[Paper Review] New Methods for Solving Large Scale Linear Programming Problems in the Windows and Linux computer operating systems

Saeed Ketabchi, Hossein Moosaei+2 more|arXiv (Cornell University)|Sep 19, 2012

Advanced Optimization Algorithms Research6 references3 citations

TL;DR

This paper proposes and compares two methods for solving large-scale linear programming problems: LIPSOL (interior-point method via MATLAB's linprog) and a novel augmented Lagrangian method combined with generalized Newton iteration and Armijo line search. The key finding is that the Linux operating system outperforms Windows 7 in solving large-scale problems, successfully handling memory-intensive instances that cause out-of-memory errors in Windows, demonstrating Linux's superior efficiency and memory management for high-performance optimization workloads.

ABSTRACT

In this study, calculations necessary to solve the large scale linear programming problems in two operating systems, Linux and Windows 7 (Win), are compared using two different methods. Relying on the interior-point methods, linear-programming interior point solvers (LIPSOL) software was used for the first method and relying on an augmented Lagrangian method-based algorithm, the second method used the generalized derivative. The performed calculations for various problems show the produced random in the Linux operating system (OS) and Win OS indicate the efficiency of the performed calculations in the Linux OS in terms of the accuracy and using of the optimum memory.

Motivation & Objective

• To address the challenge of solving large-scale linear programming problems that exceed memory limits on standard Windows systems.
• To evaluate and compare the performance of two optimization methods—LIPSOL (interior-point) and an augmented Lagrangian-based generalized Newton method—on Windows and Linux.
• To investigate the impact of operating system choice on computational efficiency, memory usage, and solution accuracy for large-scale LP problems.
• To demonstrate the feasibility of solving large-scale LP problems using a custom algorithm on Linux when standard tools fail on Windows.

Proposed method

• The first method uses LIPSOL, a MATLAB-based interior-point solver accessed via the linprog function, for solving standard form linear programs.
• The second method employs an augmented Lagrangian formulation of the dual problem, transforming it into an unconstrained minimization problem using the function $ \Phi(u,\alpha,\bar{x}) = -b^T u + \frac{1}{2\alpha} \| (\bar{x} + \alpha(A^T u - c))_+ \|^2 $.
• A generalized Newton method is applied to minimize $ \Phi $, with the Hessian approximated as $ \nabla^2\Phi(u,\alpha,\bar{x}) = A D(z) A^T $, where $ D(z) $ is a diagonal matrix based on active constraints.
• A modified Newton direction is used to handle potential singularity: $ -(\nabla^2\Phi + \delta I_m)^{-1} \nabla\Phi $, with $ \delta = 10^{-4} $.
• An Armijo rule-based line search ensures global convergence by backtracking on step size to satisfy sufficient decrease condition.
• Random sparse linear programs are generated with varying dimensions $ m, n $, and density $ d $, using MATLAB's sparse random matrix generation and problem-specific vector construction.

Experimental results

Research questions

• RQ1 Does the augmented Lagrangian method with generalized Newton iteration outperform LIPSOL in solving large-scale linear programs?
• RQ2 Can the proposed algorithm solve large-scale linear programs that cause out-of-memory errors in MATLAB's linprog on Windows 7?
• RQ3 How does the Linux operating system compare to Windows 7 in terms of memory efficiency and computational performance for solving large-scale linear programming problems?
• RQ4 To what extent does the choice of operating system affect the feasibility and accuracy of solving large-scale LP problems using interior-point and augmented Lagrangian methods?

Key findings

• The augmented Lagrangian method with generalized Newton iteration and Armijo line search successfully solved all tested large-scale problems, including those that caused out-of-memory errors in MATLAB's linprog on Windows 7.
• On Windows 7, linprog failed with 'out of memory' errors for problems with $ m=10000, n=150000 $ and $ m=100, n=10^7 $, while the proposed method completed successfully.
• On Linux, the proposed method solved all problems, including the largest ones ($ m=10000, n=150000 $) in 4215.10 seconds, whereas linprog failed on the same problems.
• For the problem $ m=10000, n=15000 $, the proposed method took 3243.50 seconds on Linux, while linprog failed due to memory limits.
• The proposed method achieved high accuracy: feasibility gaps $ \|Ax^*-b\|_\infty \leq 1.12 \times 10^{-8} $, dual infeasibility $ \|(A^T u^* - c)_+\|_\infty \leq 3.13 \times 10^{-13} $, and duality gap $ |c^T x^* - b^T u^*| \leq 1.2 \times 10^{-7} $.
• Linux demonstrated superior memory management and computational stability, solving problems that were infeasible on Windows 7, even when the same algorithm was used.