# COIN-OR SemiDefinite Programming Interface (CSDP.jl)

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

The wrapper has two components:

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

The original algorithm is described by B. Borchers. CSDP, A C Library for Semidefinite Programming. Optimization Methods and Software 11(1):613-623, 1999. DOI 10.1080/10556789908805765. Preprint.

## Installation

Install CSDP using Pkg.add:

import Pkg; Pkg.add("CSDP")


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

We highly recommend that you use the CSDP.jl package with higher level packages such as CSDP.jl.

To use CSDP with JuMP, use CSDP.Optimizer:

using JuMP, CSDP

model = Model(CSDP.Optimizer)
set_optimizer_attribute(model, "maxiter", 1000)


## CSDP 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 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:
• 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 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.

Remark: 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.

Remark: 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.

## Status

The table below shows how the different CSDP status are converted to MathOptInterface (MOI) status.

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

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.

## Options

The CSDP options are listed in the table below.

Name Default Value
axtol Tolerance for primal feasibility 1.0e-8
atytol Tolerance for dual feasibility 1.0e-8
objtol Tolerance for relative duality gap 1.0e-8
pinftol Tolerance for determining primal infeasibility 1.0e8
dinftol Tolerance for determining dual infeasibility 1.0e8
maxiter Limit for the total number of iterations 100
minstepfrac The minstepfrac and maxstepfrac parameters determine how close to the edge of the feasible region CSDP will step 0.90
maxstepfrac The minstepfrac and maxstepfrac parameters determine how close to the edge of the feasible region CSDP will step 0.97
minstepp If the primal step is shorter than minstepp then CSDP declares a line search failure 1.0e-8
minstepd If the dual step is shorter than minstepd then CSDP declares a line search failure 1.0e-8
usexzgap If usexzgap is 0 then CSDP will use the objective duality gap d - p instead of the XY duality gap ⟨Z, X⟩ 1
tweakgap If tweakgap is set to 1, and usexzgap is set to 0, then CSDP will attempt to "fix" negative duality gaps 0
affine 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 0
perturbobj 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 (e.g. 100) increase the size of the perturbation. This can be helpful in solving some difficult problems. 1
fastmode 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 0
printlevel 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 1