Hence, we will investigate sufficient conditions for this property Le. Some authors used other terms such as nondominance boundedness Yu [Y2] and domination property Benson [B8], Henig [H9], and Luc [L9] instead of external stability.
Since the external stability is closely related to the nonemptiness of efficient sets, they are discussed together in this section. Conversely, every transitive and asymmetric domination structure is acyclic. The following theorem is the existence result with a general domination structure; namely, D is not assumed to be a constant cone. Proof If we suppose, to the contrary, that t!
However, this contradicts the assumption that D is acyclic. Hence, t! A number of authors extended this condition to multiobjective optimization problems and obtained existence theorems for efficient points e.
We first introduce the cone semicompactness condition by Corley [CI5], which is a slight generalization of the definition by Wagner [WI]. Here r is some index set and the superscript c denotes the complement of a set.
Proof In view of Proposition 3. Hence, it suffices to prove the case in which D is a pointed closed convex cone. Suppose to the contrary that Y is not inductively ordered. However, this contradicts the fact yY E Y. It may be troublesome to check the cone semicompactness condition. Hartley [H5] used the slightly stronger cone compactness condition. Since from the definition of D- compactness, l - cl D n Y is compact, this subfamily has a finite sub- cover, which together with yY - cl D C constitutes a finite subcover of Y.
The proof is completed. However, a D- compact set is not necessarily compact. Some authors used conditions in terms of recession cones of feasible sets Bitran and Magnanti [B14], Henig [H7]. Here we adopt an extended definition of the recession cone by Henig, since Y is not necessarily convex. As in Theorem 3.
Thus Yl E Yt n - Yn, which con- tradicts the hypothesis. I Henig [H7] cf. Case i By taking a subsequence, if necessary, we may assume without loss of generality that akdk Since D is closed, d E D. Then Pk Xk y" Since cl D is a convex cone. Hence Y is D-semicompact. Therefore, from Lemmas 3. Thus, in view of Lemma 3. Hence, from Lemma 3. Proof This theorem can be proved easily by Theorem 3. As can be seen from these examples, the concepts introduced by the various authors are different, in general.
However, they coincide in the convex case, as shown by the following proposition. Proof The equivalence of i and iv is immediate from Proposition 2. From Lemma 3. In view of Proposition 2. Thus we have proved the equivalence of i and ii. Take any y E Y. Then y - V r, Y is compact, and from Proposition 2. From Proposition 2. In the analysis of efficient sets, the V-compactness condition was used by Hartley [H5] and Naccache [Nl], while Henig [H7] used a condition similar to V-boundedness and V-closedness.
Borwein [B16] and Bitran and Magnanti [B14] used conditions i or iv in the above proposition for closed convex Y. Corley [C15] used the V-semicompactness condition. However, as can be seen from the above propositions, they are closely related to one another. Now we can prove that if Y is closed convex, V-compactness is also necessary for 1if Y, V to be nonempty.
Sufficiency has been already estab- lished in Theorem 3. When Y is a polyhedral convex set, the existence condition of efficient solutions which are also properly efficient solutions by Theorem 3. Then c! Proof The results follow immediately from Theorem 3. We conclude this subsection with an existence theorem of efficient solutions in the decision space, which is a quite natural extension of the scalar case.
For this purpose, we introduce an extended semicontinuity concept of functions according to Corley [CIS]. Moreovei fl' Since X is compact, it has a finite subcover whose image by f clearly constitutes a finite subcover of Y. Thus Yis D-semicompact. Then there exists an efficient solution. Proof Immediate from Lemma 3. Proof Immediate from Remark 3. In section 3. Each point outside the efficient set is, therefore, dominated by some other point in the feasible set.
However, is it dominated by a point in the efficient set? If this is the case, the efficient set is said to be externally stable.
In graph theory, another type of stability concept-internal stability-is known. It is clear that the efficient set of Y is internally stable.
