椭圆锥平衡模型及其在特征值互补问题中的应用

Ellipsoidal Cone Equilibrium Models and Its Application on Eigenvalue Complementarity Problems

摘要: 锥特征值互补问题在工程与经济学中具有广泛应用，然而现有研究主要集中于对称锥情形。本文针对一类非对称锥——椭圆锥，系统研究其平衡模型及相关的特征值互补问题。首先，基于椭圆锥与二阶锥之间的转换关系和二阶锥的结构表示，将椭圆锥平衡模型转化为等价的非线性系统，进而分析其解的存在条件与类型特征，包括平凡解、非平凡解、边界型解与内部型解的等价刻画。进一步，将所得理论推广至圆锥与正椭圆锥两类非对称锥平衡模型。在应用方面，本文建立了椭圆锥特征值互补问题与二阶锥互补问题之间的等价重构，将原问题转化为可借助半光滑牛顿法、邻近点算法等现有数值方法求解的形式。本研究不仅统一并扩展了对称锥特征值互补问题的已有认识，也为非对称锥情形下的算法设计与数值实现提供了理论支撑。

Abstract: Eigenvalue complementarity problems over cones have wide applications in engineering and economics, with most existing studies focusing on symmetric cones. This paper addresses a class of non-symmetric cones—ellipsoidal cones—and systematically investigates their equilibrium models and related eigenvalue complementarity problems. First, by exploiting the transformation relationship between ellipsoidal cones and second-order cones and the structured expression of second-order cone, the ellipsoidal cone equilibrium model is reformulated into an equivalent nonlinear system. Based on this, existence conditions and characterization of solutions are analyzed, including trivial, non-trivial, boundary-type, and interior-type solutions. The theoretical results are further extended to circular cone and elliptic cone equilibrium models. In terms of applications, equivalent reformulations are established for ellipsoidal cone eigenvalue complementarity problems in terms of second-order cone complementarity problems, enabling the use of existing numerical methods such as semismooth Newton and proximal point algorithms. This work not only unifies and extends known results on symmetric cone eigenvalue complementarity problems, but also provides a theoretical foundation for algorithm design and numerical implementation in the non-symmetric cone setting.