Marco Castellani

A Dual Representation for Proper Positively Homogeneous Functions

In this paper we show that any proper positively homogeneous function annihilating at the origin is a pointwise minimum of sublinear functions (MSL function). By means of a generalized Gordans theorem for inequality systems with MSL functions, we present an application to a locally Lipschitz extremum problem without constraint qualifications.

Second Order Optimality Conditions for Differentiable Multiobjective Problems

A second order optimality condition for multiobjective optimization with a set constraint is developed; this condition is expressed as the impossibility of nonhomogeneous linear systems. When the constraint is given in terms of inequalities and equalities, it can be turned into a John type multipliers rule, using a nonhomogeneous Motzkin Theorem of the Alternative. Adding weak second order regularity assumptions, Karush, Kuhn-Tucker type conditions are therefore deduced.

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.

K-epiderivatives for set-valued functions and optimization

Exploiting different tangent cones, many derivatives for set-valued functions have been introduced and considered to study optimality. The main goal of the paper is to address a general concept of K-epiderivative and to employ it to develop a quite general scheme for necesary optimality conditions in set-valued problems.

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.

Refinements of existence results for relaxed quasimonotone equilibrium problems

We consider a general equilibrium problem in a normed vector space setting and we establish sufficient conditions for the existence of solutions in compact and non compact cases. Our approach is based on the concept of upper sign property for bifunctions, which turns out to be a very weak assumption for equilibrium problems. In the framework of variational inequalities, this notion coincides with the upper sign continuity for a set-valued operator introduced by Hadjisavvas. More in general, it allows to strengthen a number of existence results for the class of relaxed