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<p>The linear complementarity problem (LCP) serves as a crucial mathematical model for characterizing scenarios such as Nash equilibria in game theory, supply and demand balance in economic systems, and dynamic traffic flow distribution. The development of efficient numerical solutions for LCP holds significant theoretical value and practical significance. This paper proposes the modified matrix splitting iteration (MMSI) method, which introduces a double-diagonal parameter matrix and a relaxation factor to construct a novel iteration format. It innovatively integrates a projection operator to improve the algorithm’s robustness with respect to the choice of initial point. Theoretical proof shows that when the coefficient matrix <i>M</i> is a positive definite <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mo>+</mo> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{H}_{+}$</EquationSource> </InlineEquation>-matrix with positive diagonal elements, the algorithm globally converges to a unique solution. Numerical experiments fully demonstrate the efficiency and stability of this algorithm.</p>

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Parameter regulation and projection enhancement matrix splitting method: efficient solution for \(\mathcal{H}_{+}\)-matrix linear complementarity problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月25日

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  • Yajun Xie,
  • Yuting Zhang,
  • Jianfeng Li

摘要

The linear complementarity problem (LCP) serves as a crucial mathematical model for characterizing scenarios such as Nash equilibria in game theory, supply and demand balance in economic systems, and dynamic traffic flow distribution. The development of efficient numerical solutions for LCP holds significant theoretical value and practical significance. This paper proposes the modified matrix splitting iteration (MMSI) method, which introduces a double-diagonal parameter matrix and a relaxation factor to construct a novel iteration format. It innovatively integrates a projection operator to improve the algorithm’s robustness with respect to the choice of initial point. Theoretical proof shows that when the coefficient matrix M is a positive definite H + $\mathcal{H}_{+}$ -matrix with positive diagonal elements, the algorithm globally converges to a unique solution. Numerical experiments fully demonstrate the efficiency and stability of this algorithm.