Berge [Bt2] calls external and internal stability absorbency and stability, respectively. The reader should not confuse these stability concepts with stability with respect to parameter perturbation, which will be fully discussed in Chapter 4.
A subset of Y is called a kernel of Y with respect to D, denoted by f Y, D , if it is both externally and internally stable. However, this is a contradiction. Hence tB' Y, D is externally stable. Conversely, if tB' Y, D is externally stable, then it becomes a kernel because it is internally stable. As for some properties concerning the kernel, the reader may refer to White [W9, Chapter 2].
We consider in the remaining part of this section conditions for the external stability of the efficient set. Suppose that a domination structure D is transitive and upper semicontinuous as a point-to-set map on Y. Moreover, for each compact subset y' of Y, tS' Y', D is assumed to be nonempty. Then G Y, D is externally stable and so is the kernel of Y. Namely, the set Y' consists of y and all points in Y that dominate y.
Since Y' c Yand Y is bounded, Y' is also bounded. Therefore, in view of Theorem 3. Clearly Y' is closed and nonempty since Yis D-closed. Thus Y' is bounded and, hence, compact. Then, in view of Remark 3. Proof If Y is empty, the result is trivial; we assume, therefore, that Y is nonempty. In view of the lemma by Henig [H7], Lemma 3. Hence, by Theorem 3. Hence, by Lemma 3. In this theorem, the indispensability of the D-boundedness of Y can be understood by Example 3.
Then Yis D-closed but not D-bounded. The indispensability of the D-closedness of Y is illus- trated in the following example. Then Y is D-bounded but not D-closed. In this case, while. The converse of Theorem 3. Then the following are equivalent: i Any of the conditions i - v in Proposition 3.
Some examples of externally unstable sets will be given later in Chapter 4, in connection with the perturbation stability of efficient sets. Throughout this section, the domination structure is given by a constant pointed closed convex cone D. The following results are mainly due to Naccache [Nl]. The following two lemmas are well known and useful in this section.
From the results given later in Section 3. We first prove the connectedness of the set!! Proof Suppose, to the contrary, that!! Then there exist two open sets Y1 and Y2 such that li n!!
These all imply that int DO is not connected, which is a contradiction. Proof We will show that Jr. Suppose, to the contrary, that Jr. We have therefore expressed S Y, D as the union of a family of connected sets having a nonempty intersection and as such S Y, D must be connected. This completes the proof. Warburton [W4] recently obtained connectedness results for the set of Pareto optimal and weak Pareto optimal solutions in the decision space i. The domination structure is assumed to be specified by a convex cone D.
Moreover, when D is a closed convex cone, we use 8I' Y, D to denote the set of all properly efficient points of Y in the sense of Benson. Three kinds of characterization are discussed in sequence.
Some more results not discussed here can be seen in Gearhart [G4] and Jahn [12]. As in Section 3. So we can apply Proposition 2. Namely, for all y E Y. This geometric result is quite useful in the dicussion of multiobjective duality theory Chapter 5. Proof Let pe tt Y, D. From Proposition 3. Applying the separation theorem Theorem 2.
In the above theorem, the acuteness condition of D cannot be eliminated. The following example was given by Yu [Yl]. Proof In view of Proposition 2. Y, D , and the corollary directly follows from Theorems 3. However, we can at least claim the following. Next, assume that Y is nonempty and compact. Then Y E 1ff Y, D. The following theorem, which is a generalization of a theorem of Arrow et al. Proof Theorems 3. Hence, the result follows immediately from Corollary 3. However, we will give a direct proof of the relationship.
Then, for all e sufficiently small, Yand C 6 n B are both nonempty, compact, and convex, where B is the closed unit ball. Since DSO is a cone, this implies that for all f.
