L1 optimization under linear inequality constraints

An algorithmic approach is developed for the problem of L-1 optimization under linear inequality constraints based upon iteratively reweighted iterative projection (or IRIP). IRIP is compared to a linear programming (LP) strategy for L-1 minimization (Spath 1987, Chapter 5.3) using the ultrametric condition as an exemplar class of constraints to be fitted. Coded for general constraints, the LP approach proves to be faster. Both methods, however, suffer from a serious limitation in being unable to process reasonably-sized data sets because of storage requirements for the constraints. When the simplicity of vector projections is used to allow IRIP to be coded for specific (in this case, ultrametric) constraints, we obtain a fast and efficient algorithm capable of handling large data sets. It is also possible to extend IRIP to operate as a heuristic search strategy that simultaneously identifies both a reasonable set of constraints to impose and the optimally-estimated parameters satisfying these constraints. A few noteworthy characteristics of L-1 optimal ultrametrics are discussed, including other strategies for reformulating the ultrametric optimization problem.