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Alberto ZAFFARONI

Professore Ordinario
Dipartimento di Economia "Marco Biagi"

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Pubblicazioni

2022 - Separation, convexity and polarity in the space of normlinear functions [Articolo su rivista]
Zaffaroni, A.
abstract

The paper studies separation properties for subsets of the space (Formula presented.) of normlinear functions on the Banach space X, i.e. the sum of a linear function and a multiple of the norm. We use as separating functionals the ones that are generated by the duality pairing (Formula presented.). We show that normlinear functions can be used to separate points from radiant or coradiant sets in X. Then we pass to separation defined by means of the pairing (Formula presented.), seen as a function defined on (Formula presented.). Since in this case the evaluation functionals are linear, separation describes a special subclass of convex sets in (Formula presented.). We characterize X-convex sets by exploiting the isomorphism between (Formula presented.) and the space (Formula presented.). We also study polarity relations defined on (Formula presented.). Polar sets are X-convex. And we describe what further properties are needed in order to make a subset (Formula presented.) the polar, or the reverse polar, of some set in X. For polar sets (Formula presented.) it is possible to deduce the unique closed, radiant prepolar set (Formula presented.). To conclude, we emphasize some connections between upward (downward) X-convex sets in Y and sets of upper (resp. lower) bounds in Y.

2021 - Nonlinear dynamics in asset pricing: the role of a sentiment index [Articolo su rivista]
Campisi, G.; Muzzioli, S.; Zaffaroni, A.
abstract

This paper aims to contribute to the literature on the role of sentiment indices in heterogeneous asset pricing models. A new sentiment index in financial markets is proposed in which transactions take place between two groups of fundamentalists with divergent perceptions of fundamental value. It is assumed that the proportion of fundamentalists in the two groups depends on the sentiment index. After examining the analytical properties of the deterministic discrete dynamical system, stochastic components are added to the expectations of fundamentalists. First, the study measures the performance of the model in reproducing the stylized facts of financial data relying on the S&P 500 index. Second, the forecasting power of the model to predict the daily prices of the S&P 500 index is examined. For this purpose, the forecasting accuracy of the proposed dynamical model, where the sentiment index is explicitly modelled, is compared with a model where the sentiment index is not taken into account. In this case, the predictions are obtained by means of a machine learning technique (lasso regression). The results show that the sentiment index is important in explaining the stylized facts of financial returns and in forecasting prices.

2018 - New representations of sets of lower bounds and of sets with supremum in Archimedean order-unit vector spaces [Working paper]
Zaffaroni, A.; Ernst, E.
abstract

2017 - Characterizing Sets of Lower Bounds: a Hidden Convexity Result [Articolo su rivista]
Ernst, Emil; Zaffaroni, Alberto
abstract

This study addresses sets of lower bounds in a vector space ordered by a convex cone. It is easy to see that every set of lower bounds is downward (lower?), bounded from above, with the further property that it contains the supremum of any of its subsets which admits one. Our main result proves that these conditions are also sufficient, if the ordering cone is polyhedral. We provide other characterizations and properties of sets of lower bounds in primal and dual terms and show by means of simple counterexamples that such results fail when the polyhedrality assumption is dropped.

2016 - Characterizing sets of lower bounds: a hidden convexity result [Working paper]
Ernst, E.; Zaffaroni, A.
abstract

This study addresses sets of lower bounds in a vector space ordered by a convex cone. It is easy to see that every set of lower bounds must be simultaneously downward and bounded from above, and must possesses the further property that it contains the supremum of any of its subsets which admits one. Our main result proves that these conditions are also sufficient, provided that the ordering cone is polyhedral. Simple counter-examples prove that the sufficiency fails when the polyhedrality assumption is dropped

2013 - Convex Radiant Costarshaped Sets and the Least Sublinear Gauge [Articolo su rivista]
Zaffaroni, Alberto
abstract

The paper studies convex radiant sets (i.e. containing the origin) of a linear normed space X and their representation by means of a gauge. By gauge of a convex radiant set C we mean a sublinear function p such that C=[p< 1]. We characterize the class of convex radiant sets which admit a gauge different from the Minkowski gauge in two different ways: they are contained in a translate of their recession cone or, equivalently, they are costarshaped, that is complement of a starshaped set. We prove that the family of all sublinear gauges of a convex radiant set admits a least element and characterize its support set in terms of polar sets. The key concept for this study is the outer kernel of C, that is the kernel (in the sense of Starshaped Analysis) of the complement of C. We also devote some attention to the relation between costarshaped and hyperbolic convex sets.

