Application-oriented Mixed Integer Non-Linear Programming

In the most recent years there is a renovate interest for Mixed Integer Non-Linear Programming (MINLP) problems. This can be explained for different reasons: (i) the performance of solvers handling non-linear constraints was largely improved; (ii) the awareness that most of the applications from the real-world can be modeled as an MINLP problem; (iii) the challenging nature of this very general class of problems. It is well-known that MINLP problems are NP-hard because they are the generalization of MILP problems, which are NP-hard themselves. However, MINLPs are, in general, also hard to solve in practice. We address to non-convex MINLPs, i.e. having non-convex continuous relaxations: the presence of non-convexities in the model makes these problems usually even harder to solve. The aim of this Ph.D. thesis is to give a flavor of different possible approaches that one can study to attack MINLP problems with non-convexities, with a special attention to real-world problems. In Part 1 of the thesis we introduce the problem and present three special cases of general MINLPs and the most common methods used to solve them. These techniques play a fundamental role in the resolution of general MINLP problems. Then we describe algorithms addressing general MINLPs. Parts 2 and 3 contain the main contributions of the Ph.D. thesis. In particular, in Part 2 four different methods aimed at solving different classes of MINLP problems are presented. Part 3 of the thesis is devoted to real-world applications: two different problems and approaches to MINLPs are presented, namely Scheduling and Unit Commitment for Hydro-Plants and Water Network Design problems. The results show that each of these different methods has advantages and disadvantages. Thus, typically the method to be adopted to solve a real-world problem should be tailored on the characteristics, structure and size of the problem. Part 4 of the thesis consists of a brief review on tools commonly used for general MINLP problems, constituted an integral part of the development of this Ph.D. thesis (especially the use and development of open-source software). We present the main characteristics of solvers for each special case of MINLP.

Low codimensional intrinsic regular submanifolds in the Heisenberg group H^n

The thesis mainly concerns the study of intrinsically regular submanifolds of low codimension in the Heisenberg group H^n, called H-regular surfaces of low codimension, from the point of view of geometric measure theory. We consider an H-regular surface of H^n of codimension k, with k between 1 and n, parametrized by a uniformly intrinsically differentiable map acting between two homogeneous complementary subgroups of H^n, with target subgroup horizontal of dimension k. In particular the considered submanifold is the intrinsic graph of the parametrization. We extend various results of Ambrosio, Serra Cassano and Vittone, available for the case when k = 1. We prove that the uniform intrinsic differentiability of the parametrizing map is equivalent to the existence and continuity of its intrinsic differential, to the local existence of a suitable approximating family of Euclidean regular maps, and, when the domain and the codomain of the map are orthogonal, to the existence and continuity of suitably defined intrinsic partial derivatives of the function. Successively, we present a series of area formulas, proved in collaboration with V. Magnani. They allow to compute the (2n+2−k)-dimensional spherical Hausdorff measure and the (2n+2−k)-dimensional centered Hausdorff measure of the parametrized H-regular surface, with respect to any homogeneous distance fixed on H^n. Furthermore, we focus on (G,M)-regular sets of G, where G and M are two arbitrary Carnot groups. Suitable implicit function theorems ensure the local existence of an intrinsic parametrization of such a set, at any of its points. We prove that it is uniformly intrinsically differentiable. Finally, we prove a coarea-type inequality for a continuously Pansu differentiable function acting between two Carnot groups endowed with homogeneous distances. We assume that the level sets of the function are uniformly lower Ahlfors regular and that the Pansu differential is everywhere surjective.

L'inclusione lavorativa tra accesso e partecipazione

La Convenzione delle Nazioni Unite sui Diritti delle Persone con Disabilità (UNCRPD) riconosce il diritto di tutte le persone al lavoro “gli Stati Parti adottano misure adeguate a garantire alle persone con disabilità, su base di uguaglianza con gli altri, l’accesso all’ambiente fisico, ai trasporti, all’informazione e alla comunicazione, compresi i sistemi e le tecnologie di informazione e comunicazione e ad altre attrezzature e servizi aperti o forniti al pubblico”(United Nation 2016 p.14). Nonostante i progressi (in ambito politico culturale) che si stanno compiendo in ambito internazionale in termini di pari opportunità e di inclusione, le persone in situazione di disabilità continuano a incontrare barriere che limitano la loro partecipazione attiva al mondo del lavoro. A partire da questo scenario, la ricerca si propone di indagare i bisogni (es. di accoglienza, di accesso al contesto fisico e digitale, di partecipazione nella vita dell’azienda ecc.) delle persone con disabilità e di sviluppare una applicazione digitale (web app), rivolta alle imprese, finalizzata a monitorare e a promuovere l'inclusione lavorativa. Ripercorrendo il modello di progettazione del design thinking e valorizzando un processo di ricerca basato su metodi misti (qualitativi e qualitativi) è stato ideato Job inclusion for all; un ambiente digitale fondato sull’adattamento di due strumenti di “metariflessione”: l’Index for inclusion job version e l’employment role mapping. Lo strumento digitale prototipato è stato testato e validato, durante l’ultimo anno di ricerca, da parte di una equipe multidisciplinare internazionale; tale processo ha consentito di raccogliere feedback (rispetto alla rilevanza e alla chiarezza degli item, rispetto ai punti di forza e di debolezza) che hanno consentito di migliorare e implementare la versione finale del prototipo di web app.