Verified Branch and Bound for Singular Linear and Nonlinear Programs – An Epsilon-Inflation Process

Linear and nonlinear programs are often ill-posed, or approximately ill-posed. However, current commercial software often is constructed with implicit assumptions of well-posedness built in. The result is that such software returns just one of many possible solutions, without any indication that singularity or approximate singularity exists. This is illustrated by applying several commercial packages, including a commercial global optimization package with complete search, to a simple example problem. However, on that same problem, a global optimization package that does not include nonsingularity assumptions does return the entire solution set, but very inefficiently. An algorithm is therefore proposed for efficiently depicting such singular solution sets and for efficiently but rigorously verifying that all such solution sets have been enclosed. This algorithm is implemented, and computations with it are illustrated.