Guaranteed Accuracy for Conic Programming Problems in Vector Lattices

TL;DR

This paper develops rigorous, guaranteed error bounds for conic optimization problems in general vector lattices, including linear, second order cone, and semidefinite programming, accounting for floating point errors.

Contribution

It introduces a general framework for error bounds in conic programming that is applicable to infinite-dimensional spaces and non-polyhedral cones, with specialized formulas for common problems.

Findings

Guaranteed accuracy bounds for linear programming.

Error bounds for semidefinite programming.

Software implementing the bounds for practical problems.

Abstract

This paper presents rigorous forward error bounds for linear conic optimization problems. The error bounds are formulated in a quite general framework; the underlying vector spaces are not required to be finite-dimensional, and the convex cones defining the partial ordering are not required to be polyhedral. In the case of linear programming, second order cone programming, and semidefinite programming specialized formulas are deduced yielding guaranteed accuracy. All computed bounds are completely rigorous because all rounding errors due to floating point arithmetic are taken into account. Numerical results, applications and software for linear and semidefinite programming problems are described.