A quadratic penalty algorithm for linear programming and its application to linearizations of quadratic assignment problems

TL;DR

This paper analyzes a quadratic penalty algorithm designed to efficiently approximate solutions to linear programming problems, especially those arising from linearizations of quadratic assignment problems, combining augmented Lagrangian and quadratic penalty methods.

Contribution

It provides the first detailed documentation and analysis of a penalty-based technique for obtaining approximate LP solutions before applying the simplex method, with insights into its application to quadratic assignment problem linearizations.

Findings

The algorithm effectively produces approximate LP solutions.

It combines augmented Lagrangian and quadratic penalty approaches.

Potential for fast approximate solutions in quadratic assignment problems.

Abstract

This paper provides the first meaningful documentation and analysis of an established technique which aims to obtain an approximate solution to linear programming problems prior to applying the primal simplex method. The underlying algorithm is a penalty method with naive approximate minimization in each iteration. During initial iterations an approach similar to augmented Lagrangian is used. Later the technique corresponds closely to a classical quadratic penalty method. There is also a discussion of the extent to which it can be used to obtain fast approximate solutions of LP problems, in particular when applied to linearizations of quadratic assignment problems.