Massimo Pappalardo

Nash equilibria, variational inequalities, and dynamical systems

In this paper, we introduce some relationships between Nash equilibria, variational equilibria, and dynamical equilibria for noncooperative games.

Regularity conditions for constrained extremum problems via image space

Exploiting the image-space approach, we give an overview of regularity conditions. A notion of regularity for the image of a constrained extremum problem is given. The relationship between image regularity and other concepts is also discussed. It turns out that image regularity is among the weakest conditions for the existence of normal Lagrange multipliers.

A new solution method for equilibrium problems

A globally convergent algorithm for equilibrium problems with differentiable bifunctions is proposed. The algorithm is based on descent directions of a suitable family of gap functions. The novelty of the approach is that assumptions which guarantee that the stationary points of the gap functions are global optima are not required.

Regularity conditions in vector optimization

By exploiting very recent results concerning linear separation between a set and a convex cone, the image space approach allows us to achieve new regularity conditions for vector minimum problems. In particular, we obtain very general Mangasarian–Fromovitz type conditions.

Stability for equilibrium problems: from variational inequalities to dynamical systems

We study the connections between solutions of variational inequalities and equilibrium points of a generalized dynamical system. Furthermore, we analyze some stability questions arising in this field.

Image space approach to penalty methods

In this paper, we introduce a unified framework for the study of penalty concepts by means of the separation functions in the image space (see Ref. 1). Moreover, we establish new results concerning a correspondence between the solutions of the constrained problem and the limit points of the unconstrained minima. Finally, we analyze some known classes of penalty functions and some known classical results about penalization, and we show that they can be derived from our results directly.

First order cone approximations and necessary optimality conditions

In [5] it has been developed a general scheme which gives an unifying treatment for the use of generalized first-order cone approximations in establishing necessary optimality conditions. In this paper we go insight in this study and we show how to deduce new and more general necessary optimality conditions both in the case of constraints of abstract type or of inequality type. A first proposal for applying these results to duality theory is presented

On regularity for generalized systems and applications

In the field of constrained optimization the nonvacuity or the boundedness of the generalized Lagrange multiplier set is guaranteed under some regularity conditions (or constraint qualification; the difference in the terminology consisting of whether or not the condition involves the objective function). This type of analysis is now-a-days well stated also for nondifferenti able optimization. Moreover, the great development of these topics has been enforced with the recent results that establish strict relationships between regularity conditions (as well as metric regularity) and calmness, exact penalization and stability. The nature of all these conditions is of analytical type. On the other hand, a new approach has been recently proposed for establishing regularity conditions. It mainly exploits geometrical tools and takes into account that regularity conditions for optimality can be expressed as geometrical conditions for certain types of separation or more generally they are conditions which guarantee the impossibility of a system. This paper aims to give a characterization of regularity conditions for generalized systems and to apply it to the study of optimality conditions.

Existence results for nonconvex equilibrium problems

In this paper, we establish sufficient conditions for the existence of solutions of equilibrium problems in a metric space, that do not involve any convexity assumption either for the domain or for the function. To prove these results, a weak notion of semicontinuity is considered. Furthermore, some existence results for systems of equilibrium problems are provided.

Local second{order approximations and applications in optimization

A new general abstract scheme for local second-order approximations and second-order generalized directional derivatives is presented. Applications to optimization are provided.