2012 - Convex radiant costarshaped sets and the least sublinear gauge [Working paper]
Zaffaroni, A
abstract

The paper studies convex radiant sets (i.e. containing the origin) of a linear normed space X and their representation by means of a gauge. By gauge of a convex radiant set C ⊆ X we mean a sublinear function p : X → R such that C = [p ≤ 1]. Besides the most important instance, namely the Minkowski gauge µC(x) = inf{λ &gt; 0 : x ∈ λC}, the set C may have other gauges, which are necessarily lower than µC. We characterize the class of convex radiant sets which admit a gauge different from µC in two different way: they are contained in a translate of their recession cone or, equivalently, they are costarshaped, that is complement of a starshaped set. We prove that the family of all sublinear gauges of a convex radiant set admits a least element and characterize its support set in terms of polar sets. The key concept for this study is the outer kernel of C, that is the kernel (in the sense of Starshaped Analysis) of the complement of C. We also devote some attention to the relation between costarshaped and hyperbolic convex sets.

2010 - Conically equivalent convex sets and applications [Articolo su rivista]
Elisa, Caprari; Zaffaroni, Alberto
abstract

Given a normed space X and a cone K in X, two closed, convex sets A and B in X* are said to be K-equivalent if the support functions of A and B coincide on K. We characterize the greatest set in an equivalence class, analyze the equivalence between two sets, find conditions for the existence and the uniqueness of a minimal set, extending previous results. We give some applications to the study of gauges of convex radiant sets and of cogauges of convex coradiant sets. Moreover we study the minimality of a second order hypodifferential.

2008 - Convex coradiant sets with a continuous concave cogauge [Articolo su rivista]
Zaffaroni, Alberto
abstract

The paper studies convex coradiant sets and their cogauges. While the concave gauge of a convex coradiant set is superlinear but discontinuous and its Minkowski cogauge is (possibly) continuous but is not concave, we are interested in those convex coradiant sets which admit a continuous concave cogauge. These sets are characterized in primal terms using their outer kernel and in dual terms using their reverse polar set. It is shown that a continuous concave cogauge, if it exists, is not unique; we prove that the class of continuous concave cogauges of some set C admits a greatest element and characterize its support set as the intersection of the reverse polar of C and the polar of its outer kernel.

2008 - Robust Optimization of Conditional Value at Risk and Portfolio Selection [Articolo su rivista]
A. G., Quaranta; Zaffaroni, Alberto
abstract

This paper deals with a Portfolio Selection model in which the methodologies of Robust Optimization are used for the minimization of the Conditional Value at Risk of a portfolio of shares. Conditional Value at Risk, being in essence the mean shortfall at a specified confidence level, is a coherent risk measure which can hold account of the so called "tail risk" and is therefore an efficient and synthetic risk measure, which can overcome the drawbacks of the most famous and largely used VaR. An important feature of our approach consists in the use of techniques of Robust Optimization to deal with uncertainty, in place of Stochastic Programming as proposed by Rockafellar and Uryasev. Moreover we succeeded in obtaining a linear robust copy of the bi-criteria minimization model proposed by Rockafellar and Uryasev. We suggest different approaches for the generation of input data, with special attention to the estimation of expected returns and finally implement the model as a Linear Program. The relevance of out methodology is illustrated by a portfolio selection experiment on the italian market.

2007 - Superlinear separation for radiant and coradiant sets [Articolo su rivista]
Zaffaroni, Alberto
abstract

The paper studies radiant and coradiant sets of some normed space X from the point of view of separation properties between a set A of X and a point x not included in A; indeed they show striking similarities with the ones holding for convex sets and can be obtained by simply changing halfspaces (level sets of linear continuous functions), with level sets of continuous superlinear functions. In a geometric perspective we can say that radiant sets are separated by means of convex coradiant sets and coradiant sets are separated by means of convex radiant sets. The identification between the geometric and the analytic approach passes through the well-known Minkoski gauge and the study of concave continuous gauges of convex coradiant sets. The results are then applied to the study of abstract convexity with respect to the family L of continuous superlinear functions, to the characterization of evenly coradiant convex sets and to the subdifferentiability of positively homogeneous functions.

