SCS
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 BinaryProvider.jl to automatically install the SCS binaries. 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.
Custom Installation
Custom build binaries will allow to use e.g. the indirect solver on (a CUDA-enabled) gpu,
however special caution is required during the compilation of the scs
libraries to ensure proper options and linking:
libscsdir
andlibscsindir
need to be compiled withDLONG=1
.- (optional)
libscsgpuindir
needs to be compiled withDLONG=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)[https://github.com/cvxgrp/scs/commit/e6ab81db115bb37502de0a9917041a0bc2ded313]) 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 ofscs
, andJULIA_BLAS_PATH
is the path to the directory containing BLAS library used byjulia
. (Beforejulia-1.3
: the path to julia-shipped libraries e.g.abspath(joinpath(Sys.BINDIR, "..", "lib", "julia"))
), afterwards the path toBLAS
library artifact).
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.
Usage
High-level interfaces
SCS implements the solver-independent MathOptInterface interface, and so can be used within modeling softwares like Convex and JuMP. The optimizer constructor is SCS.Optimizer
.
A legacy MathProgBase interface is available as well, in maintanence mode only.
Options
All SCS solver options can be set through the direct interface(documented below), through Convex.jl
or MathOptInterface.jl
.
The list of options follows the glbopts.h
header in lowercase.
To use these settings you can either pass them as keyword arguments to SCS_solve
(high level interface) or using the SCS.Optimizer
constructor (MathOptInterface), e.g.
# Direct
solution = SCS_solve(m, n, A, ..., psize; max_iters=10, verbose=0);
# via MathOptInterface:
optimizer = SCS.Optimizer()
MOI.set(optimizer, MOI.RawParameter("max_iters"), 10)
MOI.set(optimizer, MOI.RawParameter("verbose"), 0)
or via specific helper functions:
problem = ... # JuMP problem
optimizer_constructor = optimizer_with_attributes(SCS.Optimizer, "max_iters" => 10, "verbose" => 0)
set_optimizer(problem, optimizer_constructor)
optimize!(problem)
Moreover, you may select one of the linear solvers to be used by SCS.Optimizer
via linear_solver
keyword. The options available are SCS.IndirectSolver
(the
default) and SCS.DirectSolver
. 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. The
latter depends on CUDA_jll
version 9.0
, which must be installed and loaded
*beforeSCS
.
julia> import Pkg
julia> Pkg.add(Pkg.PackageSpec(name = "CUDA_jll", version = "9.0"))
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> solver = SCS.Optimizer(linear_solver = SCS.GpuIndirectSolver)
High level wrapper
The file c_wrapper.jl
is thoroughly commented. Here is the basic usage.
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 colsb
is of length m x 1c
is of length n x 1f
is the number of primal zero / dual free cones, i.e. primal equality constraintsl
is the number of linear conesq
is the array of SOCs sizess
is the array of SDCs sizesep
is the number of primal exponential conesed
is the number of dual exponential conesp
is the array of power cone parameters (±1, with negative values for the dual cone)options
is a dictionary of options (see above).
The function is
function SCS_solve(linear_solver::Type{<:LinearSolver},
m::Integer, n::Integer,
A::SCS.VecOrMatOrSparse, 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<:SCSInt}
x::Array{Float64, 1}
y::Array{Float64, 1}
s::Array{Float64, 1}
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.
E.g. SCS.raw_status(::SCSInfo)::String
describes the status, e.g. 'Solved', 'Intedeterminate', 'Infeasible/Inaccurate', etc.
Convex and JuMP examples
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]
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