https://github.com/mariohsouto/proxsdp.jl
Semidefinite programming optimization solver
https://github.com/mariohsouto/proxsdp.jl
conic-programs convex-optimization julia optimization sdp semidefinite-programming solver
Last synced: 8 days ago
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Semidefinite programming optimization solver
- Host: GitHub
- URL: https://github.com/mariohsouto/proxsdp.jl
- Owner: mariohsouto
- License: mit
- Created: 2018-01-09T14:02:23.000Z (about 8 years ago)
- Default Branch: master
- Last Pushed: 2025-05-25T16:36:45.000Z (9 months ago)
- Last Synced: 2026-01-27T22:11:35.850Z (about 1 month ago)
- Topics: conic-programs, convex-optimization, julia, optimization, sdp, semidefinite-programming, solver
- Language: Julia
- Homepage:
- Size: 12.1 MB
- Stars: 99
- Watchers: 9
- Forks: 13
- Open Issues: 14
-
Metadata Files:
- Readme: README.md
- License: LICENSE
- Citation: CITATION.bib
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[ProxSDP](https://github.com/mariohsouto/ProxSDP.jl) is an open-source
semidefinite programming ([SDP](https://en.wikipedia.org/wiki/Semidefinite_programming))
solver based on the paper ["Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting"](https://arxiv.org/abs/1810.05231).
The main advantage of ProxSDP over other state-of-the-art solvers is the ability
to exploit the **low-rank** structure inherent to several SDP problems.
## Overview of problems ProxSDP can solve
* General conic convex optimization problems with the presence of the
[positive semidefinite cone](https://web.stanford.edu/~boyd/papers/pdf/semidef_prog.pdf),
[second-order cone](https://web.stanford.edu/~boyd/papers/pdf/socp.pdf) and
[positive orthant](https://www.math.ucla.edu/~tom/LP.pdf);
* Semidefinite relaxation of nonconvex problems, for example,
[max-cut](http://www-math.mit.edu/~goemans/PAPERS/maxcut-jacm.pdf),
[binary MIMO](https://arxiv.org/pdf/cs/0606083.pdf),
[optimal power flow](http://authorstest.library.caltech.edu/141/1/TPS_OPF_2_tech.pdf),
[sensor localization](https://web.stanford.edu/~boyd/papers/pdf/sensor_selection.pdf),
[sum-of-squares](https://en.wikipedia.org/wiki/Sum-of-squares_optimization);
* Control theory problems with [LMI](https://en.wikipedia.org/wiki/Linear_matrix_inequality)
constraints;
* Nuclear norm minimization problems, for example,
[matrix completion](https://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf);
## License
ProxSDP is licensed under the [MIT License](https://github.com/mariohsouto/ProxSDP.jl/blob/master/LICENSE).
## Installation
Install ProxSDP using Julia package manager:
```julia
import Pkg
Pkg.add("ProxSDP")
```
## Documentation
The online documentation is available at [https://mariohsouto.github.io/ProxSDP.jl/latest/](https://mariohsouto.github.io/ProxSDP.jl/latest/).
## Usage
Consider the semidefinite programming relaxation of the
[max-cut](http://www-math.mit.edu/~goemans/PAPERS/maxcut-jacm.pdf) problem, as
in:
```
max 0.25 * W•X
s.t. diag(X) = 1,
X ≽ 0,
```
### JuMP
This problem can be solved by the following code using **ProxSDP** and
[JuMP](https://github.com/JuliaOpt/JuMP.jl).
```julia
# Load packages
using ProxSDP, JuMP, LinearAlgebra
# Number of vertices
n = 4
# Graph weights
W = [
18.0 -5.0 -7.0 -6.0
-5.0 6.0 0.0 -1.0
-7.0 0.0 8.0 -1.0
-6.0 -1.0 -1.0 8.0
]
# Build Max-Cut SDP relaxation via JuMP
model = Model(ProxSDP.Optimizer)
set_optimizer_attribute(model, "log_verbose", true)
set_optimizer_attribute(model, "tol_gap", 1e-4)
set_optimizer_attribute(model, "tol_feasibility", 1e-4)
@variable(model, X[1:n, 1:n], PSD)
@objective(model, Max, 0.25 * LinearAlgebra.dot(W, X))
@constraint(model, LinearAlgebra.diag(X) .== 1)
# Solve optimization problem with ProxSDP
optimize!(model)
# Retrieve solution
Xsol = value.(X)
```
### Convex.jl
Another alternative is to use **ProxSDP** via
[Convex.jl](https://github.com/jump-dev/Convex.jl):
```julia
# Load packages
using Convex, ProxSDP
# Number of vertices
n = 4
# Graph weights
W = [
18.0 -5.0 -7.0 -6.0
-5.0 6.0 0.0 -1.0
-7.0 0.0 8.0 -1.0
-6.0 -1.0 -1.0 8.0
]
# Define optimization problem
X = Semidefinite(n)
problem = maximize(0.25 * dot(W, X), diag(X) == 1)
# Solve optimization problem with ProxSDP
solve!(problem, ProxSDP.Optimizer(log_verbose=true, tol_gap=1e-4, tol_feasibility=1e-4))
# Get the objective value
problem.optval
# Retrieve solution
evaluate(X)
```
## Citing this package
The published version of the paper can be found [here](https://doi.org/10.1080/02331934.2020.1823387)
and the arXiv version [here](https://arxiv.org/pdf/1810.05231.pdf).
We kindly request that you cite the paper as:
```bibtex
@article{souto2020exploiting,
author = {Mario Souto and Joaquim D. Garcia and \'Alvaro Veiga},
title = {Exploiting low-rank structure in semidefinite programming by approximate operator splitting},
journal = {Optimization},
pages = {1-28},
year = {2020},
publisher = {Taylor & Francis},
doi = {10.1080/02331934.2020.1823387},
URL = {https://doi.org/10.1080/02331934.2020.1823387}
}
```
The preprint version of the paper can be found [here](https://arxiv.org/abs/1810.05231).
## Disclaimer
* ProxSDP is a research software, therefore it should not be used in production.
* Please open an issue if you find any problems, developers will try to fix and
find alternatives.
* There is no continuous development for 32-bit systems, the package should
work, but might reach some issues.
* ProxSDP assumes primal and dual feasibility.
## ROAD MAP
* Support for exponential and power cones
* Warm start