Immunity to Increasing Condition Numbers of Linear Superiorization versus Linear Programming

TL;DR

This work investigates how Linear Superiorization (LinSup) performs relative to traditional linear programming (LP) solvers when the underlying constraint system becomes ill-conditioned, characterized by a high condition number $κ$. The authors implement LinSup using an AMS projection-based feasibility-seeking core with bounded perturbations that steer iterates toward lower objective values, and compare it against SciPy simplex, revised simplex, interior-point, and Gurobi primal simplex, under a fair stopping rule based on infeasibility. Results show LinSup is robust to increasing problem size and conditioning, often terminating faster than LP solvers at the same infeasibility threshold, and in large ill-conditioned problems can outperform interior-point methods, while commercial Gurobi remains highly competitive. The findings suggest LinSup as a practical alternative when the goal is a feasible point with a reduced objective value rather than exact optimality, particularly for large-scale ill-conditioned problems, with potential extensions to nonlinear settings.

Abstract

Given a family of linear constraints and a linear objective function one can consider whether to apply a Linear Programming (LP) algorithm or use a Linear Superiorization (LinSup) algorithm on this data. In the LP methodology one aims at finding an optimal point, i.e., a point that fulfills the constraints and has the minimal value of the objective function over these constraints. The Linear Superiorization approach considers the same data as linear programming problems but instead of attempting to solve those with linear programming methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward a feasible point with reduced (not necessarily minimal) objective function value. This aim of the superiorization method (SM) is less demanding than aiming to reach full-fledged constrained optimality and it places more importance on reaching feasibility than on reaching optimality. Previous studies (e. g. [12]) compared LP and LinSup in terms of their respective outputs and the resources they use. This paper is a follow-up analysis of [12], where we investigate classical LP approaches and LinSup in terms of their sensitivity to condition numbers of the system of linear constraints. Condition numbers are a measure for the impact of deviations in the input data on the output of a problem and, in particular, they describe the factor of error propagation when given wrong or erroneous data. Therefore, the ability of LP and LinSup to cope with increased condition numbers, thus with ill-posed problems, is an important matter to consider which was not studied until now. We investigate experimentally the advantages and disadvantages of both LP and LinSup on exemplary problems of linear programming with multiple condition numbers and different problem dimensions.

Figures (6)

• Figure 1: Dimensions $80\times100$ - $400\times500$. Figures \ref{['fig:Dim80']}--\ref{['fig:Dim4000']} show the behavior of the different algorithms on problems with increasing dimension for fixed condition number $\kappa=1000$. On the left we see the maximum violation of the constraints $\max_{i}(\langle a^{i},x\rangle-b_{i})$ plotted against the run-time. On the right we have the corresponding objective function values $\langle c,x\rangle$ plotted against the run-time.
• Figure 2: Dimensions $800\times1000$ - $4000\times5000$.
• Figure 3: Figures \ref{['fig:accDim80']}--\ref{['fig:accDim4000']} demonstrate well the robustness with respect to increasing condition numbers. On the left we see the averaged run-times of $5$ problem instances for each algorithm, plotted against the condition numbers, on the right we see the averaged objective function values at termination, plotted against the condition numbers. Notice the severely suboptimal objective function value of scipy's simplex for high condition numbers. This explains the low run-time as the algorithm realizes that these problems are hard and quickly “ surrenders".
• Figure 4: The trend continues in Figures \ref{['fig:accDim800']} and \ref{['fig:accDim4000']} and the revised simplex too starts to reach its limits. Superiorization on the other hand proves to be quite stable in terms of its run-time with regards to increasing condition numbers, at times surpassing the Gurobi implementation in both runtime and objective function value. While the total runtime does increase significantly with the condition number, it remains within reasonable timeframes, considering the increased difficulty of the problems that accompanies higher condition numbers
• Figure 5: Average infeasibilities at termination of the problem instances. Notice how with increasing condition number the general trend is towards higher infeasibility, except for superiorization, which remains constant throughout. Like before, scipy.simplex and scipy.revised simplex turn out to be unreliable in moderate dimension already. Additionally, we can see that scipy.interior-point reaches infeasibilities in the order of magnitude $10^0-10^1$.