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

A conic linear system is a system of the form (FPd ) Ax = b x ∈ CX , where A : X −→ Y is a linear operator between nand m-dimensional linear spaces X and Y , b ∈ Y , and CX ⊂ 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 d = ( A, b) needed to create a data instance d + d 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 C(d) of the data instance d as the scale-invariant reciprocal of ρ(d): C(d) = ‖d‖ ρ(d) . In this paper we develop an elementary algorithm that computes a solution of (FPd) when it is feasible, or demonstrates that (FPd ) 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 ( c̃ C(d)2 ln(C(d)) ) where the constant c̃ depends only on the properties of the cone CX 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 CX . The algorithm is “elementary” in the sense that it performs only a few relatively simple computations at each iteration. The solution x̂ of the system (FPd ) generated by the algorithm has the property of being “reliable” in the sense that the distance from x̂ to the boundary of the cone CX , dist(x̂, ∂CX ), and the size of the solution, ‖x̂‖, satisfy the following inequalities: ‖x̂‖ ≤ c1C(d), dist(x̂, ∂CX ) ≥ c2 1 C(d) , and ‖x̂‖ dist(x̂, ∂CX ) ≤ c3C(d), where c1, c2, c3 are constants that depend only on properties of the cone CX and are independent of the data d (with analogous results for the alternative system when the system (FPd ) is infeasible).