Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system

Abstract.

A conic linear system is a system of the form¶¶(FP _d )Ax=b¶x∈C _X ,¶¶where A:X?Y is a linear operator between n- and m-dimensional linear spaces X and Y, b∈Y, and C _X ⊂X is a closed convex cone. The data for the system is d=(A,b). This system is “well-posed” to the extent that (small) changes in the data d=(A,b) do not alter the status of the system (the system remains feasible or not). Renegar defined the “distance to ill-posedness,”ρ(d), to be the smallest change in the data Δd=(ΔA,Δb) needed to create a data instance d+Δd that is “ill-posed,” i.e., that lies in the intersection of the closures of the sets of feasible and infeasible instances d ^′=(A ^′,b ^′) of (FP_(·)). Renegar also defined the condition number ?(d) of the data instance d as the scale-invariant reciprocal of ρ(d) : ?(d)=.¶In this paper we develop an elementary algorithm that computes a solution of (FP _d ) when it is feasible, or demonstrates that (FP _d ) has no solution by computing a solution of the alternative system. The algorithm is based on a generalization of von Neumann’s algorithm for solving linear inequalities. The number of iterations of the algorithm is essentially bounded by¶¶O( ?(d)²ln(?(d)))¶¶where the constant depends only on the properties of the cone C _X and is independent of data d. Each iteration of the algorithm performs a small number of matrix-vector and vector-vector multiplications (that take full advantage of the sparsity of the original data) plus a small number of other operations involving the cone C _X . The algorithm is “elementary” in the sense that it performs only a few relatively simple computations at each iteration.¶The solution of the system (FP _d ) generated by the algorithm has the property of being “reliable” in the sense that the distance from to the boundary of the cone C _X , dist(,∂C _X ), and the size of the solution, ∥∥, satisfy the following inequalities:¶¶∥∥≤c ₁?(d),dist(,∂C _X )≥c ₂ , and ≤c ₃?(d),¶¶where c ₁, c ₂, c ₃ are constants that depend only on properties of the cone C _X and are independent of the data d (with analogous results for the alternative system when the system (FP _d ) is infeasible).