COIN-OR SemiDefinite Programming Interface (CSDP.jl)

CSDP.jl is an interface to the COIN-OR SemiDefinite Programming solver. It provides a complete interface to the low-level C API, as well as an implementation of the solver-independent MathProgBase and MathOptInterface API's.

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

Build Status

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.

Installing CSDP

You can either use the system LAPACK and BLAS libaries or the libraries shipped with Julia. First, make sure that you have a compiler available, e.g. on Ubuntu do

$ sudo apt-get install build-essential

To use the libraries shipped by Julia, simply do

$ CSDP_USE_JULIA_LAPACK=true julia -e 'import Pkg; Pkg.add("CSDP"); Pkg.build("CSDP")'

To use the system libaries, first make sure it is installed, e.g. on Ubuntu do

$ sudo apt-get install liblapack-dev libopenblas-dev

and then do

$ CSDP_USE_JULIA_LAPACK=false julia -e 'import Pkg; Pkg.add("CSDP"); Pkg.build("CSDP")'

Note that if the environment variable CSDP_USE_JULIA_LAPACK is not set, it defaults to using the system libraries if available and the Julia libraries otherwise.

To use CSDP with JuMP v0.19 and later, do

using JuMP, CSDP
model = Model(with_optimizer(CSDP.Optimizer))

and with JuMP v0.18 and earlier, do

using JuMP, CSDP
model = Model(solver=CSDPSolver())

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 MathProgBase status.

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. Their value can be specified in the constructor of the CSDP solver, e.g. CSDPSolver(axtol=1e-7, printlevel=0).

Getting the CSDP Library

The original make-file build system only provides a static library. The quite old (September 2010) pycsdp interface by Benjamin Kern circumvents the problem by writing some C++ code to which the static library is linked. The Sage module by @mghasemi is a Cython interface; I don't know how the libcsdp is installed or whether they assume that it is already available on the system.

That is why this package tries to parse the makefile and compiles it itself on Unix systems (so gcc is needed).

For Windows, a pre-compiled DLL is downloaded (unless you configure the build.jl differently).

Next Steps (TODOs)

• Maybe port libcsdp to use 64bit Lapack, aka replace “some ints” by long int (the variables used in a Lapack call). Started in brach julias_openblas64
• Maybe think about an own array type to circumvent the 1-index problems in libcsdp.
• Map Julia's sparse arrays to sparsematrixblock.
• Upload libcsdp.dll for Windows via Appveyor deployment as described at JuliaCon. Currently we use a separate repository.