = (
        qtqp.EquilibrationStrategy.RUIZ
    ),
    collect_stats: bool = False,
    refinement_strategy: qtqp.RefinementStrategy = (
        qtqp.RefinementStrategy.GMRES
    ),
    gmres_restart: int = 20,
    max_centrality_correctors: int = 1,
    adaptive_step_size: bool = True,
    warm_start: tuple[np.ndarray, np.ndarray, np.ndarray] | None = None,
    warm_start_threshold: float = 100.0,
) -> qtqp.Solution

Key parameters:

• tol_feas, tol_gap_abs, tol_gap_rel: Stopping tolerances for optimality (Clarabel's definitions; see below).
• tol_infeas_abs, tol_infeas_rel: Thresholds for (primal/dual) infeasibility detection.
• certificate_ktratio: Embedding ratio kappa / tau above which certificates are considered (default 1e9, as in Clarabel).
• max_iter: Iteration cap.
• step_size_scale (0,1): Scale for line search step size to stay strictly interior.
• min_static_regularization: Diagonal regularization on KKT for robustness. Dense Gram backends (SCIPY_DENSE, CUPY_DENSE) require a positive value when initializing a problem with equality rows (z > 0); zero raises ValueError because Gram elimination cannot invert a zero equality diagonal.
• max_iterative_refinement_steps, linear_solver_atol/rtol: Control iterative refinement of the linear solve. The default is 20 refinement steps, counting the initial solve.
• linear_solver: (qtqp.LinearSolver) Choose the KKT solver backend (see below).
• verbose: Print per-iteration table with key metrics.
• equilibration_strategy: Choose how problem data is scaled before the IPM iterations. Defaults to qtqp.EquilibrationStrategy.RUIZ.
• collect_stats: If True, populate Solution.stats with per-iteration diagnostics (sy, s/y statistics, complementarity, etc.). Defaults to False for faster throughput. Solution.iterations always reports completed IPM steps, excluding initialization and failed step attempts, even with collect_stats=False. A solution found at initialization reports zero. In collected rows, iter remains the zero-based label and iterations is the completed-step count at that row; the final log footer reports the total completed count. Collected rows also include delta_path, an estimate of ||T_mu(u)|| / mu. In exact arithmetic with positive mu, this bounds the distance to the central-path point by strong monotonicity. The recorded value uses floating-point denominator guards, so it is a diagnostic rather than a certified numerical bound. Its local-norm companion delta_path_local approximates the residual norm lambda = ||T_mu(u)||_(H^-1), where H = mu*I + mu*hess(Phi). This score also uses denominator guards and is not itself a distance bound. lambda_init (also an attribute on the solver) is the same guarded local score at the chosen initial point, warm or cold, before the first step. For an accepted warm start, it reuses the screening score.
• refinement_strategy: Choose the iterative-refinement method used for KKT solves. Defaults to qtqp.RefinementStrategy.GMRES.
• gmres_restart: Restart length for qtqp.RefinementStrategy.GMRES. Defaults to 20: one uninterrupted Krylov cycle spanning the full refinement budget, avoiding restart stagnation. Ignored by Richardson refinement.
• max_centrality_correctors: Maximum Gondzio-style centrality correctors per iteration, each one extra back-solve on the existing factorization, recentering the aspirational trial point's outlier complementarity products and accepted only when the step size improves. Default 1 (validated on Maros-Meszaros + NETLIB + MIPLIB: 14-16% fewer median iterations on every dataset with unchanged robustness); 0 disables.
• adaptive_step_size: If True (the default), once mu < 1e-3 the fraction-to-boundary scale follows min(0.9999, max(step_size_scale, 1 - 10*mu)): the margin to the cone boundary shrinks proportionally to mu, unlocking the superlinear endgame that a constant haircut caps at a linear rate. Set False for the constant legacy schedule.
• warm_start: Optional (x, y, s) from a nearby problem (original scale, e.g. a previous solution's arrays). The point is equilibrated into the operating scale, its inequality duals and slacks are floored at each of 1e-6, 1e-4, 1e-2, and 1.0, and each candidate is normalized. The candidate with the smallest guarded local score is accepted when lambda <= warm_start_threshold; otherwise the solver uses the standard initialization. This is empirical screening: acceptance does not certify distance to the path or guarantee fewer iterations or less time than a cold solve. Screening uses no factorization and six matrix-vector products across the four candidates: shared A @ x and P @ x, plus one A.T @ y per candidate. An accepted start skips the cold initialization factorization and reuses its score as lambda_init; a rejected start adds screening overhead before the cold solve. After solve, the measured warm_lambda and the warm_accepted decision are attributes on the solver. A warm start is ignored on an all-equality problem, which the initialization solves outright.
• warm_start_threshold: Empirical screening threshold (default 100.0), not a certified distance tolerance. For the exact local score, the theoretical distance bound instead requires eta = lambda / sqrt(mu) < 1 and is eta / (1 - eta). The solver does not use this test for acceptance, and its guarded score must not be substituted into that bound as a numerical certificate.

