Improved linear programming methods for checking avoiding sure loss

TL;DR

This paper introduces novel improvements to linear programming methods, especially the primal-dual approach, for efficiently checking avoiding sure loss in desirable gambles, outperforming traditional simplex methods in large problems.

Contribution

The paper proposes new enhancements to interior-point methods tailored for avoiding sure loss problems, including structure exploitation, stopping criteria, and feasible start calculations.

Findings

Affine scaling and primal-dual methods outperform simplex in most scenarios.

Improved primal-dual method is at least three times faster for large problems.

Small problems show no significant performance difference among methods.

Abstract

We review the simplex method and two interior-point methods (the affine scaling and the primal-dual) for solving linear programming problems for checking avoiding sure loss, and propose novel improvements. We exploit the structure of these problems to reduce their size. We also present an extra stopping criterion, and direct ways to calculate feasible starting points in almost all cases. For benchmarking, we present algorithms for generating random sets of desirable gambles that either avoid or do not avoid sure loss. We test our improvements on these linear programming methods by measuring the computational time on these generated sets. We assess the relative performance of the three methods as a function of the number of desirable gambles and the number of outcomes. Overall, the affine scaling and primal-dual methods benefit from the improvements, and they both outperform the simplex method in most scenarios. We conclude that the simplex method is not a good choice for checking avoiding sure loss. If problems are small, then there is no tangible difference in performance between all methods. For large problems, our improved primal-dual method performs at least three times faster than any of the other methods.