A new view of some fundamental results in optimization (Q2208496)

The authors point out that some of the basic results of the optimization theory can be proved in a simpler way than it is usually done in the literature. They show that for instance separation theorems, Farkas lemma, contraction mapping principle, implicit function theorems can be avoided and present new elementary proofs of the Lagrange theorem in the case of equality constraints and a Kuhn-Tucker theorem. It is proved that the Farkas theorem and the cone closedness theorem follow from each other. Besides, the authors simplify the Farkas condition in the Farkas theorem and formulate a general version of the Farkas theorem. The equivalence of several optimality conditions is proved. A further part of the paper deals with the reducibility of optimization problems with inequality constraints to optimization problems with equality constraints. The last section of the paper studies the solution of a singular Kuhn-Tucker system using the 2-factor Newton method.