Equilibration strategies

Choose one with the equilibration_strategy argument:

• qtqp.EquilibrationStrategy.RUIZ: Default. Ruiz equilibration on the KKT block [P, A'; A, 0], plus the two scalars a QP admits freely: a joint scale on b and c (a rescale of the solution) taking ||b||_inf to 1, and a scale on the objective P, c (a rescale of the duals) taking max(||c||_inf, max |P_ij|) to 1, both kept within [1e-4, 1e4]. b and c never enter the row/column scalings, so the factorized block keeps unit rows and columns.
• qtqp.EquilibrationStrategy.AUGMENTED: Ruiz equilibration on the symmetric augmented matrix containing P, A, b, and c, so that b and c also inform the row and column scalings. Kept for experimentation: it splits the magnitude of large entries of b and c into the factorized block, and RUIZ solved more problems on every benchmark collection.
• qtqp.EquilibrationStrategy.NONE: Disable equilibration.

Initialization

The initial iterate is Clarabel's initialization: one solve of [P, A'; A, -H][x; y] = [-c; b] for QPs (two, with a shared factorization, for LPs - primal from feasibility, dual from optimality), with H the identity on inequality rows and zero on equality rows, so equality rows are satisfied exactly by the initial point. It runs through the same linear-solver backend, ordering, static regularization, and iterative refinement as the main loop, then the inequality components are shifted into the strict interior. The initial point is graded before the first step, so an exact initialization or an accepted warm start can terminate at iteration 0. If the initialization solve produces non-finite values, the solver falls back to a trivial unit start.

Refinement strategies

Choose one with the refinement_strategy argument:

• qtqp.RefinementStrategy.GMRES: Default. Restarted right-preconditioned GMRES on the true KKT system. Each Arnoldi step consumes one factor-solve, and gmres_restart controls the restart length.
• qtqp.RefinementStrategy.RICHARDSON: Classical iterative refinement using the factorized regularized KKT matrix as a preconditioner (a smoother; its best-effort iterates behave differently from GMRES's residual-optimal ones in the deep endgame).

Advanced numerical options

• Termination criteria are Clarabel's, evaluated on the returned point, so results are directly comparable: SOLVED requires ||Ax + s - b|| / max(1, ||b||_inf + ||x|| + ||s||) < tol_feas, ||Px + A'y + c|| / max(1, ||c||_inf + ||x|| + ||y||) < tol_feas (2-norms), and a duality gap |pcost - dcost| below tol_gap_abs or below tol_gap_rel * max(1, min(|pcost|, |dcost|)), with the iterate on the solution side of the embedding (kappa / tau < 1). A certificate requires kappa / tau > certificate_ktratio, an objective slope (b'y or c'x) below -tol_infeas_abs, and violations relative to max(1, ||ray||) below tol_infeas_rel * |slope|.

• x: (n) Primal variable or certificate of unboundedness.

• y: (m) Dual variable or certificate of infeasibility.

• s: (m) Slack variable or certificate of unboundedness.

• status: (qtqp.SolutionStatus) One of SOLVED, INFEASIBLE, UNBOUNDED, ALMOST_SOLVED, HIT_MAX_ITER, FAILED.

  Inequality rows removed by presolve (RHS at or above the 1e20 infinity sentinel) are restored in s as +inf with dual 0 on solution outputs, and as NaN on certificates; verifiers should mask them. ALMOST_SOLVED is returned in place of HIT_MAX_ITER (or of a numerical breakdown of the linear solver, which never raises) when the best iterate over the trajectory (by max normalized residual) meets the solved criteria at Clarabel's reduced tolerances (1e-4 feasibility, 5e-5 gap): the returned solution is that best iterate, honestly labeled as not meeting the full SOLVED contract. SOLVED semantics are unchanged. A breakdown whose best iterate does not qualify returns FAILED, as does a numeric failure in the initialization factorization, with NaN arrays because no iterate exists yet.

