On Regularity for Constrained Extremum Problems

The fundamental necessary optimality criterion of nonlinear mathematical programming is the Kuhn-Tucker stationary point optimality criterion. Both for the Kuhn-Tucker optimality criterion and for the saddle point sufficient optimality criterion there is no guarantee that the multiplier associated to the objective function is strictly positive. If the multiplier is equal to zero, then the respective condition has been achieved without the contribution of the objective function. Even if this degenerate case is mathematically correct, this fact means perhaps that there is a gap between the mathematical model and the real problem represented by it. A condition ensuring that the multiplier is strictly positive involves only the constraints it is called constraint qualification(CQ); it is called regularity condition if it involves also the objective function. This thesis deals with regularity conditions ensuring the existence of Lagrange multipliers both for finite-dimensional and infinite-dimensional problems; the approach used is that of the Image Space approach, i.e. the space where the values of the functions run. For the finite-dimensional problems a new condition for linear separation based on Caratheodory's Theorem and having a Helly like form is given. This condition makes possible a subtle treatment of the distinction between various constraint qualifications as well as an exact characterisation of the existence of a saddle point for the Lagrangian function. Comparison with calmness and metric regularity are performed. In a framework of C-differentiabiltiy, which encompasses the separate cases of classical differentiability and the subdifferentiability of convex analysis, we give a regularity condition necessary and sufficient for the existence of regular Lagrange multipliers is provided in terms of the nonintersection of two special sets in the Image Space. Comparison with calmness, metric regularity, Slater CQ, Mangasarian-Fromowitz CQ, Guignard CQ, Basic CQ, Abadie CQ together with a new CCQ condition are elucidated. Examples and graphical representations, both for sufficient and for necessary conditions, demonstrate all the implications found. For infinite-dimensional problems two approaches are analysed. First, for a classic geodesic type problem of Calculus of Variations, a selection approach is adopted in which, by means of a selection multifunction, an equivalent finite-dimensional problem is introduced. The tools of the finite-dimensional are used to obtain necessary and sufficient optimality conditions. It is shown that the classic Euler Equation is equivalent to the regular separation of two sets in the Image Space. Following the classical approach, convexity is needed in order to obtain strong duality for a general infinite-dimensional problem. The regular separation of two special sets in the infinite-dimensional Image Space is proved to be equivalent to the existence of Lagrange multipliers. Exploiting the quasirelative interior of convex sets, a new Slater type CQ is given