This article introduces a method for local sensitivity analysis of practical interest. A theorem is given that provides a general and neat manner to obtain all sensitivities of a general nonlinear programming problem (around a local minimum) with respect to any parameter irrespective of it being a right-hand side, objective function or constraint constant. The method is based on the well-known duality property of mathematical programming, which states that the partial derivatives of the primal objective function with respect to the constraints' right-hand side parameters are the optimal values of the dual problem variables. For the parameters or data for which sensitivities are sought to appear on the right-hand side, they are converted into artificial variables and set to their actual values, thus obtaining the desired constraints. If the problem is degenerated and partial derivatives do not exist, the method also permits obtaining the right, left, and also directional derivatives, if they exist. In addition to its general applicability, the method is also computationally inexpensive because the necessary information becomes available without extra calculations. Moreover, analytical relations among sensitivities, locally valid, are straightforwardly obtained. It is also shown how the roles of the objective function and any of the active constraints (equality or inequality) can be exchanged leading to equivalent optimization problems. This permits obtaining the sensitivities of any constraint with respect to the parameters without the need of repeating the calculations. The method is illustrated by its application to two examples, one degenerated and the other one of a competitive market.

In this paper we discuss the power of a pivoting transformation introduced by Castillo, Cobo, Jubete, and Pruneda [Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations and Inequalities, and Linear Programming, John Wiley, New York, 1999] and its multiple applications. The meaning of each sequential tableau appearing during the pivoting process is interpreted. It is shown that each tableau of the process corresponds to the inverse of a row modified matrix and contains the generators of the linear subspace orthogonal to a set of vectors and its complement. This transformation, which is based on the orthogonality concept, allows us to solve many problems of linear algebra, such as calculating the inverse and the determinant of a matrix, updating the inverse or the determinant of a matrix after changing a row (column), determining the rank of a matrix, determining whether or not a set of vectors is linearly independent, obtaining the intersection of tw...