J Optim Theory Appl

DOI 10.1007/s10957-015-0744-6

Existence of Optimal Points Via Improvement Sets

Maurizio Chicco1 · Anna Rossi1

Received: 9 October 2014 / Accepted: 20 April 2015 © Springer Science+Business Media New York 2015

Abstract According to the recent definition of efficiency via improvement sets, the aim of this paper is to characterize the set of optimal points for a set. New existence results are proved in multicriteria situations, and their novelty is illustrated via several examples. Moreover, the study of an economic model is provided as an example of application of our achievements.

Keywords Vector optimization · Improvement sets · Optimal points ·

Existence results

Mathematics Subject Classification 90C26 · 90C29 1 Introduction

Most real-life problems are subject to decisions that must be taken according to appropriate optimality criteria. The original concepts of proper efficiency, proposed by

Pareto in a pioneering paper, and approximate proper efficiency were later on modified and formulated in a more general framework by many authors.

The unification of the various definitions of efficiency has recently received a special attention in the literature on vector optimization. To our knowledge, the most general definition is reported in [1]. From the economic point of view, the research of such

B Anna Rossi rossi@dime.unige.it

Maurizio Chicco chicco@dime.unige.it 1 DIME, Scuola Politecnica, Università di Genova, Piazzale Kennedy, pad. D, 16129 Genoa, Italy 123

J Optim Theory Appl points is very general since the preference relation can be given on a Banach space and determined by a utility function or/and on a topological vector space by a order set, which is not necessarily a cone. For other details about mathematical economics, see [2] and references therein. When a convex cone is given, by specializing the order set, it is possible to recover the classical definition of efficient, weak efficient, strict efficient, and Henig proper efficient solutions (see [3]). There exist other attempts to unify the definition of efficiency. One of these is due to Ha, who in [4] introduced the notion of minimal point of a set with respect to a proper open (not necessarily convex) cone with apex at the origin. We point out that the definition given by Ha preceded the one by Flores-Bazàn and Hernàndez [1], but it can be deduced from the latter.

Another attempt to unify was carried out first in [5], where the concept of efficiency for maximization in a finite dimensional setting is presented, then it has been generalized to a real locally convex Hausdorff topological vector space by Gutiérrez et al. in [6]. It is based on special sets called improvement sets, which have two properties: the exclusion property and the comprehensive property. We observe that the definition of efficiency, given in [5], follows from the one quoted in [1] with suitable choices of the function and of the order set.

In order to formulate a definition that unifies the concepts of efficiency, in most of the papers mentioned so far, it is possible to find optimality conditions for vector optimization problems, carried out via scalarizations and via approximate subdifferentials in the convex case or via the Mordukhovich subdifferential, when nonconvexity is assumed (see e.g., [1,3,7,8] and references therein), and existence results for efficient points via topological properties of the sets (see e.g., [5,9,10]).

Inspired by such results, in this paper, we study the existence of efficient points via topological properties of the sets, currently in a finite dimensional setting. To emphasize the importance of the topic, we quote the recent paper [11], that in this framework applies the equilibria concept to study multicriteria games.

The outline of this article is as follows. In Sect. 2, we recall the definition of improvement set and some preliminary results. Section 3 introduces the characterization of the improvement sets ”close” to zero. In Sect. 4, we present some conditions (necessary, sufficient or both) for the existence of optimal points, giving several examples to illustrate our results and their novelty. Moreover, we provide an example, which shows that our results can be applied to an economic model. Section 5 provides an answer to a question proposed by Zhao and Yang in [12]. Finally, some research topics and the main conclusions are presented in Sects. 6 and 7, respectively. 2 Definitions and Preliminary Results

We write x = (x1, . . . , xn) ∈ Rn and denote by ei the vector of Rn with the i-th component equal to 1 and the others 0, so that, for every x ∈ Rn with components x1, x2, . . . , xn , it turns out x = ∑ni=1 xi ei . In case we have a sequence {xm}m∈N ⊂ Rn , we denote by (xm)i the i-th component of the vector xm , in such a way that xm = n∑ i=1 (xm)i ei (m ∈ N). 123

J Optim Theory Appl

By Rn+, we mean the points in Rn with all coordinates positive or null, by Rn++ those with all coordinates strictly positive (analogously for Rn− and Rn−−) and, given x, y ∈ Rn , for every i = 1, . . . , n: x y ⇔ xi ≥ yi ; x > y ⇔ xi > yi (analogous definitions for , <).

For A ⊂ Rn : int (A) and bd(A) are the interior and the boundary of A, respectively; cl(A) is the closure of A; d(x, A) = inf{d(x, a) : a ∈ A}, where d(x, a) is the usual distance in Rn , i.e., d(x, a) := {∑ni=1(xi − ai )2}1/2.

We say that V ⊂ Rn is upper-bounded iff there exists b ∈ Rn such that x b ∀ x ∈ V .

In the following, we will write 〈a, b〉, a, b ∈ Rn , to indicate the internal (scalar) product of two vectors, i.e., 〈a, b〉 := ∑ni=1 ai bi .

Definition 2.1 Let A ⊂ Rn . We define the upper comprehensive set of A as follows: u-compr(A) := {x ∈ Rn : there exists a ∈ A s.t. a x}

It is easy to prove:

Proposition 2.1 u-compr(A) = ⋃a∈A ( a + Rn+ ) .

Definition 2.2 A subset E of Rn is called ”upper comprehensive set” iff u-compr(E) = E .

Definition 2.3 Let E ⊂ Rn \ {0} be an upper comprehensive set. We shall call E an improvement set of Rn , and we will denote by n the family of the improvement sets in Rn .