SCS

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Julia wrapper for the SCS splitting cone solver. SCS can solve linear programs, second-order cone programs, semidefinite programs, exponential cone programs, and power cone programs.

Installation

You can install SCS.jl through the Julia package manager:

julia> Pkg.add("SCS")

SCS.jl will use SCS_jll and binaries built by the Yggdrasil infrastructure. Note that if you are not using the official Julia binaries from https://julialang.org/downloads/ you may need a custom install of the SCS binaries (see below for more information).

Usage

Use with Convex.jl

This example shows how we can model a simple knapsack problem with Convex and use SCS to solve it.

using Convex, SCS
items  = [:Gold, :Silver, :Bronze]
values = [5.0, 3.0, 1.0]
weights = [2.0, 1.5, 0.3]

# Define a variable of size 3, each index representing an item
x = Variable(3)
p = maximize(x' * values, 0 <= x, x <= 1, x' * weights <= 3)
solve!(p, SCS.Optimizer)
println([items x.value])
# [:Gold 0.9999971880377178
#  :Silver 0.46667637765641057
#  :Bronze 0.9999998036351865]

Use with JuMP

This example shows how we can model a simple knapsack problem with JuMP and use SCS to solve it.

using JuMP, SCS
items  = [:Gold, :Silver, :Bronze]
values = Dict(:Gold => 5.0,  :Silver => 3.0,  :Bronze => 1.0)
weight = Dict(:Gold => 2.0,  :Silver => 1.5,  :Bronze => 0.3)

model = Model(SCS.Optimizer)
@variable(model, 0 <= take[items] <= 1)  # Define a variable for each item
@objective(model, Max, sum(values[item] * take[item] for item in items))
@constraint(model, sum(weight[item] * take[item] for item in items) <= 3)
optimize!(model)
println(value.(take))
# 1-dimensional DenseAxisArray{Float64,1,...} with index sets:
#     Dimension 1, Symbol[:Gold, :Silver, :Bronze]
# And data, a 3-element Array{Float64,1}:
#  1.0000002002226671
#  0.4666659513182934
#  1.0000007732744878

Options

All SCS solver options can be set through Convex.jl or MathOptInterface.jl.

For example:

model = Model(optimizer_with_attributes(SCS.Optimizer, "max_iters" => 10))

# via MathOptInterface:
optimizer = SCS.Optimizer()
MOI.set(optimizer, MOI.RawParameter("max_iters"), 10)
MOI.set(optimizer, MOI.RawParameter("verbose"), 0)

Common options are:

  • max_iters: the maximum number of iterations to take
  • verbose: turn printing on (1) or off (0) See the glbopts.h header for other options.

Select one of the linear solvers using the linear_solver option. The values available are SCS.DirectSolver (the default) and SCS.IndirectSolver. A third option for using a GPU is experimental, see the section below.

SCS on GPU

An experimental SCS.GpuIndirectSolver can be used by either providing the appropriate libraries in a custom installation, or via the default binaries. SCS_jll-2.1.3 depends on CUDA_jll version 10.1, while prior versions require CUDA_jll in version9.0. In both cases CUDA_jll must be installed and loaded *before SCS.

julia> import Pkg

julia> Pkg.add(Pkg.PackageSpec(name = "CUDA_jll", version = "10.1"))

julia> using CUDA_jll  # This must be called before `using SCS`.

julia> using SCS

julia> SCS.available_solvers
3-element Array{DataType,1}:
 SCS.DirectSolver
 SCS.IndirectSolver
 SCS.GpuIndirectSolver

julia> optimizer = SCS.Optimizer();

julia> MOI.set(
           optimizer,
           MOI.RawParameter("linear_solver"),
           SCS.GpuIndirectSolver
       )

Low-level wrapper

SCS provides a low-level interface to solve a problem directly without interfacing through MathOptInterface.

