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Variational inequality formulation of circular cone eigenvalue complementarity problems. (English) Zbl 06563710
Summary: In this paper, we study the circular cone eigenvalue complementarity problem (CCEiCP) by variational inequality technique, prove the existence of a solution to CCEiCP, and investigate different nonlinear programming formulations of the symmetric and asymmetric CCEiCP, respectively. We reduce CCEiCP to a variational inequality problem on a compact convex set, which guarantees that CCEiCP has at least one solution. Based on the variational inequality formulation of CCEiCP, the symmetric CCEiCP can be reformulated as a nonlinear programming problem NLP\(_{1}\), and solved by computing a stationary point of the Rayleigh quotient function on a compact set. We formulate the asymmetric CCEiCP as another nonlinear programming problem NLP\(_{2}\), and show that any global minimum of NLP\(_{2}\) with an objective function value equal to zero is a solution of the asymmetric CCEiCP. Moreover, a stationary point of NLP\(_{2}\) is a solution of the asymmetric CCEiCP, if and only if the Lagrange multipliers associated with the equalities in NLP\(_{2}\) are equal to zero. The different formulations of CCEiCP provide alternative approaches for solving CCEiCP, which will play an important role in designing efficient algorithms to solve CCEiCP.

47 Operator theory
54 General topology
PATH Solver
Full Text: DOI
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