A Condition Number for Multifold Conic Systems

Abstract

Let $A:Y\to X$ be a linear map and $K\subseteq X$ be a regular closed convex cone. Consider the problem of finding a nontrivial solution to the conic feasibility problem $Ay\in K$. Condition numbers for this problem (as well as for related ones) are studied to quantify various issues concerning properties of the conic feasibility problem. Some issues especially relevant are the behavior of the problem under data perturbations, the geometry of the set of solutions, and the complexity analyses of algorithms that solve the problem. In this paper we define and characterize a condition number that exploits the possible factorization of K as a product of simpler cones. This condition number extends both Renegar's condition number and the one we defined in [Math. Program., 91 (2001), pp. 163–174] for polyhedral conic systems. We see these results as a step in developing a theory of conditioning that takes into account the structure of the problem.