A Graph Model Proposal for Convex Non Linear Separable Problems with Linear Constraints

Abstract—The solution of non linear problems are based on very well grounded mathematical theories. This document proposes a Newton step graph-based model for convex non linear separable problems (NLP) with linear constraints. The Newton step is well suited for this kind of problems, but when the problem size grows the NLSP model will grow in a non linear manner. Furthermore, the constraints handling becomes the main problem as we have to select the right constraints in the different solution steps. When this happens, the sparse matrix representation is the path to follow, but very little has been made in order to fully explode the sparsity structure. Indeed, the Hessian matrix for the NLSP model has a very particular structure which can be exploited by using the graph underlying the problem, this is the approach taken in this document. To this end a graph is built derived from the components involved in the Newton step. which describes the solution for the NLSP problem. Based on this graph, the gradient can be evaluated directly based on the graph topology, as it will be shown, the information needed for such evaluation is embedded within the graph. Furthermore, the explicit graph model derived from the mathematical model allows us to think about it in terms of its structure which will be used in further works.