Generating linear programming instances with controllable rank and condition number

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.