Generating Linear programming Instances with Controllable Rank and Condition Number

TL;DR

This paper introduces a flexible framework for generating linear programming instances with controllable properties like rank and condition number, aiding algorithm testing and development.

Contribution

It presents a novel matrix decomposition-based method for generating instances with preset optimal solutions and explores neighborhood operators to create diverse, complex LP instances.

Findings

Generated instances have controllable rank and condition number.

Neighborhood operators increase instance complexity.

Framework covers the entire feasible and bounded case space.

Abstract

Instances generation is crucial for linear programming algorithms, which is necessary either to find the optimal pivot rules by training learning method or to evaluate and verify corresponding algorithms. This study proposes a general framework for designing linear programming instances based on the preset optimal solution. First, we give a constraint matrix generation method with controllable condition number and rank from the perspective of matrix decomposition. Based on the preset optimal solution, the bounded feasible linear programming instance is generated with the right-hand side and objective coefficient satisfying the original and dual feasibility. In addition, we provide three kind of neighborhood exchange operators and prove that instances generated under this method can fill the whole feasible and bounded case space of linear programming. We experimentally validate that the proposed schedule can generate more controllable linear programming instances, while neighborhood exchange operator can construct more complex instances.