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Quadratic Objective Conic Optimizer

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# Quadratic Objective Conic Optimizer (QOCO)

QOCO is a C implementation of a primal-dual interior point method to solve second-order cone programs with quadratic objectives of the following form

$$
  \begin{split}
      \underset{x}{\text{minimize}}
      \quad & \frac{1}{2}x^\top P x + c^\top x \\
      \text{subject to}
      \quad & Gx \preceq_\mathcal{C} h \\
      \quad & Ax = b
  \end{split}
$$

with optimization variable $x \in \mathbb{R}^n$ and problem data $P = P^\top \succeq 0$, $c \in \mathbb{R}^n$, $G \in \mathbb{R}^{m \times n}$, $h \in \mathbb{R}^m$, $A \in \mathbb{R}^{p \times n}$, $b \in \mathbb{R}^p$, and $\preceq_\mathcal{C}$
is an inequality with respect to cone $\mathcal{C}$, i.e. $h - Gx \in \mathcal{C}$. Cone $\mathcal{C}$ is the Cartesian product of the non-negative orthant and second-order cones, which can be expressed as

$$\mathcal{C} =  \mathbb{R}^l_+ \times \mathcal{Q}^{q_1}_1 \times \ldots \times \mathcal{Q}^{q_N}_N$$

where $l$ is the dimension of the non-negative orthant, and $\mathcal{Q}^{q_i}_i$ is the $i^{th}$ second-order cone with dimension $q_i$ defined by

$$\mathcal{Q}^{q_i}_i = \\{(t,x)  \in \mathbb{R} \times \mathbb{R}^{q_i - 1} \\; | \\; norm(x) \leq t \\}$$

## Bug reports

File any issues or bug reports using the [issue tracker](https://github.com/qoco-org/qoco/issues).

## Citing

If you find QOCO useful please cite the [QOCO paper](https://arxiv.org/abs/2503.12658) as follows

```
@article{chari2026qoco,
  title = {{QOCO}: a quadratic objective conic optimizer with custom solver generation},
  author = {Chari, Govind M. and A\c{c}ıkmeşe, Beh\c{c}et},
  journal = {Mathematical Programming Computation},
  issn = {1867-2957},
  url = {http://dx.doi.org/10.1007/s12532-026-00311-8},
  doi = {10.1007/s12532-026-00311-8},
  publisher = {Springer Science and Business Media LLC},
  year = {2026},
  month = mar,
}
```

If you find the GPU backend useful please cite the [QOCO-GPU](https://arxiv.org/abs/2603.29197) paper as follows

```
@article{chari2026qoco,
  title={{QOCO}-{GPU}: A Quadratic Objective Conic Optimizer with {GPU} Acceleration},
  author={Chari, Govind M and A{\c{c}}{\i}kme{\c{s}}e, Beh{\c{c}}et},
  journal={arXiv preprint arXiv:2603.29197},
  year={2026}
}
```

## License

QOCO is licensed under the BSD-3-Clause license.

qoco

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