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CSDP.jl is a wrapper for the COIN-OR SemiDefinite Programming solver.

The wrapper has two components:


This wrapper is maintained by the JuMP community and is not a COIN-OR project.

The original algorithm is described by B. Borchers (1999). CSDP, A C Library for Semidefinite Programming. Optimization Methods and Software. 11(1), 613-623. [preprint]


CSDP.jl is licensed under the MIT License.

The underlying solver, coin-or/Csdp, is licensed under the Eclipse public license.


Install CSDP using Pkg.add:

import Pkg

In addition to installing the CSDP.jl package, this will also download and install the CSDP binaries. You do not need to install CSDP separately.

Use with JuMP

To use CSDP with JuMP, use CSDP.Optimizer:

using JuMP, CSDP
model = Model(CSDP.Optimizer)
set_attribute(model, "maxiter", 1000)

MathOptInterface API

The CSDP optimizer supports the following constraints and attributes.

List of supported objective functions:

List of supported variable types:

List of supported constraint types:

List of supported model attributes:


The CSDP options are listed in the table below.

Name Default Value Explanation
axtol 1.0e-8 Tolerance for primal feasibility
atytol 1.0e-8 Tolerance for dual feasibility
objtol 1.0e-8 Tolerance for relative duality gap
pinftol 1.0e8 Tolerance for determining primal infeasibility
dinftol 1.0e8 Tolerance for determining dual infeasibility
maxiter 100 Limit for the total number of iterations
minstepfrac 0.90 The minstepfrac and maxstepfrac parameters determine how close to the edge of the feasible region CSDP will step
maxstepfrac 0.97 The minstepfrac and maxstepfrac parameters determine how close to the edge of the feasible region CSDP will step
minstepp 1.0e-8 If the primal step is shorter than minstepp then CSDP declares a line search failure
minstepd 1.0e-8 If the dual step is shorter than minstepd then CSDP declares a line search failure
usexzgap 1 If usexzgap is 0 then CSDP will use the objective duality gap d - p instead of the XY duality gap ⟨Z, X⟩
tweakgap 0 If tweakgap is set to 1, and usexzgap is set to 0, then CSDP will attempt to "fix" negative duality gaps
affine 0 If affine is set to 1, then CSDP will take only primal-dual affine steps and not make use of the barrier term. This can be useful for some problems that do not have feasible solutions that are strictly in the interior of the cone of semidefinite matrices
perturbobj 1 The perturbobj parameter determines whether the objective function will be perturbed to help deal with problems that have unbounded optimal solution sets. If perturbobj is 0, then the objective will not be perturbed. If perturbobj is 1, then the objective function will be perturbed by a default amount. Larger values of perturbobj (for example, 100) increase the size of the perturbation. This can be helpful in solving some difficult problems.
fastmode 0 The fastmode parameter determines whether or not CSDP will skip certain time consuming operations that slightly improve the accuracy of the solutions. If fastmode is set to 1, then CSDP may be somewhat faster, but also somewhat less accurate
printlevel 1 The printlevel parameter determines how much debugging information is output. Use a printlevel of 0 for no output and a printlevel of 1 for normal output. Higher values of printlevel will generate more debugging output

Problem representation

The primal is represented internally by CSDP as follows:

max ⟨C, X⟩
      A(X) = a
         X ⪰ 0

where A(X) = [⟨A_1, X⟩, ..., ⟨A_m, X⟩]. The corresponding dual is:

min ⟨a, y⟩
     A'(y) - C = Z
             Z ⪰ 0

where A'(y) = y_1A_1 + ... + y_mA_m

Termination criteria

CSDP will terminate successfully (or partially) in the following cases:

  • If CSDP finds X, Z ⪰ 0 such that the following 3 tolerances are satisfied:
    • primal feasibility tolerance: ||A(x) - a||_2 / (1 + ||a||_2) < axtol
    • dual feasibility tolerance: ||A'(y) - C - Z||_F / (1 + ||C||_F) < atytol
    • relative duality gap tolerance: gap / (1 + |⟨a, y⟩| + |⟨C, X⟩|) < objtol
      • objective duality gap: if usexygap is 0, gap = ⟨a, y⟩ - ⟨C, X⟩
      • XY duality gap: if usexygap is 1, gap = ⟨Z, X⟩ then it returns 0.
  • If CSDP finds y and Z ⪰ 0 such that -⟨a, y⟩ / ||A'(y) - Z||_F > pinftol, it returns 1 with y such that ⟨a, y⟩ = -1.
  • If CSDP finds X ⪰ 0 such that ⟨C, X⟩ / ||A(X)||_2 > dinftol, it returns 2 with X such that ⟨C, X⟩ = 1.
  • If CSDP finds X, Z ⪰ 0 such that the following 3 tolerances are satisfied with 1000*axtol, 1000*atytol and 1000*objtol but at least one of them is not satisfied with axtol, atytol and objtol and cannot make progress, then it returns 3.

In addition, if the printlevel option is at least 1, the following will be printed:

  • If the return code is 1, CSDP will print ⟨a, y⟩ and ||A'(y) - Z||_F
  • If the return code is 2, CSDP will print ⟨C, X⟩ and ||A(X)||_F
  • Otherwise, CSDP will print
    • the primal/dual objective value,
    • the relative primal/dual infeasibility,
    • the objective duality gap ⟨a, y⟩ - ⟨C, X⟩ and objective relative duality gap (⟨a, y⟩ - ⟨C, X⟩) / (1 + |⟨a, y⟩| + |⟨C, X⟩|),
    • the XY duality gap ⟨Z, X⟩ and XY relative duality gap ⟨Z, X⟩ / (1 + |⟨a, y⟩| + |⟨C, X⟩|)
    • and the DIMACS error measures.

In theory, for feasible primal and dual solutions, ⟨a, y⟩ - ⟨C, X⟩ = ⟨Z, X⟩, so the objective and XY duality gap should be equivalent. However, in practice, there are sometimes solution which satisfy primal and dual feasibility tolerances but have objective duality gap which are not close to XY duality gap. In some cases, the objective duality gap may even become negative (hence the tweakgap option). This is the reason usexygap is 1 by default.

CSDP considers that X ⪰ 0 (resp. Z ⪰ 0) is satisfied when the Cholesky factorizations can be computed. In practice, this is somewhat more conservative than simply requiring all eigenvalues to be nonnegative.


The table below shows how the different CSDP statuses are converted to the MathOptInterface statuses.

CSDP code State Description MOI status
0 Success SDP solved MOI.OPTIMAL
1 Success The problem is primal infeasible, and we have a certificate MOI.INFEASIBLE
2 Success The problem is dual infeasible, and we have a certificate MOI.DUAL_INFEASIBLE
3 Partial Success A solution has been found, but full accuracy was not achieved MOI.ALMOST_OPTIMAL
4 Failure Maximum iterations reached MOI.ITERATION_LIMIT
5 Failure Stuck at edge of primal feasibility MOI.SLOW_PROGRESS
6 Failure Stuck at edge of dual infeasibility MOI.SLOW_PROGRESS
7 Failure Lack of progress MOI.SLOW_PROGRESS
8 Failure X, Z, or O was singular MOI.NUMERICAL_ERROR
9 Failure Detected NaN or Inf values MOI.NUMERICAL_ERROR