Exterior point simplex-type algorithms for linear and network optimization problems

Two decades of research led to the development of a number of efficient algorithms that can be classified as exterior point simplex‐type. This type of algorithms can cross over the infeasible region of the primal (dual) problem and find an optimal solution reducing the number of iterations needed. The main idea of exterior point simplex‐type algorithms is to compute two paths/flows. Primal (dual) exterior point simplex‐type algorithms compute one path/flow which is basic but not always primal (dual) feasible and the other is primal (dual) feasible but not always basic. The aim of this paper is to explain to the general OR audience, for the first time, the developments in exterior point simplex‐type algorithms for linear and network optimization problems, over the recent years. We also present other approaches that, in a similar way, do not preserve primal or dual feasibility at each iteration such as the monotonic build‐up Simplex algorithms and the criss‐cross methods.