^{1}

^{*}

^{2}

We propose a new scalarization method which consists in constructing, for a given multiobjective optimization problem, a single scalarization function, whose global minimum points are exactly vector critical points of the original problem. This equivalence holds globally and enables one to use global optimization algorithms (for example, classical genetic algorithms with “roulette wheel” selection) to produce multiple solutions of the multiobjective problem. In this article we prove the mentioned equivalence and show that, if the ordering cone is polyhedral and the function being optimized is piecewise differentiable, then computing the values of a scalarization function reduces to solving a quadratic programming problem. We also present some preliminary numerical results pertaining to this new method.

Scalarization is one of the most commonly used methods of solving multiobjective optimization problems. It consists in replacing the original multiobjective problem by a scalar optimization problem, or a family of scalar optimization problems, which is, in a certain sense, equivalent to the original problem. The existing scalarization methods can be divided into two groups:

1) Methods that use some representation of a given multiobjective problem as a parametrized family of scalar optimization problems. Such scalarization methods should have the following two properties (see [

2) Methods that use local equivalence of a multiobjective optimization problem and some scalar optimization problem whose formulation depends on a given point. Such equivalence enables one to solve the multiobjective problem locally by using necessary and/or sufficient optimality conditions formulated for the scalar problem (for examples of such an approach, see [

There are also scalarization approaches which combine properties of both groups such as the Pascoletti-Serafini scalarization [

In this paper, we propose a new scalarization method different from the above-mentioned ones. It consists in constructing, for a given multiobjective optimization problem, a single scalarization function, whose global minimum points are exactly vector critical points in the sense of [

So far, the term “scalarization function” has been used for a scalar-valued function defined on the image space of an optimization problem, which transforms a vector-valued objective function into a scalar-valued one (see [

The purpose of this research is to describe the idea of our new scalarization method and to present some underlying theory for the case of an unconstrained multiobjective optimization problem. The extension to constrained optimization is also possible and will be the subject of further investigations.

Let ^{+} the positive polar cone to C, i.e.,

where

for all

Definition 1 [

where

We will denote by

Definition 2 Let Ω be an open subset of

(i) A function

(ii) A function ^{1}-function if for every ^{1}-functions

(iii) Let ^{1}-function and let

Proposition 3 ( [^{1}-function with C^{1} selection functions

Definition 4 [

(i)

where

(ii)

(iii)

(iv)

It is obvious that implications

Definition 5 [

Remark 6 If B is a base of the nontrivial convex cone C, then

Lemma 7 (a finite-dimensional version of [

is a compact base for

In the sequel, we consider a fixed vector

Lemma 8 A point

Proof. If

so that (13) holds. Conversely, if (14) is true for some

For a nonempty subset S of

where

Note that

Theorem 9 A point

Proof. If

Having defined the scalarization function s, we can now replace problem (3) by the following scalar optimization problem:

Obviously, problems (3) and (17) are not equivalent because there may exist vector critical points which are not (weakly) efficient solutions for (3). Nevertheless, by solving problem (17) we can obtain some approximation of the set of solutions to (3).

Computing the distance function in (16) is not easy in the general case, but under additional assumptions on both C and f, it is possible to apply some existing algorithms to perform this task. The details are described below.

Definition 10 ( [

A convex cone which is a polyhedral set is called a polyhedral cone.

Theorem 11 Suppose that the ordering cone C in ^{1}. Let

Proof. It follows from ( [

where

In particular, if D is bounded, then no

By assumption, C is polyhedral, hence, by [

Theorem 11 reduces the problem of computing the values

For two objectives, under differentiability assumptions, it is possible to find some representation of the scalarization function s in terms of the gradients _{i} at x (i = 1, 2). Then (4) implies

The following theorem will help to compute the scalarization function (16) for bi-objective problems.

Theorem 12 Let p = 2,

Proof. It follows from (8) that B is a subset of some line in

We now consider the case of classical Pareto optimization, i.e., when

is a compact base for

hence, the scalarization function has the form

For any point

(i)

(ii)

We now consider case (ii). The line L passing through

Using the same parametrization, we can represent the line segment S as follows:

Therefore, if

Taking into account the definitions of

Example 13 (problem FON in [

The authors of [

Here the set

We have designed a program in Maple to compute

There are no other points at which

This example shows that one must be careful when using global optimization algorithms to minimize s because points like the ones appearing in (29) can be easily misclassified as vector critical points.

We have presented a new scalarization method for solving multiobjective optimization problems which is based on computing the Euclidean distance from the origin to some subset determined by the generalized Jacobian of the mapping being optimized. This article contains the main underlying theory and only some preliminary numerical computations pertaining to this method. More numerical results will be presented in another research.

The authors are grateful to an anonymous referee for his/her comments which have improved the quality of the paper.

Rahmo, E.-D. and Studniarski, M. (2017) A New Global Scalarization Method for Multiobjective Optimization with an Arbitrary Ordering Cone. Applied Mathematics, 8, 154-163. https://doi.org/10.4236/am.2017.82013