Well-posed saddle point problems

We provide a new well-posedness concept for saddle-point problems. We characterize it by means of the behavior of the sublevel sets of an associated function. We then study the concave-convex case in Euclidean spaces. Applying these results in the setting of Convex Programming, we get a result on the convergence of the pair solution-Lagrange multiplier of approximating problems to the pair solution-Lagrange multiplier of the limit problem

Nonsmooth Duality, Sandwich and Squeeze Theorems

Given a nonlinear function h separating a convex and a concave function, we provide various conditions under which there exists an affine separating function whose graph is somewhere almost parallel to the graph of h. Such results blend Fenchel duality with a variational principle, and are closely related to the Clarke-Ledyaev mean value inequality. 1 Introduction The central theorems in this paper blend two completely distinct types of result, both fundamental in optimization theory: Fenchel duality and variational principles. The simplest version of Fenchel duality states that for any convex functions f and g on R n satisfying f \Gammag, a regularity condition implies the set L def = fy 2 R n : f (y) + g (\Gammay) 0g is nonempty (where f is the Fenchel conjugate of f ). Geometrically, this means there exists an affine function sandwiched between f and \Gammag. On the other hand, one of the easiest examples of a variational principle states Keywords. Sandwich The..

Nonsmooth duality, sandwich and Squeez theorems

Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Stability of critical points for vector valued functions and Pareto efficiency

In this work we consider the crtical points of a vector-valued functions, as defined by S. Smale. We study their stability in order to obtain a necessary conditions for Pareto efficiency. We point out, by an example, that the classical notions of stability (concerning a single point) are not suitable in this setting. We use a stability notion for sets to prove that the counterimage of a minimal point is stable. This result is based on the study of a dynamical system defined by a differential inclusion. In the vector case this inclusion plays the same role as gradient system in the scalar setting