Large scale convex optimization solvers in julia
Author mfalt
20 Stars
Updated Last
2 Years Ago
Started In
May 2017


Build Status Coverage Status

Package for large scale convex optimization solvers in julia. This package is intended to allow for easy implementation, testing, and running of solvers through the Convex.jl interface. The package is currently under active development and uses the ProximalOperators.jl package to do the low level projections.


To run the solvers you need to have the following packages



Define an optimization problem in the format supported by Convex.jl, and supply the desired solver to the solve! function. Exaple using DR for feasibility problems with the GAP solver

using Convex, FirstOrderSolvers
m = 40;  n = 50
A = randn(m, n); b = randn(m, 1)
x = Variable(n)
problem = minimize(sumsquares(A * x - b), [x >= 0])

solve!(problem, GAP(0.5, 2.0, 2.0, max_iters=2000))


Currently, the available solvers are

Solver Description Reference
GAP(α=0.8, α1=1.8, α2=1.8; kwargs...) Generalized Alternating Projections
DR(α=0.5; kwargs...) Douglas-Rachford (GAP(α, 2.0, 2.0)) Douglas, Rachford (1956)
AP(α=0.5; kwargs...) Alternating Projections (GAP(α, 1.0, 1.0)) Agmon (1954), Bregman (1967)
GAPA(α=1.0; kwargs...) GAP Adaptive Fält, Giselsson (2017)
FISTA(α=1.0; kwargs...) FISTA Beck, Teboulle (2009)
Dykstra(; kwargs...) Dykstra Boyle, Dykstra (1986)
GAPP(α=0.8, α1=1.8, α2=1.8; iproj=100; kwargs...) Projected GAP Fält, Giselsson (2016)

Keyword Arguments

All solvers accept for the following keyword arguments:

Argument Default Description (Values)
max_iters 10000 Maximum number of iterations
verbose 1 Print verbosity level 0,1
debug 1 Level of debug data to save 0,1,2
eps 1e-5 Accuracy of solution
checki 100 Interval for checking convergence


If the keyword argument debug is set to 1 or 2 the following values will be stored in a ValueHistories.MVHistory in problem.model.history, for each iteration the convergence check is run:

Name Debug Level Required Description
:p 1 Relative Primal Residual
:d 1 Relative Dual Residual
:g 1 Relative Duality Gap
:ctx 1 cᵀx
:bty 1 bᵀy
1 κ
1 τ
:x 2 x
:y 2 y
:s 2 s

These values correspond to the values in the paper Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding (O'Donoghue

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