Spectral classification of convex cones

Abstract

Closed convex cones can be classified according to their capacity to produce complementarity eigenvalues. Let K be a closed convex cone in a Euclidean space E. The K - spectrum of a linear map $$A: E\rightarrow E$$ is the set $$\sigma _K(A)$$ of all $$\lambda \in {\mathbb {R}}$$ for which the complementarity problem $$\begin{aligned} K \ni x\perp (Ax-\lambda x) \in K^*\end{aligned}$$has a nonzero solution $$x\in E$$. Here, $$K^*$$ is the dual cone of K and $$\perp $$ stands for orthogonality. It is known that if K is a polyhedral cone, then $$\sigma _K(A)$$ has finite cardinality for all A. This work identifies a special class of non-polyhedral cones with the same property. It also identifies some classes of cones for which $$\sigma _K(A)$$ contains an interval for a particular A. The cardinality analysis of the spectral map $$\sigma _K$$ induced by K provides valuable information on the structure of the cone itself.