2006 - Monotonicity along rays and consumer duality with nonconvex preferences [Relazione in Atti di Convegno]
Zaffaroni, Alberto
abstract

The goal of the paper is to prove a duality relation between the direct and indirect utility function without any reference to convexity of preferences, nor quasiconcavity of the utility function. To reach this result we allow for the pricing system to be sublinear, rather than linear. The relevance of this assumption has been illustrated both in Consumer Theory, to keep account of the presence of intermediation or of bundling, and in Mathematical Finance. The main tool is a nonlinear separation theory, which uses sublinear functionals to separate points from radiant or coradiant sets. This yields a characterization of the class of functions for which the duality can be proved, namely those whose upper level sets are evenly coradiant. Such functions are nondecreasing along each rays emanating from the origin, a very weak requirement of nonsatiation of preferences, and satisfy a further technical requirement. We underline that this further requirement is always satisfied if u is upper semicontinuous hence, in particular, if u is continuous or differentiable. The conditions that we obtain are necessary and sufficient and consequently they offer the minimal assumption under which a utility function coincides with the dual of the indirect utility.

2006 - Superlinear separation and dual properties of radiant functions [Articolo su rivista]
Zaffaroni, Alberto
abstract

Superlinear functionals are used to separate points from a radiant set according to both a strict and a weak version. Strict separation characterizes closed radiant sets; weak separation is used to define evenly radiant sets, which are characterized by means of a property of the tangent cone to the set at points of the boundary. The separation properties can be described via a polarity relation between a normed space X and the set L of continuous superlinear functionals defined on X. Radiant functions are the ones which are increasing along rays, i.e. the ones whose lower level sets are radiant and so they extend the class of quasiconvex functions with minimum at the origin. We study two particular subclasses: the one of l.s.c. radiant functions, whose lower level sets are closed and radiant and the one of evenly radiant functions, whose lower levels are evenly radiant. We introduce a conjugate function (defined on L), in two different versions, and prove the coincidence between a function and its second conjugate when the function belongs to one of the classes mentioned above. The conjugate function is then used to give global optimality conditions for problems described by radiant objective and constraints.

2004 - IS EVERY RADIANT FUNCTION THE SUM OF QUASICONVEX FUNCTIONS? [Articolo su rivista]
Zaffaroni, Alberto
abstract

An open question in the study of quasiconvex function is the characterization of the class of functions which are sum of quasiconvex functions. In this paper we restrict attention to quasiconvex radiant functions, i.e. those whose level sets are radiant as well as convex and deal with the claim that a function can be expressed as the sum of quasiconvex radiant functions if and only if it is radiant. Our study is carried out in the framework of Abstract Convex Analysis: the main tool is the description of a supremal generator of the set of radiant functions, i.e. a class of elementary functions whose sup-envelope gives radiant functions, and of the relation between the elementary generators of radiant functions and those of quasiconvex radiant functions. An important intermediate result is a nonlinear separation theorem in which a superlinear function is used to separate a point from a closed radiant set.

2003 - Degrees of efficiency and degrees of minimality [Articolo su rivista]
Zaffaroni, Alberto
abstract

In this work we characterize different types of solutions of a vector optimization problem by means of a scalarization procedure.Usually different scalarizing functions are used in order to obtain thevarious solutions of the vector problem. Here we consider different kinds ofsolutions of the \emph{same} scalarized problem. Our results allow us toestablish a parallelism between the solutions of the scalarized problem andthe various efficient frontiers: stronger solution concepts of the scalarproblem correspond to more restrictive notions of efficiency. Besides the usual notions of weakly efficient and efficient points, which are characterized as global (or strict global) solution of the scalarized problem, we also consider some restricted notions of efficiency, such as strict and proper efficiency, which are characterized as Tikhonov well-posed minima and sharp minima for the scalarized problem.

