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  1. Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone
  2. Submission » DergiPark
  3. Ordered Cones and Approximation (Lecture Notes in Mathematics)

The book is addressed to researchers in functional analysis and approximation theory as well as to those that want to applythese methods in other fields. It is largely self- contained, but the readershould have a solid background in abstract functional analysis. The unified approach is based on a new notion of locally convex ordered cones that are not embeddable in vector spaces but allow Hahn-Banach type separation and extension theorems.

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Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone

Klaus Keimel. Publisher: Springer Verlag , This specific ISBN edition is currently not available. Learn more. 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 which no convexity hypotheses are required.

Moreover, the construction of these dilating cones allows us to obtain scalarization results that are easier to handle, and attractive from a computational point 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. 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.

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Submission » DergiPark

In the setting of Pareto multiobjective optimization i. Shortly after that, Benson gave a concept of proper efficiency that extends the notion of Geoffrion and implies the one given by Borwein. Another remarkable concept of proper efficiency is due to Henig , 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. 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. 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, On the other hand, the definition of the ordering cone in terms of a matrix leads to characterizations 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 ordering cone through the Tammer—Iwanow functional and a type of dilating cones introduced by Kaliszewski see, for instance, Kaliszewski, These cones are defined by means of a perturbation 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, ; Jahn, ; Kaliszewski, The paper is structured as follows. In Section 2. In Section 3. 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.


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Finally, in Section 4. In this paper, the notation refers to two sets such that A is included or equal to B. Given a nonempty set , we denote by , and the topological interior, the closure, the boundary, and the cone generated by F , respectively. The nonnegative orthant of is denoted by. In the sequel, we consider an order in defined by an ordering cone in the usual way, that is, We suppose that D is pointed and polyhedral, that is, 1 where the set of matrices with p rows and n columns and. Let us note that by definition D is convex and closed.

Thus, D induces a partial order in. Along this paper, we consider the following multiobjective optimization problem: where is the objective function , and is the feasible set. Frequently, the feasible set is defined by a cone constraint, that is, in the following way: 2 where , and is a convex cone. Whenever we consider that S is defined as in 2 , we suppose that K is solid and polyhedral, that is, , with. We say that is a Pareto problem when. A point is an efficient solution of problem if there is not such that. We denote the set of efficient solutions by. Moreover, in the case when D is solid, it is said that a feasible point x 0 is a weak efficient solution of if there is not such that.

The set of weak efficient solutions of will be denoted by. From now on, we assume that D is solid whenever we consider this type of solutions. Let us observe that 3. On the other hand, when , it is said that is a Geoffrion proper efficient solution of , denoted by , if it is efficient and there exists such that for all and for all satisfying that there exists such that and.

From that definition, we see that the set contains the efficient solutions for which an improvement in one objective function implies a considerable worsening in another objective function. The common idea of these concepts, introduced by Benson and Henig , respectively, is to replace one of the sets that take part in 3 by another bigger one, in order to obtain a more restrictive notion of efficiency and to avoid, in this way, anomalous efficient solutions of.

A point is a Benson proper efficient solution of problem , and we denote it by , if On the other hand, a feasible point x 0 is a Henig proper efficient solution of , and we denote it by , if there exists a solid and convex cone , with , such that and. Remark 2. Let us observe that the concept of Henig proper efficient solution can be redefined in the following equivalent way: a point is a proper efficient solution of problem in the sense of Henig if there exists a solid and convex cone , with , such that and. Moreover, see Benson, , Theorem 3. Given a function , we denote by.

The cone was introduced by Kaliszewski see Kaliszewski, It is clear that is convex and closed, and for all. Also, observe that. The following result provides an equivalent formulation for Henig proper efficient solutions in terms of efficient solutions with respect to the cones.


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It was proved in Theorem 3. Theorem 2. It follows that if and only if there exists such that.

Ordered Cones and Approximation (Lecture Notes in Mathematics)

From Theorem 2. Let and. The nonlinear functional defined as was introduced by Gerstewitz Tammer and Iwanow in Gerstewitz and Iwanow We are going to start the next section with a characterization of the weak efficient solutions of problem with respect to the cones , for , by using the functional.

For this aim, let us note that when we are assuming that is solid. Because of that, we can use the following notation: The next lemma will be needed in the following section. In this lemma, we determine explicitly the functional , for and. Lemma 2. It follows that 5 where. By definition, , for all. We have that if and only if 6 where , and I is the identity matrix in. Thus, statement 6 is equivalent to 7 On the other hand, since , it follows that and then 7 is equivalent to Therefore, and formula 5 is proved. By formula 5 or applying directly Corollary 2.


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Moreover, for all it follows that 8. In this section, we are going to characterize proper efficient solutions of problem through scalarization. For this aim, first, we need the following result, in which we establish a characterization of weak efficient solutions of problem with respect to the cones. Let us observe that in this result no convexity assumptions are needed. Theorem 3.

The cone spectral radius

Let , and. Then, if and only if 9 where.

Approximation - 2nd Application of Derivatives

We know that if and only if and by 8 this last statement is equivalent to recall that we denote 10 Let defined by. Thus, 10 is equivalent to , for all , that is, attains its minimum value on S at x 0 and this minimum value is 0. Therefore, by Lemma 2. Remark 3. Let us note that 9 is equivalent to say that Let us observe that Equation 9 involves , since it is derived via the nonlinear functional.

However, the characterization stated in Theorem 3. Corollary 3. We have the following equality:. Reciprocally, let and 11 Suppose by contradiction that.