Regularization and normal solutions of systems of linear equations and inequalities
The paper provides unconstrained dual formulations of unconstrained optimization problems which arise from the regularization of the problems of solving a system of linear equations, a system of linear equations with nonnegativity constraints on the variables, and a system of linear inequalities. It is first shown that the regularized problem for a system of linear equations is equivalent to some constrained optimization problems as well as to a special least squares problem. By means of a theorem of the alternative the latter version of the problem allows the set-up of a dual problem which, in turn, can be equivalently stated as an unconstrained optimization problem. Then similar problems are shown to be dual ones for a system of linear equations with nonnegative variables and a system of linear inequalities. For each of these pairs of mutually dual problems, the solution of one of the problems can be easily computed from the solution of the other one so that the lower-dimensional problem of both may be solved in practice.
system of linear equations with nonnegativity constraints
scientific article; zbMATH DE number 6474028