DFMethods.jl

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

Solve

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

where $X$ is closed convex and $\psi : \mathbb{R}^n \to \mathbb{R}^n$ is continuous and satisfies

\[\psi(x)^\top (x - u^*) \geq 0 \quad \forall x \in \mathbb{R}^n,\]

a property automatically satisfied if $\psi$ is monotone or pseudo-monotone. No derivatives of $\psi$ are required.

Why DFMethods

DFMethods.jl is the first Julia implementation of the Solodov–Svaiter hyperplane-projection family with derivative-free CG-style search directions.

Quick navigation

• Quickstart
• Algorithm
• Constraint Sets
• Extending
• API Reference

Public API in one paragraph

using NonlinearSolve, DFMethods

prob = NonlinearProblem(F, u0, p)                  # SciML problem (in-place or out-of-place)
alg  = DFProjection(;                              # configure components
    direction  = SpectralThreeTerm(),                        # search direction (or custom)
    linesearch = LSII(),                           # one of LSI..LSVII (or custom)
    inertial   = Inertial(0.25),                   # or NoInertial()
    set        = BoxSet(lower, upper),             # or RealSpace / HalfSpace / CappedBox / UserSet
    abstol     = 1e-6,
    maxiters   = 2000,
)
sol = solve(prob, alg)                             # NonlinearSolution with retcode, u, resid, stats

Reference

Ibrahim, A. H., Alshahrani, M., & Al-Homidan, S. (2026). A Unified Derivative-Free Projection Framework for Convex-Constrained Nonlinear Equations. Journal of Optimization Theory and Applications, 208:11. DOI:10.1007/s10957-025-02826-x.