Error Bounds and Numerical Analysis for Linear Complementarity Problems

The linear complementarity problem (LCP) is a fundamental mathematical formulation that encapsulates a wide array of equilibrium and optimisation models, including frictional contact mechanics, market equilibrium and network flow. Given a matrix M and a vector q, the LCP seeks vectors x and w satisfying w = Mx + q, x ≥ 0, w ≥ 0 and xᵀw = 0. Beyond existence and uniqueness conditions—most notably the requirement that M be a P-matrix—practitioners must assess how close a computed iterate lies to the true solution. Error bounds provide quantitative estimates of the distance from an approximate vector to the solution set in terms of a residual function, often ‖min{x,Mx+q}‖. Global error bounds guarantee this estimate across the entire domain, underpinning robust stopping criteria for iterative solvers, while local error bounds sharpen convergence rates near a solution. Numerical analysis of LCPs combines spectral properties of M—such as those of H-matrices and B-matrices—with norm estimates (particularly the infinity norm) to derive practical, computable bounds. Modern advances consider block-structured decompositions, perturbation analysis and weaker regularity conditions to extend applicability to large sparse systems and more general complementarity extensions.

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Recent studies in the wider literature have refined error bounds by exploiting specialised matrix classes and extended problem formats. A 2023 investigation into matrices with only one non-strictly diagonally dominant row introduced a new subclass of H-matrices and derived spectrum-based bounds and block-wise inverse estimates that yield sharper error estimates for high-dimensional LCPs. In 2021, a novel subclass of P-matrices—termed CKV-type B-matrices—was shown to admit tighter infinity-norm bounds for the inverse, leading to improved error bounds for the corresponding LCP compared with earlier results for DZ-type B-matrices. Earlier work in 2018 on extended linear complementarity problems (ELCP) established a residual-based global error bound under weaker feasibility conditions, and demonstrated its use in vertical and mixed LCP variants, thus broadening the scope of guaranteed convergence monitoring in more general complementarity models.

Error Bounds and Numerical Analysis for Linear Complementarity Problems publication trend

The graph below shows the total number of articles in error bounds and numerical analysis for linear complementarity problems across all publications each year (not limited to Nature Index journals).

Technical terms

Linear complementarity problem (LCP): Given M and q, find non-negative x such that w = Mx + q, w ≥ 0 and xᵀw = 0.

Error bound: A computable estimate of the distance from an approximate solution to the true solution set, typically expressed in terms of a residual.

Residual function: A measure of infeasibility, for example r(x) = ‖min{x, Mx+q}‖∞, used to quantify error.

P-matrix: A matrix all of whose principal minors are positive, ensuring existence and uniqueness of the LCP solution.

H-matrix: A generalisation of diagonally dominant matrices for which comparison matrices are nonsingular, facilitating norm bounds on inverses.

B-matrix: A subclass of P-matrices characterised by sign and dominance conditions that allow specialised error estimates.

References

1. On Matrices with Only One Non-SDD Row. Mathematics (2023).
2. CKV-type $ B $-matrices and error bounds for linear complementarity problems. AIMS Mathematics (2021).
3. New global error bound for extended linear complementarity problems. Journal of Inequalities and Applications (2018).