Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization

Reviewer: Michael Minkoff

The authors provide an excellent algorithmic description of the software known as L-BFGS-B, an extension of a well-known limited-memory BFGS algorithm and software (due to Liu and Nocedal), L-BFGS. Bound constraints are often not treated thoroughly, yet the effective handling of simple bounds requires addressing most of the issues that arise in general linear constraints. The treatment of constrained optimization, even in the bound constrained case, provides an important practical tool for the application of optimization techniques to a wide range of application areas, such as simulation and optimization. Thus, this software provides a much-needed quality tool. Additionally, the authors provide theoretical, implementation, and benchmarking treatments of the software development process. After the problem statement, a summary of related algorithms clarifies the relative position of the limited-memory BFGS within the literature. The paper also relates the unconstrained algorithm to the simple bound version of the algorithm. The paper presents three drivers for the software, which appear to be well designed. A thorough presentation of termination criteria follows. Implementation aspects include details regarding the bound algorithm (its predecessor unconstrained algorithm). The use of a line search is then presented. In order to obtain high accuracy in nonlinear optimization, attention in adapting algorithms to treat optimization techniques in the presence of rounding errors is necessary. The authors address this issue and summarize implementation aspects. The implementation section also includes a treatment of machine constants and scale dependency, including limitations on scale dependency. The paper concludes with numerical results. The testing presented is quite thorough (something I do not always see). A comparison with the LANCELOT package is presented, and the results are analyzed. The paper concludes with issues for future study.