A Complementarity Partition Theorem for Multifold Conic Systems

Consider a homogeneous multifold convex conic system $$Ax=0,x\in K_1\times \mathrm{.}\mathrm{.}\mathrm{.}\times K_r$$ and its alternative system $$A\text{transp}y\in K_1^{\ast }\times \mathrm{.}\mathrm{.}\mathrm{.}\times K_r^{\ast },$$ where $K_1,\mathrm{.}\mathrm{.}\mathrm{.},K_r$ are regular closed convex cones. We show that there is canonical partition of the index set $1,\mathrm{.}\mathrm{.}\mathrm{.},r$ determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming.