A Complementarity Partition Theorem for Multifold Conic Systems

Consider a homogeneous multifold convex conic system $$Ax = 0, \; x\in K_1\times...\times K_r$$ and its alternative system A\transp y \in K_1^*\times...\times K_r^*, where $K_1,..., K_r$ are regular closed convex cones. We show that there is canonical partition of the index set ${1,...,r}$ 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.Comment: 12 pages, 2 figure

A Condition Number for Multifold Conic Systems

The analysis of iterative algorithms solving a conic feasibility problem Ay ∈ K, with A a linear map and K a regular, closed, convex cone, can be conveniently done in terms of Renegar’s condition number C(A) of the input data A. In this paper we define and characterize a condition number which exploits the possible factorization of K as a product of simpler cones. This condition number, which extends the one defined in [Math. Program., 91:163–174, 2001] for polyhedral conic systems, captures better the conditioning of the problem by filtering out, e.g., differences in scaling between components corresponding to different factors of K. We see these results as a step in developing a theory of conditioning that takes into account the structure of the problem