Piera Mazzoleni

Generalized Convexity and Fractional Programming with Economic Applications

I. Generalized Convexity.- to generalized convexity.- Structural developments of concavity properties.- Projectively-convex models in economics.- Convex directional derivatives in optimization.- Differentiable (? , ?)-concave functions.- On the bicriteria maximization problem.- II. Fractional Programming.- Fractional programming - some recent results.- Recent results in disjunctive linear fractional programming.- An interval-type algorithm for generalized fractional programming.- A modified Kelleys cutting plane algorithm for some special nonconvex problems.- Equivalence and parametric analysis in linear fractional programming.- Linear fractional and bicriteria linear fractional programs.- III. Duality and Conjugation.- Generalized conjugation and related topics.- On strongly convex and paraconvex dualities.- Generalized convexity and fractional optimization.- Duality in multiobjective fractional programming.- An approach to Lagrangian duality in vector optimization.- Rubinstein Duality Scheme for Vector Optimization.- IV. Applications of Generalized Convexity in Management Science and Economics.- Generalized convexity in economics: some examples.- Log-Convexity and Global Portfolio Immunization.- Improved analysis of the generalized convexity of a function in portfolio theory.- On some fractional programming models occurring in minimum-risk problems.- Quasi convex lower level problem and applications in two level optimization.- Problems of convex analysis in economic dynamical models.- Recent bounds in coding using programming techniques.- Logical aspects concerning Shephards axioms of production theory.- Contributing Authors.

About Derivatives of some Generalized Concave Functions

Abstract The definitions of (α, λ)-concave and (G, h)-concave functions, introduced in a previous work, are reconsidered. The first one is simplified by symmetrizing λ. A theorem characterizing (α, λ)-concave differentiable functions is given. Moreover it is shown that it is possible to define a new directional derivative so that the usual properties of classical concave functions still hold. A similar approach is also developed for (G, h)-concave functions: the new directional derivative can be expressed in two equivalent ways and can also be used to define a sort of directional derivative for quasi concave functions intended as a limiting case of (G, h)-concave functions.

Generalized connectedness for families of Arcs

In this paper the topological property of are wise connectedness is generalized both for sets and functions: a family of “modified secants” is defined, not referring to particular kinds of functions, quadratic, logarithmic, exponential, etc., but fixing the extreme configurations of the generalized means. Such a theory is developed by proving properties which generalize convexity and quasi-convexity in a topological rather than a purely algebraic framework. A new separation theorem is given and a theory of the alternative is stated for the new functions introduced in order to unify the convex and quasi-convex results.

Order preserving functions and generalized convexity

In order to better understand the monotonicity properties which characterize the gradient of pseudoconvex and quasi convex funzions, psedomonotonicity and quasi monotonicity can be introduced.A quite different approach is proposed in this paper, by defining new order relations, whose preservation leads precisely to several kinds of pseudoconvexity and quasi convexity.Some general properties of order preservation are proved; they are useful to state necessary and sufficient conditions of monotonicity for particular orders related with generalized convexity.RiassuntoPer meglio intendere le proprietà di monotonia che caratterizzano i gradienti delle funzioni pseudoconvesse e quasi convesse, si sono introdotti i concetti di pseudomonotonia e di quasi monotonia.In questo lavoro si propone un approccio diverso definendo alcune relazioni di preordine parziale la cui preservazione conduce appunto ai vari tipi di pseudoconvessità e di quasi convessità.Si mostrano alcune proprietà generali della preservazione di un ordinamento che si rivelano utili per formulare condizioni necessarie e condizioni sufficienti di monotonia per ordinamenti particolari legati alla convessità generalizzata.

Towards a unified type of concavity

Abstract Starting from the pioneering definition of concavity, many extensions have been suggested in the literature, some of them based on the first and second order approximations. They are mainly linked to practical problems: see, for instance, the case of utility and production functions, risk analysis and, in mathematical programming problems, the ones related to fractional and geometric programming. In our opinion it is very worth looking for the basic idea underlying these properties and a tool may be represented by generalized means as in Ben-Tal (1977). In that paper most of the attention has been given to analyze the properties of (G, h)- functions. Our aim is devoted to look for a unifying tool so that most of the known definitions can be considered as a continuous development from the classical case to the quasi concave one. In this effort we propose a new definition showing that it preserves the most important properties gradually going to more general cases.

Proprietà degli operatori monotoni per problemi di ottimizzazione vettoriale

AbstractIn questa nota si assegnano condizioni per la monotonia della funzione di ottimo e si dimostrano alcune relazioni fra problemi reciproci, sia nel caso in cui si considera un problema di ottimizzazione scalare vincolata, che in quello in cui si introduce una funzione obiettivo vettoriale, in presenza di uno o piu vincoli. Si formulano infine alcune applicazioni economiche.AbstractFirst of all we consider a scalar problem withk inequality constraints max {f(x):gi(x)≤bi,i=1, ...,k, x∈D} and analyze the conditions under which the optimal solution is not strictly dominated with respect to the constraint operatorG(x)=[g1(x),...,gk(x)] in the sense of the vector programming. Furthermore we link optimal solutions for “reciprocal” problems such as

Consensus Measures for Qualitative Order Relations

\phi (b)\underline{\underline {der}} \max \{ f(x):G(x) \leqslant b, x \in D\} and min \{ G(x):f(x) = \phi (b), x \in D\}

Verso un unico tipo di concavità

and min {G(x):f(x)=ϕ(b), x∈D}. Since the minimization problem is a vector one we pass to study monotonicity properties for the optimum-value correspondence individuated by a vector optimization problem with one or more constraints in order to establish on one hand that the constraint is active at the solution point, on the other hand that the optimal solution is not strictly dominated with respect to the constraint operator.These results allow us to give an interesting interpretation to some economic problems.