Continuous Optimization
Improvement sets and vector optimization

https://doi.org/10.1016/j.ejor.2012.05.050Get rights and content

Abstract

In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize well-known results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems.

Highlights

► Nondominated solutions of convex/nonconvex vector optimization problems are studied. ► The preference relation is defined via an improvement set E. ► The improvement sets are characterized via sublevel sets of scalar functions. ► E-optimality conditions are obtained through scalarization processes.

Introduction

A current research line related with vector optimization problems is to develop concepts and settings in order to unify the more important solution notions of these problems, like the so-called efficiency, proper efficiency, weak efficiency, strict efficiency, strong efficiency, and ε-efficiency (see [1], [4], [6], [12], [13], [14], [15], [22]).

Probably, the first attempt to unify classical solution concepts in vector optimization problems via a more general solution notion that collapses all of them was due to Mordukhovich (see [22]) through the notion of generalized order optimality of vector optimization problems. This notion works when the objective space is normed, it was motivated by the well-known set extremality concept due to Kruger and Mordukhovich (see [18]), extends the concepts of Slater minimizer, efficient solution and weak efficient solution (see [27]) and it has been used to obtain necessary optimality conditions of vector optimization problems (see [22]).

Recently, in the framework of a normed space or a Hausdorff locally convex topological linear space Y, several authors have introduced and studied concepts of nondominated point of a set based on binary relations defined as usual, but from a general ordering set Θ  Y:y1,y2Y,y1Θy2y2-y1Θ.

This ordering set could not be convex neither a cone and so the relation ⩽Θ could not be transitive. Analogously, it could happen that 0  Θ and then ⩽Θ could not be reflexive. Both facts are interesting from a practical point of view, since preference relations that are not necessarily a preorder are usual in Economics (see [19], [24] and the references therein). Sometimes, the closedness of the ordering set is not assumed either, and so certain important preference relations like the lexicographical order can be dealt with. Moreover, some ordering sets could not be solid, i.e., the topological interior of Θ could be empty, in order to deal with objective spaces as ℓp and Lp, 1  p < ∞, where the natural ordering is given by a non-solid convex cone.

To compensate these generalizations, other mild conditions are assumed on the ordering set. For example, in [16], [24] the authors considered nondominated point notions via conic ordering sets, and in [12], [13], [14], [15] we introduced and studied a concept of nondominated point based on a coradiant ordering set (αΘ  Θ, for all α > 1), that generalizes the more important exact and approximate concepts of nondominated point in linear spaces.

To our knowledge, the more general approaches in this line are due to Bao and Mordukhovich [1] and Flores–Bazán and Hernández [6], since they consider the usual notion of nondominated point without imposing any condition on the ordering set Θ. These notions reduce to the concepts of ideal, efficient, weak efficient, relative efficient, quasi relative efficient and Henig proper efficient point among others.

However, in order to obtain optimality conditions in vector optimization problems, some conditions on the ordering set are assumed. To be exact, Flores–Bazán and Hernández suppose that zero is in the boundary of Θ and there exists a point q  Y⧹{0} such that (cl Θ and int Θ denote the closure and the interior of Θ, respectively)clΘ+(0,)qintΘ,

which implies that the ordering set Θ is solid (see [6]), and Bao and Mordukhovich assume the so-called local closedness property (see [1, Definition 3.3]), which is compatible with non-solid ordering sets.

A similar contribution has recently been introduced by Chicco et al. in [4]. Specifically, by considering the finite dimensional setting Y=Rp and via an improvement set E (i.e., 0  E and E+R+p=E, where R+p is the non-negative orthant), these authors have defined the notion of E-optimal point (nondominated with respect to the binary relation (1) given by Θ = E), that collapses the concepts of Pareto minimality and weak Pareto minimality, and that is useful to deal with approximate minimal points too. This paper focus on this notion.

Essentially, in a non-necessarily finite dimensional ordered linear space, an improvement set E is a free disposal set, i.e., its cone expansion E + K is itself, where K is the ordering cone. This kind of sets was introduced by Debreu in [5] and they have frequently been used in mathematical economics and optimization. So, one can find in the literature several previous concepts very close to the notion of improvement set.

For example, in the setting of ordered linear spaces where the ordering relation (1) is defined by a convex cone Θ, the concepts of downward set and upward set (see [20], [26]) are the same as the notion of improvement set, but relaxing the condition 0  E. To be precise, an improvement set E is an upward set such that 0  E.

On the other hand, downward sets can be considered as a certain analog of normal subsets of the non-negative orthant (see [20], [26]). Moreover, let us observe that in mathematical economics and game theory, the upward sets are also called free disposal sets and the convex downward sets are known as comprehensive sets.

This work has two objectives. The first one is to study the notion of improvement set by Chicco et al. [4] in the setting of an arbitrary quasi ordered linear space. In particular, we are interested in characterizing these sets in terms of sublevel sets of scalar functions. The second one is to state necessary and sufficient conditions for E-optimal solutions of vector optimization problems via scalarization. In the whole paper it is implicit that a lot of ideas and techniques of the works [12], [13], [14], [15] can be used to obtain results about E-optimal points and E-optimal solutions.

