Chebyshev and euclidean projections of point on linear manifold

Results of research of properties and interrelations of Chebyshev and Euclidean projections of the origin on linear manifold are considered in the article. Many problems of applied mathematics can be presented in the such view. They are problems of linear approximations, problems of search solutions of balance models closed to the given infeasible solutions, search of pseudosolutions of the models with inconsistent conditions. Euclidean projections are corresponded to application of the least square method. Chebyshev projections are corresponded to minimization of a maximal deviation. We developed and theoretical justified algorithm of searching of Chebyshev projections. The algorithm gives single-valued result and allows to dispense without the difficult verified and sometimes violated Haar condition. The algorithm is based on using of lexicographic optimization. The relative interior point of set of optimal solutions is found on each stage of lexicographic optimization. The property of producing of relative interior points is the main property of algorithms of interior point method. The sets of Chebyshev and Euclidean projections of the origin on linear manifold are formed by way of varying of positive coefficients corresponding to components of vectors in Chebyshev and Euclidean norms. We justified that closure of these sets are equal with the set of vectors of the linear manifold with Pareto-efficient absolute meanings of the components. Consequently, any Chebyshev and Euclidean projection can be get with any required accuracy through choosing the weight coefficients. It was also proved any Euclidean projection with any set of positive weight coefficients in Euclidian norm can be get for the account of choosing the weight coefficients in the form of Chebyshev projection.