DFMethods.jl

Derivative-free projection methods for constrained nonlinear equations, plugged into NonlinearSolve.jl.

Solve

\[\text{Find } u^* \in X \subset \mathbb{R}^n \text{ such that } F(u^*) = 0,\]

where $X$ is closed convex and $F : \mathbb{R}^n \to \mathbb{R}^n$ is continuous. The algorithm uses no derivative information of $F$. Precise convergence assumptions are stated on the Algorithm page.

What it does

DFMethods.jl provides solvers for nonlinear equations $F(u) = 0$ with $u$ constrained to a closed convex set $X$. The methods are derivative-free (only function evaluations of $F$ are needed) and projection-based (each iteration produces a feasible iterate by projecting onto $X$).

The package is organized as a configurable framework: each step of an outer iteration — inertial extrapolation, search direction, line search, trial point, iterate update, stopping rule — is an independently pluggable component. Users can swap in custom rules at each step without touching the solver core. See Algorithm for the structure and Extending for the contracts.

When to use it

DFMethods.jl is appropriate when:

the problem is a system of equations $F(u) = 0$, not an optimization problem (use Optimization.jl for the latter);
a feasibility set $X \subset \mathbb{R}^n$ must be respected at every iteration, and $X$ is more general than box bounds;
evaluating $F$ is feasible but evaluating $\nabla F$ is not (or not worthwhile).

For problems outside these conditions, see Comparisons for a discussion of related packages.

	Constrained?	Derivative-free?	Approach
NonlinearSolve.jl `DFSane` / `SimpleDFSane`	✗	✓	spectral residual
NonlinearSolve.jl `Broyden` / `Klement` / `LimitedMemoryBroyden`	✗	✓ (secant)	quasi-Newton
NonlinearSolve.jl JFNK (Newton + Krylov)	✗	✓ (matrix-free)	Newton–Krylov
`NLboxsolve.jl`	box only	✗	Newton/QN
`ProximalAlgorithms.jl`, `SPGBox.jl`	✓	✗	minimization
DFMethods.jl	any closed convex (incl. ℝⁿ)	✓	projection-based

Installation

From the Julia General registry:

using Pkg
Pkg.add("DFMethods")

Or, equivalently, from the REPL's Pkg mode:

] add DFMethods

To track unreleased changes directly from GitHub:

using Pkg
Pkg.add(url = "https://github.com/mmogib/DFMethods.jl")

Public API in one example

F(u, p) = u .- sin.(u)
u0      = ones(100)
prob    = NonlinearProblem(F, u0)            # SciML problem (in-place or out-of-place)
alg     = DFProjection(;                     # configure components
    direction      = SpectralThreeTerm(),    # search direction (or custom)
    linesearch     = ResidualNormBacktrack(),# or ConstantBacktrack / AdaptiveClampedBacktrack
    inertial       = Inertial(0.25),         # or NoInertial()
    iterate_update = SolodovSvaiterProjection(), # or DirectUpdate / HalpernUpdate(β)
    abstol         = 1e-6,
    maxiters       = 2000,
)
sol = solve(prob, alg)                       # NonlinearSolution with retcode, u, resid, stats
sol.retcode

ReturnCode.Success = 1

Box constraints are passed to the problem (NonlinearProblem(F, u0; lb, ub)); other convex sets use ConstrainedNonlinearProblem. See Constraint Sets.

Theoretical lineage