• stats: (list of dicts) Per-iteration diagnostics. Empty unless collect_stats=True. When enabled, includes primal/dual objective, residuals, gap, mu, elapsed time, and complementarity statistics.

Linear solvers

The backend linear system solver can be changed by passing a qtqp.LinearSolver to the solve method via the linear_solver argument. By default linear_solver=qtqp.LinearSolver.AUTO. AUTO resolves to qtqp.LinearSolver.PARDISO first on Linux / Windows and to qtqp.LinearSolver.ACCELERATE first on macOS, then falls back through the other sparse CPU backends before finally using qtqp.LinearSolver.SCIPY. The enum qtqp.LinearSolver contains values corresponding to the following backend solvers:

Recommended starting points:

System / problem type	Recommended solver
Default choice	qtqp.LinearSolver.AUTO
Linux / Windows	qtqp.LinearSolver.PARDISO
macOS	qtqp.LinearSolver.ACCELERATE
NVIDIA GPU available	qtqp.LinearSolver.CUDSS
Dense data	qtqp.LinearSolver.SCIPY_DENSE
Tiny problems (n + m < 50)	qtqp.LinearSolver.QDLDL

Automatic selection: qtqp.LinearSolver.AUTO

Runtime selection for sparse CPU backends.

• Linux / Windows preference order starts with PARDISO.
• macOS preference order starts with ACCELERATE.
• The default install brings in py-mkl-pardiso on Linux / Windows x86_64 and macldlt on macOS arm64.
• If the preferred backend is unavailable, QTQP tries the remaining sparse CPU backends and finally falls back to SCIPY.

scipy SuperLU: qtqp.LinearSolver.SCIPY

Baseline sparse CPU backend using scipy.sparse.linalg.factorized. No additional dependencies required.

MKL Pardiso: qtqp.LinearSolver.PARDISO

Recommended sparse CPU backend on Linux and Windows. Available via the py-mkl-pardiso package (Linux and Windows, x86_64). To install

python -m pip install py-mkl-pardiso

Accelerate: qtqp.LinearSolver.ACCELERATE

Apple Accelerate sparse LDL^T factorization via macldlt (macOS only). Recommended sparse CPU backend on macOS. Published wheels are currently Apple Silicon only. To install

python -m pip install macldlt

Nvidia cuDSS: qtqp.LinearSolver.CUDSS

Recommended sparse GPU backend when an NVIDIA GPU is available. To install

python -m pip install nvidia-cudss-cu12
python -m pip install nvmath-python[cu12]
python -m pip install cupy-cuda12x

Dense Cholesky: qtqp.LinearSolver.SCIPY_DENSE

Recommended backend for dense data. Uses a dense Schur-complement / Cholesky factorization. No additional dependencies required.

QDLDL: qtqp.LinearSolver.QDLDL

Sparse LDL^T backend via qdldl. To install

python -m pip install qdldl

UMFPACK: qtqp.LinearSolver.UMFPACK

Sparse LU backend via scikit-umfpack. To install

conda install scikit-umfpack -c conda-forge

CHOLMOD: qtqp.LinearSolver.CHOLMOD

Sparse Cholesky / LDL^T backend via scikit-sparse. To install

conda install suitesparse -c conda-forge
python -m pip install 'scikit-sparse>=0.5'

Eigen: qtqp.LinearSolver.EIGEN

Sparse LDL^T backend via nanoeigenpy. To install

conda install nanoeigenpy -c conda-forge

MUMPS: qtqp.LinearSolver.MUMPS

Sparse direct solver backend via petsc4py / MUMPS. To install

conda install petsc4py -c conda-forge

cupy dense GPU: qtqp.LinearSolver.CUPY_DENSE

GPU counterpart of SCIPY_DENSE: dense Schur-complement / Cholesky on GPU via cupy/cuSOLVER. To install

python -m pip install cupy-cuda12x

Citing this work

Coming soon, in the meantime the closest work is:

@article{odonoghue:21,
    author       = {Brendan O'Donoghue},
    title        = {Operator Splitting for a Homogeneous Embedding of the Linear Complementarity Problem},
    journal      = {{SIAM} Journal on Optimization},
    month        = {August},
    year         = {2021},
    volume       = {31},
    issue        = {3},
    pages        = {1999-2023},
}

License and disclaimer

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All software is licensed under the Apache License, Version 2.0 (Apache 2.0); you may not use this file except in compliance with the Apache 2.0 license. You may obtain a copy of the Apache 2.0 license at: https://www.apache.org/licenses/LICENSE-2.0

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