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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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).