A new conical internal evolutive LP algorithm

In this paper we develop extensions of the conical techniques exposed in the book [3], and we develop new algorithms as well. We make an effort to keep this paper enough self-contained, however, the book [3] (hereafter often referred to as ”the book” for the sake of brevity) can be useful for further details. Such book is an outgrow of two papers published on the journal Optimization, namely [1] and [4]. To begin with the conical approach has his own nomenclature. Terms like duality have a different meaning than those of standard approaches (see e.g. [2]). Other terms like strict tangency are peculiar to the conical approach. The term bounded refers to a LP problem (maximization of a linear functional on a polyhedron) that is feasible and has maximum. The specification internal for a primal algorithm has also a meaning related to the conical context, and, as explained below, refers to the particular approach followed here to reach optimality, as opposed to the primal algorithm presented in the book. In the book we stressed the particular interest of conical conditions that are at one time purely conical ( that is, involve only cones) and purely pointed (that is, the involved cones are pointed). Dual conical feasibility and optimality conditions are of this kind. Although they have lead in the book to an enumerative algorithm, the dual conical methodology is not only useful for its peculiar properties (e.g. it gives a closed form expression of the maximum,solves a whole class of problems and is particularly efficient in handling parameter variations), but it has provided an useful tool for applications in fields like control and game theory. The related literature is by now very extended and relevant, and has provided many interesting new results (see e.g [5] and the bibliography therein). As to the primal conical conditions given in the book, the first is conical but not pointed. The second involves a pointed cone and an affine space, and has been the starting point for the development an evolutive primal external conical algorithm, enjoying exact finite convergence.