IOE 519: NLP, Winter 2012 c Marina A. Epelman 55. 8 Barrier Methods for Constrained Optimization. In this subsection, we will restrict our attention to instances of constrained problem (P) that have. inequality constraints only: S = {x 2 Rn : gi(x) ï£¿ 0,i=1,...,n}. The algorithm we develop can be easily extended to problems that also have linear equality con-.