Decision rule-based method for multiparametric linear programming

Multiparametric linear programming involves the solution of linear programming problems with parametric uncertainty. The optimal exact decisions and cost values are defined, in each region, as affine functions of the uncertain parameters. The solution, developed off-line, provides an attractive alternative for conventional on-line solution methods. Nevertheless, exact methods face significant challenges in solving problems with large number of decision variables and uncertain parameters. As the solution time and the required memory resources grow to levels that exceed practical or manageable limits, their reliability for large-scale applications comes into question. In this work, we propose a novel method to approximate the solution of multiparametric linear programming problems, under right-hand uncertainty, using a decision rule-based method. We introduce a decision rule leveraging the rectified linear units (ReLUs) found in artificial neural networks. The computational cost of the approximation technique is shown to be significantly less than that of the exact parametric solution with a reduction of one to two orders of magnitude. Notably, the memory resource required by the approximate solution is significantly less. For a given instance, the exact method required 519.9 GB of memory estimate, not accessible by common computing machines, to process the complete solution, while the approximate solution required less than 16 GB. We present a branching algorithm to enhance the approximation quality for specific subregions of the uncertainty space. We empirically show that our branching algorithm yields exponential increases in the solution quality at the expense of only a linear increase in the computational cost.