Algorithm

DFMethods implements UIDFPAF — the Unified Inertial Derivative-Free Projection Algorithmic Framework of Ibrahim, Alshahrani, Al-Homidan (JOTA 2026).

Problem class

Find $u^* \in X$ such that $\psi(u^*) = 0$, with $X \subset \mathbb{R}^n$ closed convex and $\psi$ continuous satisfying

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

This holds whenever $\psi$ is monotone or pseudo-monotone — the framework's convergence theorem (Ibrahim 2026 Thm 3.1) requires no more.

One outer iteration

Each call to step! (or one pass of solve_df!) performs:

1. Inertial extrapolation.

   \[w_k = x_k + \theta_k (x_k - x_{k-1}), \quad \theta_k = \min\!\Bigl\{\theta,\ \tfrac{1}{k^2 \|x_k - x_{k-1}\|}\Bigr\}\]

2. Function eval. Compute $\psi(w_k)$. Early-stop if $\|\psi(w_k)\| \leq \epsilon$.

3. Direction. Compute $d_k$ via the chosen AbstractSearchDirection. Must satisfy sufficient descent

   \[-\psi(w_k)^\top d_k \geq c \|\psi(w_k)\|^2\]

   and boundedness $\|d_k\| \leq \bar c \|\psi(w_k)\|$ (Ibrahim 2026 eqs. 3–4).

4. Backtracking line search. Find the smallest $i \geq 0$ such that $\alpha_k = \rho^i$ satisfies

   \[-\psi(w_k + \alpha_k d_k)^\top d_k \geq \sigma\, \alpha_k\, \gamma_k\, \|d_k\|^2\]

   with $\gamma_k$ supplied by the chosen AbstractDFLineSearch.

5. Trial point. $z_k = w_k + \alpha_k d_k$. Early-stop if $\|\psi(z_k)\| \leq \epsilon$ and $z_k \in X$.

6. Hyperplane. The hyperplane $H_k = \{x : \psi(z_k)^\top(x - z_k) \leq 0\}$ separates the current iterate from the solution set.

7. Approximate projection.

   \[x_{k+1} = \tilde P_{X \cap H_k}\!\bigl(w_k - \lambda_k \psi(z_k);\ \epsilon_k\bigr)\]

   with $\lambda_k = \psi(z_k)^\top(w_k - z_k) / \|\psi(z_k)\|^2$ and $\epsilon_k = \tfrac{1}{2} \zeta^2 \|\lambda_k \psi(z_k)\|^2$.

Pluggable components

See Extending for how to define your own direction or line search.

Default configuration

DFProjection()   # equivalent to:
DFProjection(;
    direction  = SpectralThreeTerm(; r = 0.1, alpha_bar = 1.0),
    linesearch = LSII(; σ = 0.01, ρ = 0.6),
    inertial   = Inertial(0.25),
    set        = RealSpace(),
    abstol     = 1e-6,
    maxiters   = 2000,
    ζ          = 0.5,
    inner_maxiter = 500,
    maxbt      = 50,
)

The defaults reproduce Ibrahim 2026's experimental setup with the LSII line search (best-performing variant in Tables 1–5 of the paper).

Convergence traits

Each algorithm declares its assumptions via three trait predicates:

monotonicity_required(::AbstractDFProjectionAlgorithm)::Bool
pseudomonotonicity_sufficient(::AbstractDFProjectionAlgorithm)::Bool
convex_set_required(::AbstractDFProjectionAlgorithm)::Bool

DFProjection inherits the defaults (true for all three) from AbstractDFProjectionAlgorithm. Future algorithms (e.g., non-Lipschitz variants from Ibrahim 2023) override these to widen their assumed scope.

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. https://doi.org/10.1007/s10957-025-02826-x