This work is structured as follows. In Section 2, some notations are fixed, several well-known concepts and mathematical tools are recalled and the notion of improvement set by Chicco et al. [4] is extended to a linear space ordered via a convex cone as usual. This notion is showed to be very close to the concepts of free disposal set, downward set, upward set and coradiant set and several general classes of improvement sets are introduced. Moreover, the relation between an improvement set and the ordering cone K is clarified.

In Section 3, the improvement sets are characterized. First, by the separation theorem, closed and convex improvement sets are showed to be such that the positive polar cone of their recession cone is included into the positive polar cone of the order cone. Second, nonconvex closed (resp. open) improvement sets are characterized through sublevel sets of decreasing lower semicontinuous (resp. continuous) functionals. The main mathematical tool for proving these last results is the well-known nonconvex separation functional by Tammer and Weidner [8], [10].

In Section 4, the notion of nondominated solution of a vector optimization problem with respect to an ordering set given by an improvement set is defined. It is showed that this notion collapses the more popular exact and approximate solution concepts of a vector optimization problem. Moreover, it is characterized through approximate solutions of associated scalar optimization problems without considering any convexity assumption. These scalarization results are obtained in two steps. First, by considering generic scalar functionals whose sublevel set at zero is the improvement set. Second, by applying these generic characterizations to some specific scalarization functionals.

In Section 5, other characterizations for this kind of nondominated solutions via linear scalarization processes are obtained in K-convexlike vector optimization problems, i.e., in problems where the objective function is closely K-convexlike and the improvement set is convex. As a consequence, some necessary and sufficient approximate optimality conditions via linear scalarizations obtained by ourselves in [12] for K-convexlike Pareto optimization problems are improved and extended to vector optimization problems. Finally, in Section 6, some conclusions are presented which summarize this work.

Section snippets

Notations and preliminaries

Let Y be a Hausdorff locally convex topological linear space and K  Y be a convex cone (we consider that 0  K), which is assumed to be proper, i.e., {0}  K  Y. In the sequel, we suppose that Y is ordered through the following quasi order:y1,y2Y,y1Ky2y2-y1K.

Let R¯R{±}. Given a functional φ:YR¯, we denoteepiφ={(y,r)Y×R:φ(y)r}.

Moreover, φ is said to be ⩽K-increasing (resp. ⩽K-decreasing) on a nonempty set M  Y if φ(y)  φ(y + d) (resp. φ(y)  φ(y + d)), for all y  M, d  K. We say that φ is ⩽K

Characterization of ⩽K-i.s.

Next we characterize the notion of ⩽K-i.s. by means of the recession cone of E and also by using ⩽K-decreasing functionals.

We denote by M the so-called recession cone of a nonempty set M  Y (see [9], [23]), i.e.,M={uY:M+R++uM}.It is well-known that M is a convex cone for any set M (see, for instance, [9], [23]). Moreover, if M is convex, then M = {u  Y:y0 + tu  M, ∀ t > 0}, where y0 is an arbitrary point of M, and M is free disposal if and only if K  M.

In the sequel we obtain a necessary and

E-optimality and nonlinear scalarization

Consider the following vector optimization problem:Min{f(x):xS},where f:X  Y, X is an arbitrary decision space and the feasible set S  X is nonempty. We say that (7) is a Pareto problem if Y=Rp and K=R+p.

From now on we assume that E  Y is ⩽K-i.s. By translating the E-optimality notion by Chicco et al. (see [4, Definition 3.1]) to problem (7), the following optimality notion is defined.

Definition 4.1

A point x0  S is said to be an E-optimal (resp. weak E-optimal) solution of problem (7), denoted x0  Op(f, S; E)

E-optimality and linear scalarization

Recall that a function f:X  Y is closely K-convexlike on S if cl(f(S) + K) is a convex set (see [7]). For λ  Y, we denoteσE(λ)=infeEλ(e).Let us observe that if λ  E+, then σE(λ)  0.

Theorem 5.1

Assume that f is closely K-convexlike on S, and E is a solid convex ⩽K-i.s. ThenWOp(f,S;E)λE+{0}argminS(λf,σE(λ)).

Proof

Choose x0  WOp(f, S; E), then(f(S)-f(x0))(-intE)=.From here, it follows that(f(S)+K-f(x0))(-intE)=.Indeed, if statement (11) was false, then there exist x  S, d  K and e  int E such that f(x) + d  f(x0) = e.

Conclusions

In this work we have studied nondominated points of a set and nondominated solutions of vector optimization problems where the ordering relation is given by an improvement set in the framework of linear spaces.

This kind of minimality has been shown to be very suitable to deal in a unified way with well-known exact and approximate nondominated concepts. Moreover, it works with very general ordering sets (non-necessarily cones neither pointed, convex, solid or closed sets) and so it can be used

Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments and suggestions.

References (27)

  • J.B.G. Frenk et al.

    On classes of generalized convex functions, Gordan–Farkas type theorems, and Lagrangian duality

    Journal of Optimization Theory and Applications

    (1999)
  • C. Gerth et al.

    Nonconvex separation theorems and some applications in vector optimization

    Journal of Optimization Theory and Applications

    (1990)
  • G. Giorgi et al.

    Mathematics of Optimization: Smooth and Nonsmooth Case

    (2004)
  • This research was partially supported by the Ministerio de Ciencia e Innovación (Spain) under project MTM2009-09493.

    View full text