SDPLR

SDPLR.jl is a wrapper for the SDPLR semidefinite programming solver.

License

SDPLR.jl is licensed under the MIT License.

The underlying solver, SDPLR, is licensed under the GPL v2 license.

Installation

Install SDPLR.jl using Pkg.add:

import Pkg
Pkg.add("SDPLR")

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

To use a custom binary, read the Custom solver binaries section of the JuMP documentation.

Use with JuMP

To use SDPLR with JuMP, use SDPLR.Optimizer:

using JuMP, SDPLR
model = Model(SDPLR.Optimizer)

MathOptInterface API

The SDPLR optimizer supports the following constraints and attributes.

List of supported objective functions:

• MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}

List of supported variable types:

• MOI.Nonnegatives
• MOI.PositiveSemidefiniteConeTriangle

List of supported constraint types:

• MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}

List of supported model attributes:

• MOI.ObjectiveSense()

Attributes

The algorithm is parametrized by the attributes that can be used both with JuMP.set_attributes and JuMP.get_attributes and have the following types and default values:

rho_f::Cdouble = 1.0e-5
rho_c::Cdouble = 1.0e-1
sigmafac::Cdouble = 2.0
rankreduce::Csize_t = 0
timelim::Csize_t = 3600
printlevel::Csize_t = 1
dthresh_dim::Csize_t = 10
dthresh_dens::Cdouble = 0.75
numbfgsvecs::Csize_t = 4
rankredtol::Cdouble = 2.2204460492503131e-16
gaptol::Cdouble = 1.0e-3
checkbd::Cptrdiff_t = -1
typebd::Cptrdiff_t = 1
maxrank::Function = default_maxrank

The following attributes can be also be used both with JuMP.set_attributes and JuMP.get_attributes, but they are also modified by optimize!:

• majiter
• iter
• lambdaupdate
• totaltime
• sigma

When they are set, it provides the initial value of the algorithm. With get, they provide the value at the end of the algorithm. totaltime is the total time in second. For the other attributes, their meaning is best described by the following pseudo-code.

Given values of R, lambda and sigma, let vio = [dot(A[i], R * R') - b[i]) for i in 1:m] (vio[0] is dot(C, R * R') in the C implementation, but we ignore this entry here), val = dot(C, R * R') - dot(vio, lambda) + sigma/2 * norm(vio)^2, y = -lambda - sigma * vio, S = C + sum(A[i] * y[i] for i in 1:m) and the gradient is G = 2S * R. Note that norm(G) used in SDPLR when comparing with rho_c which has a 2-scaling difference from norm(S * R) used in the paper.

The SDPLR solvers implements the following algorithm.

sigma = inv(sum(size(A[i], 1) for i in 1:m))
origval = val
while majiter++ < 100_000
    lambdaupdate = 0
    localiter = 100
    while localiter > 10
        lambdaupdate += 1
        localiter = 0
        if norm(G) / (norm(C) + 1) <= rho_c / sigma
            break
        end
        while norm(G) / (norm(C) + 1) - rho_c / sigma > eps()
            localiter += 1
            iter += 1
            D = lbfgs(G)
            R += linesearch(D) * D
            if norm(vio) / (norm(b) + 1) <= rho_f || totaltime >= timelim || iter >= 10_000_000
                return
            end
        end
        lambda -= sigma * vio
    end
    if val - 1e10 * abs(origval) > eps()
        return
    end
    if norm(vio) / (norm(b) + 1) <= rho_f || totaltime >= timelim || iter >= 10_000_000
        return
    end
    sigma *= 2
    while norm(G) / (norm(C) + 1) < rho_c / sigma
        sigma *= 2
    end
    lambdaupdate = 0
end