mariohsouto / ProxSDP.jl

Build Status

ProxSDP is an open-source semidefinite programming (SDP) solver based on the paper "Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting". 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, second-order cone and positive orthant;
• Semidefinite relaxation of nonconvex problems, e.g. max-cut, binary MIMO, optimal power flow, sensor localization, sum-of-squares;
• Control theory problems with LMI constraints;
• Nuclear norm minimization problems, e.g. matrix completion;

Installation

You can install ProxSDP through the Julia package manager:

] add ProxSDP

Usage

Let's consider the semidefinite programming relaxation of the max-cut 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.

# 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(with_optimizer(ProxSDP.Optimizer, log_verbose=true, tol_gap=1e-4, tol_feasibility=1e-4))
@variable(model, X[1:n, 1:n], PSD)
@objective(model, Max, 0.25 * dot(W, X))
@constraint(model, diag(X) .== 1)

# Solve optimization problem with ProxSDP
JuMP.optimize!(model)

# Retrieve solution
Xsol = JuMP.value.(X)

Convex.jl

Another alternative is to use ProxSDP via Convex.jl as the following

# 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 and the arXiv version here.

We kindly request that you cite the paper as:

@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.

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.