2000 - Continuous approximations, codifferentiable functions and minimization methods [Capitolo/Saggio]
Zaffaroni, Alberto
abstract

Our starting point relies on the observation that, for a nondifferentiable function, the classical directional derivative fails to be continuous with respect to the initial point; this is also related to the lack of continuity properties of the quasidifferential or other differential objects obtained as linearization of the directional derivative. In this paper we describe the notion of codifferentiability as a mean to obtain a continuous approximation for a nonsmooth function. Particular emphasis is given to applications to optimization theory: necessary optimality conditions, minimization methods, extensions of the Newton method for a system of nonsmooth equations.We also describe how the main ideas behind codifferentiability can be extended to mappings between Banach spaces. In the last section we discuss the concept of continuous approximation without linearization and show how the conceptual study of a number of topics in nonsmooth optimization can satisfactorily be treated in this more general setting.

1999 - Continuous approximations of nonsmooth mappings [Capitolo/Saggio]
Rubinov, A.; Zaffaroni, Alberto
abstract

We introduce the concept of approximator, i.e. a first order local approximation of a mapping, which depends on some point and on the direction without any assumptions of homogeneity. This is done in order to overcome the tight connection, shown by the directional derivative, between the continuity with respect to the point and linearity with respect to the direction. We obtain in this way an exact approximation of nonsmooth mappings which depends continuously on the point. Various examples of mappings admitting continuous approximators are given and some known results of classical analysis are extended to this nondifferentiable setting. Moreover we show how the theoretical machinery can be applied in various fields such as fixed point theory, optimization theory, and Newton's methods.

1999 - On the existence of maximal elements for partial preorders [Articolo su rivista]
Carosi, L.; Zaffaroni, Alberto
abstract

The paper presents a general approach to find conditions which ensure the existence of maximal elements in a partially preordered set. We generalize some known results and establish new ones; moreover we show that our conditions extend recent results in the economic literature and in the theory of vector optimization, which hold under more specific assumptions on the topological and algebraic structure of the space.

1998 - Codifferentiable mapings with applications to vector optimization. [Articolo su rivista]
Zaffaroni, Alberto
abstract

Codifferentiable mappings are defined as the ones which can be locally approximated by a particular type of difference convex mappings, adapting an analogous notion recently introduced for scalar functions. Some calculus rules are proved and some applications to vector optimization problems described by codifferentiable criteria and constraints are given.

1998 - Quasiconcavity of sets and connectedness of the efficient frontier in ordered vector spaces [Relazione in Atti di Convegno]
Molho, E.; Zaffaroni, Alberto
abstract

We introduce new notions of quasiconcavity of sets in ordered vector spaces, extending the properties of sets which are images of convex sets by quasiconcave functions. This allows us to generalize known results and obtain new ones on the connectedness of the sets of various types of efficient solutions.

1997 - Vector subdifferentials via recession mappings [Articolo su rivista]
Zaffaroni, Alberto; Glover,
abstract

A vector subdifferential is defined for a class of directionally differentiable mappings between ordered topological vector spaces. The method used to derive the subdifferential is based on the existence of a recession mapping for a positively homogeneous operator. The properties of the recession mapping are discussed and they are shown to be similar to those in the real valued case. In addition a calculus for the vector subdifferential is developed. Finally these results are used to develop first order necessary optimality conditions for a class of vector optimization problems involvingeither proper or weak minimality concepts.

1996 - Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization [Articolo su rivista]
V., Jeyakumar; Zaffaroni, Alberto
abstract

We establish necessary and sufficient dual conditions for weak and proper minimality of infinite dimensional vector convex programming problems without any regularity conditions. The optimality conditions are given in asymptotic form using epigraphs of conjugate function and subdifferentials. It is shown how these asymptotic conditions yield standard Lagrangian conditions under appropriate regularity conditions.The main tool used to obtain these results is a new solvability result of Motzkin type for cone convex systems. We also provide local Lagrangian conditions for certain nonconvex problems using convex approximations.

1994 - ON THE NOTION OF PROPER EFFICIENCY IN VECTOR OPTIMIZATION [Articolo su rivista]
Guerraggio, A.; Molho, E.; Zaffaroni, Alberto
abstract

We consider the main definitions of proper efficiency for a vector optimization problem in topological linear spaces. The implications among these definitions generalize the inclusion structure holding in Euclidean spaces with componentwise ordering.