Nonlinear Extensions of Farkas' Lemma with Applications to Global Optimization and Least Squares

A nonlinear extension of Farkas' lemma for systems involving the difference of sublinear functions is presented. This extension of Farkas' lemma is applied to give a complete characterization of global optimality for constrained global optimization problems in which the objective function is the difference of a convex and sublinear function and the constraints are systems of difference sublinear functions. An application to certain fractional programming problems is also given. These results are achieved with merely a stronger consistency condition which reduces to the usual feasibility requirement for problems with sublinear constraints. Further generalizations of Farkas' lemma for systems involving convex and difference sublinear functions are also presented. Moreover, a generalized Farkas' lemma for certain specially structural convex inequality systems is shown to be related to the solution of appropriate constrained least squares problems.