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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.

##### MSC:
 47 Operator theory 54 General topology
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