Iterative Methods for Complementarity Problems

Complementarity problems arise in optimisation, game theory, contact mechanics and economic modelling, where one seeks a vector that satisfies a system of inequalities coupled by a zero‐product condition. In its linear form, the task is to find x such that x ≥ 0, Mx + q ≥ 0 and xᵀ(Mx + q) = 0 for a given matrix M and vector q. Variational inequalities generalise this by replacing Mx + q with a nonlinear operator. Classical direct solvers struggle when the problem dimension grows or when the operator lacks strong smoothness. Iterative methods address these challenges by producing a sequence converging to a solution under suitable conditions. Core approaches include fixed‐point iterations, Gauss–Seidel and Jacobi schemes, Newton‐type updates and smoothing techniques. Recent advances exploit operator splitting, acceleration by momentum, adaptive step‐sizes and neural‐net formulations. Convergence analysis typically rests on monotonicity or spectral properties of the underlying operator, with M-matrix structure playing a central role for linear cases. Practical implementations demonstrate that well‐tuned iterative solvers can handle large‐scale systems encountered in network flows, mechanical equilibrium and complementarity‐based equilibrium models.

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Recent studies have introduced two novel fixed‐point schemes specifically tailored for absolute value equations, a class of complementarity problems where the operator involves a term |x|. These schemes decompose the problem into tractable subproblems and establish convergence under mild assumptions on the spectral radius of an associated matrix. Numerical experiments highlight faster convergence and greater robustness compared with classical Picard and Mann iterations.

Another line of development proposes a Newton-type two-step technique that treats absolute value equations via a predictor–corrector framework. The first stage applies a generalised Newton iteration to approximate the nonsmooth term, while the second employs a Simpson-based correction to refine the update. Under reasonable Lipschitz conditions, the method enjoys local superlinear convergence. Benchmarks on large‐scale systems, including heat­equation discretisations, attest to its computational efficiency.

A complementary contribution presents two generalised Gauss–Seidel iteration methods for absolute value equations where the coefficient matrix belongs to the M-matrix class. By carefully selecting relaxation parameters, these methods guarantee global convergence and monotone reduction of the residual. Comparative studies demonstrate notable gains over standard Gauss–Seidel, particularly when the underlying M-matrix exhibits strong diagonal dominance.

Iterative Methods for Complementarity Problems publication trend

The graph below shows the total number of articles in iterative methods for complementarity problems across all publications each year (not limited to Nature Index journals).

Technical terms

Complementarity problem: A system requiring a vector x and its image under an operator to be nonnegative and orthogonal in each component.

Fixed‐point iteration: A scheme that generates a sequence x_{k+1} = T(x_k) aiming to find x* such that x* = T(x*).

Newton‐type method: An approach that uses derivative or generalized derivative information to produce iterates with high local convergence rates.

M-matrix: A real matrix characterised by nonpositive off-diagonals and a spectrum lying in the positive half-plane, ensuring certain monotonicity properties.

Smoothing technique: A method that approximates a nonsmooth or discontinuous component with a continuously differentiable function to enable gradient-based updates.

References

1. The study of new fixed-point iteration schemes for solving absolute value equations. Heliyon (2024).
2. A Newton-type technique for solving absolute value equations. Alexandria Engineering Journal (2023).
3. Two new generalized iteration methods for solving absolute value equations using $ M $-matrix. AIMS Mathematics (2022).