We assume we are solving a problem of the form

minimize        c' * x
subject to      A * x + s = b
                s in K

where K is a product cone of:

  • zero cones,
  • positive orthant { x | x >= 0 },
  • second-order cones (SOC) { (t,x) | ||x||_2 <= t },
  • semi-definite cones (SDC) { X | X is psd },
  • exponential cones { (x,y,z) | y e^(x/y) <= z, y>0 },
  • power cone { (x,y,z) | x^a * y^(1-a) >= |z|, x>=0, y>=0 }, and
  • dual power cone { (u,v,w) | (u/a)^a * (v/(1-a))^(1-a) >= |w|, u>=0, v>=0 }.

The problem data are:

  • A is the matrix with m rows and n cols
  • b is of length m x 1
  • c is of length n x 1
  • f is the number of primal zero / dual free cones, i.e. primal equality constraints
  • l is the number of linear cones
  • q is the array of SOCs sizes
  • s is the array of SDCs sizes
  • ep is the number of primal exponential cones
  • ed is the number of dual exponential cones
  • p is the array of power cone parameters (±1, with negative values for the dual cone)
  • options is a dictionary of options (see above).

To solve this problem with SCS, call SCS.scs_solve. It has the following signature:

function scs_solve(
    linear_solver::Type{<:LinearSolver},
    m::Integer,
    n::Integer,
    A::AbstractMatrix,
    b::Vector{Float64},
    c::Vector{Float64},
    f::Integer,
    l::Integer,
    q::Vector{<:Integer},
    s::Vector{<:Integer},
    ep::Integer,
    ed::Integer,
    p::Vector{Float64},
    primal_sol::Vector{Float64}=zeros(n),
    dual_sol::Vector{Float64}=zeros(m),
    slack::Vector{Float64}=zeros(m);
    options...,
)

and it returns an object of type Solution, which contains the following fields

mutable struct Solution{T}
    x::Vector{Float64}
    y::Vector{Float64}
    s::Vector{Float64}
    info::ScsInfo{T}
    ret_val::T
end

where x stores the optimal value of the primal variable, y stores the optimal value of the dual variable, s is the slack variable, and info contains various information about the solve step.

SCS.raw_status(::ScsInfo)::String describes the status, e.g. 'Solved', 'Indeterminate', 'Infeasible/Inaccurate', etc.

Custom Installation

Custom build binaries are needed when using non-standard compile options, or non-official julia binaries. Special caution is required during the compilation of the scs libraries to ensure proper options and linking:

  • libscsdir and libscsindir need to be compiled with DLONG=1.
  • (optional) libscsgpuindir needs to be compiled with DLONG=0

All of these libraries should be linked against the OpenBLAS library which julia uses. For the official julia binaries this can be achieved (starting from this commit) by e.g.

cd SCS_SOURCE_DIR
make purge
make USE_OPENMP=1 BLAS64=1 BLASSUFFIX=_64_ DLONG=1 BLASLDFLAGS="-L$JULIA_BLAS_PATH -lopenblas64_" out/libscsdir.so out/libscsindir.so
make clean
make USE_OPENMP=1 BLAS64=1 BLASSUFFIX=_64_ DLONG=0 BLASLDFLAGS="-L$JULIA_BLAS_PATH -lopenblas64_" out/libscsgpuindir.so

where

  • SCS_SOURCE_DIR is the main directory of the source of scs, and
  • JULIA_BLAS_PATH is the path to the directory containing BLAS library used by julia.
    • Before julia-1.3: abspath(joinpath(Sys.BINDIR, "..", "lib", "julia"))),
    • afterwards: the path to BLAS library artifact, e.g. joinpath(OpenBLAS_jll.artifact_dir, "lib", "julia")

To use custom built SCS binaries with SCS.jl set the environment variable JULIA_SCS_LIBRARY_PATH to SCS_SOURCE_DIR/opt and build SCS.jl:

ENV["JULIA_SCS_LIBRARY_PATH"]="SCS_SOURCE_DIR/out"
using Pkg; Pkg.build("SCS")

To switch back to the default binaries delete JULIA_SCS_LIBRARY_PATH from ENV and call Pkg.build("SCS") again.