Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone

• DOI: http://doi.org/10.1111/itor.12398
• Author: Vicente Novo,César Gutiérrez,Lidia Huerga
• Published date: 01 May 2018
• Date: 01 May 2018

Intl. Trans. in Op. Res. 25 (2018) 763–779

DOI: 10.1111/itor.12398

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Nonlinear scalarization in multiobjective optimization with a

polyhedral ordering cone

C´

esar Guti´

erreza, Lidia Huergaband Vicente Novoc

aDepartamento de Matem´

atica Aplicada, E.T.S.de Ingenieros de Telecomunicaci´

on, Universidad de Valladolid, Valladolid,

Spain

bDepartamento de Econom´

ıa, Facultad de Ciencias Sociales y Jur´

ıdicas, Universidad Carlos III de Madrid, Madrid, Spain

cDepartamento de Matem´

atica Aplicada, E.T.S.I.Industriales, Universidad Nacional de Educaci ´

on a Distancia, Madrid,

Spain

E-mail: cesargv@mat.uva.es [Guti´

errez]; lhuerga@eco.uc3m.es [Huerga]; vnovo@ind.uned.es[Novo]

Received 3 July2016; received in revised form 14 December 2016; accepted 18 January 2017

Abstract

In this work, we consider a multiobjective optimization problem in which the ordering cone is assumed to

be polyhedral. In this framework, we characterize proper efficient solutions through nonlinear scalarization

and a kind of polyhedral dilating cones. The main results are based on a characterization of weak efficient

solutions, for whichno convexity hypotheses are required. Moreover, the construction of these dilating cones

allows us to obtain scalarization results thatare easier to handle, and attractive from a computationalpoint of

view,since they are formulated in terms of a perturbation of the matrix that defines the ordering cone.Finally,

when the feasible set is given by a cone constraint, we derive necessary and sufficient optimality conditions

via a kind of scalar nonlinear Lagrangian.

Keywords: multiobjective optimization; proper efficiency; weak efficiency; polyhedral cone; dilating cone; nonlinear

scalarization; nonlinear Lagrangian

1. Introduction

The notions of proper efficient solution have been studied in depth by many researchers in multiob-

jective and vector optimization (see, for instance,Kuhn and Tucker, 1951; Geoffrion,1968; Borwein,

1977; Hartley, 1978; Benson, 1979; Henig, 1982; Choo and Atkins, 1983; Sawaragi et al., 1985; Bor-

wein and Zhuang, 1993; Guerraggio et al., 1994; Kaliszewski, 1994; Zheng, 1997; Miettinen, 1999;

G¨

opfert et al., 2003; Giorgi et al., 2004; Qiu, 2008; Engau, 2015; Khan et al., 2015). These types of

solution are defined with the aim of selecting suitably efficient solutions, in order to avoid, in this

way, efficient solutions with undesirable properties.

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation of OperationalResearch Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

USA.

764 C. Guti´

errez et al. / Intl. Trans.in Op. Res. 25 (2018) 763–779

In the setting of Pareto multiobjective optimization (i.e., when the final space is ordered by

components), the first concept of proper efficiency was introduced by Kuhn and Tucker (1951) and

modified later by Geoffrion (1968). After that, Borwein (1977) defined a notion of proper efficiency

for vector optimization problems (not necessarily finite-dimensional problems), which works for a

general ordering cone. Shortly after that, Benson (1979) gave a concept of proper efficiency that

extends the notion of Geoffrion and implies the one given byBorwein. Another remarkable concept

of proper efficiency is due to Henig (1982), which is more restrictive than that given by Benson

and it is defined in terms of cones that contain the ordering one in their interior. The cited proper

efficiency concepts are the most consolidated. For the study of other concepts see, for instance,

Hartley (1978), Borwein and Zhuang (1993), Zheng (1997), and Guti´

errez et al. (2012).

In multiobjective optimization problems, the notions of proper efficiency in the senses of Henig

and Benson are equivalent whenever the ordering cone is closed and pointed, and they are also

equivalent to the concept given by Geoffrion in the Pareto case (see, for instance, Sawaragi et al.,

1985). In this paper, we pay our attention to the study of multiobjective optimization problems in

which the ordering cone is polyhedral, and we characterize proper efficient solutions in the sense

of Henig (and directly, Benson and Geoffrion) through nonlinear scalarization. For this aim, we

characterize previously the weak efficient solutions of the problem.

Several of the best known nonlinear scalarization techniques employed for solving multiobjective

optimization problems are based on the Tammer–Iwanow functional (see Gerstewitz and Iwanow,

1985; G¨

opfert et al., 2003; Khan et al., 2015), the Pascoletti and Serafini functional (see Pascoletti

and Serafini, 1984; Eichfelder, 2009), or on different norms, for instance, the block norms (in

particular, the Tchebycheff norm) and the oblique norms (see Choo and Atkins, 1983; Jahn, 1984;

Wierzbicki, 1986; Kaliszewski, 1994; Schandl et al., 2002; Tammer and Winkler, 2003). By using

these scalarization techniques, no convexity assumptions are needed. Moreover, let us observe that

the scalarization methods based on norms are usually supported by approximations to the utopia

point or by some reference point (see Skulimowski, 1988).

On the other hand, the definition of the ordering cone in terms of a matrix leads to characteriza-

tions of proper efficient solutions that are attractive from a computational point of view.

In this work, we characterize proper efficient solutions with respect to a general polyhedral or-

dering cone through the Tammer–Iwanow functional and a type of dilating cones introduced by

Kaliszewski (see, for instance, Kaliszewski, 1994). These cones are defined by means of a pertur-

bation of the matrix that defines the ordering cone, so the results are obtained in terms of this

perturbed matrix. Moreover, for this characterization we do not need to assume any boundedness

condition on the feasible set (given in terms of a utopia point), which is usually required in this type

of results (see, for instance, Choo and Atkins, 1983; Jahn, 1984; Kaliszewski, 1994).

The paper is structured as follows. In Section 2, we state the main notations, definitions and some

previous results. In Section 3, we characterize the set of weak efficient solutions of a multiobjective

optimization problemthrough nonlinear scalarization, with respect to the ordering cone and certain

dilating cones, and then by means of this result we derive a characterization of proper efficient

solutions. Analogously, necessary and sufficient conditions are also obtained in the particular case

when the feasible set is given by a cone constraint, and these results can be viewed as a sort of

nonlinear Lagrangian optimality conditions, since they are formulated in terms of the objective and

constraint functions. Finally, in Section 4, we state the conclusions.

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation of OperationalResearch Societies