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Abstract. This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush-Kuhn-Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn.

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Quinoa is a highly diverse crop domesticated in the Andean region of South America with broad adaptation to a wide range of marginal environments. Quinoa has garnered interest worldwide due to its nutritional and health benefits. Over the last decade, quinoa production has expanded outside of the Andean region, prompting multiple studies investigating the potential for quinoa cultivation in novel environments. Currently, quinoa is grown in countries spanning five continents, including North America, Europe, Asia, Africa, and Oceania. Here, we update the advances of quinoa research in Ecuador across different topics, including (a) current quinoa production situation with a focus on breeding progress, (b) traditional seed production, and (c) the impact of the work of the nongovernment organization “European Committee for Training and Agriculture” with quinoa farmers in Chimborazo province. Additionally, we discuss genetic diversity, primary pests and diseases, actions for adapting quinoa to tropical areas, and recent innovations in quinoa processing in Ecuador. Finally, we report a case study describing a participatory breeding project between Washington State University and the Association of Andean Seed and Nutritional Food Producers Mushuk Yuyay in the province of Cañar.

Quinoa (Chenopodium quinoa Willd.) is a pseudocereal originated from the Andes important for small farmers' food security as well as for commercial production. Recently, it has been claimed that in Bolivia genetic erosion could result from the marginalization of the crop in the north and from its commercial standardization in the south. The aim of this study was to quantify the hierarchical structure of the genetic variation present in eight quinoa field populations, consisting of cultivated and weedy individuals, representative of the altiplano and interandean valleys of Bolivia. Randomly amplified polymorphic DNA markers show that quinoa has a strong population structure and a high intra-population variation. An effect of geographical structure of the populations was highlighted, due to population isolation, not simply linked to distance but more probably to climatic and orographic barriers present in the studied zone. The population structure is also reinforced by the limited seed exchanges among farmers as revealed by field interviews. This population structure appears related to three major biogeographic zones: the northern and central altiplano, the interandean valley, and the southern Salar. Intrapopulation genetic diversity was higher than that expected for a mainly autogamous species, and higher than that reported in anterior studies based on germplasm collections. These results are commented in view of current knowledge on phylogeny and reproductive biology of the species, and their implications regarding genetic resources management are discussed.

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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 two linear subspaces, solving systems of linear equations, etc. When the process is applied to inverting a matrix and calculating its determinant, not only is the inverse of the final matrix obtained, but also the inverses and the determinants of all its block main diagonal matrices, all without extra computations.