Now suppose Y is not necessarily compact but closed and convex. Since Y n B is a nonempty, compact, convex set and o E g y r. When Y is a polyhedral convex set, we obtain the following result. Proof The theorem is obvious from Theorems 3. In other words, we pay attention to the Pareto optimal points. The idea of compromise solutions is based on a procedure that calculates the Pareto optimal points as approximations to an ideal or utopia point Yu and Leitmann [Y3], Zeleny [Z3], Bowman [B20], Gearhart [G3], Salukvadze [S3].
We investigate the relationships between compromise solutions and efficient or properly efficient points partly according to Gearhart [G3]. Therefore, our aim is to find the set of all Pareto optimal points or solutions. This point will serve as the ideal point. L, IX - norm. However, this implies that liP -. This is also a contradiction and therefore 9 E. Thus, we have a contradiction.
Let y a be a best approximation to y out of Y in the 11, a -norm. However, as has been established as Theorem 3. Possibly, the magnitude of a that is required can be related to a measure of the nonconvexity of Y. This fact will be shown later in Theorem 3. In subsection 3. If we use the [I-norm in Example 3. Then, as in the proof of the latter half of Theorem 3.
Therefore Ilyoo -. The families of norms in Example 3. In view of the last part of the proof of Theorem 3. Therefore, we may assume without loss of generality that 1- y for some y.
The kth objective constraint problem is formulated by taking the kth objective functionj, as the objective function and letting all the other objective functionsfj U k be inequality constraints Benson and Morin [BIO], Haimes et al. A similar type of equality-constrained problems was studied by Lin [L6-L8].
The following theorems provide the relationships between the Pareto optimal solutions of the original multiobjective optimization problem P and the solutions of the kth objective constraint problems Pk e. This x x implies that does not solve pk e. Conversely if does not solve Pk e for some k, then there exists x E X such thatf,. Hence x is a Pareto optimal solution of P. As has been already stated, every properly efficient solution in the sense of Geoffrion is a Pareto optimal solution, but not vice versa in general.
Benson and Morin showed that the converse holds when the constraint problems are stable as scalar optimization problems. I' miminize J,. Conversely, suppose that x is a properly efficient solution of P. Furthermore, since x is an optimal solution of Pk,ll with the optimal objective value J,. By Geoffrion [G6], Pk e is stable. Slater's constraint quali- fication is satisfied. See Geoffrion [G6]. Details concerning the constraint problems can be found, for example, in Chankong and Haimes [C6].
The interested reader may refer to it. Results under the convexity assumption are closely related to the duality theory and so will be discussed fully in later chapters. Proof Immediate from Theorem 3. Proof The condition in Theorem 3. Proof Let x be a weak Pareto optimal solution of P. Suppose, to the contrary, that there exists an h satisfying the above inequalities. This contradicts the weak Pareto optimality of X, as was to be proved.
Recent development of nonsmooth analysis is enabling us to obtain a similar type of efficiency condition with nondifferentiable functions. See,for example, Clarke [Cll], Minami [M4]. Several sufficient conditions that ensure the semicontinuity of the solution set maps will be provided along with a series of examples. The following results are well known.
Theorem 4. In order to extend the above results to multiobjective optimization, we will define a multiobjective optimization problem having two kinds of para- meters in this section.
One parameter u, which varies over a set U, specifies the set of possible decisions or feasible solutions and the objective function as before. Another parameter v, which varies over a set V, specifies the domination structure of the decision maker in the objective space.
Here, U and V may be arbitrary subsets of linear topological spaces in which the concept of convergence is defined in terms of sequences. However, in this book, we assume that U and V are subsets of the Euclidean spaces for simplicity.
Now, we can define a parametrized multiobjective optimization problem in the following way. First, the set of all feasible solutions and the vector- valued function, which changes depending on the parameter u are denoted by X u and f x, u , respectively.
Second, the domination structure of the decision maker in the objective space is specified by the parameter v as a point-to-set map D v from RP into RP.
The first one is concerned with the stability of the efficient sets N u, v and M u, v in the case in which the parameter u varies; that is, in which the set of feasible solutions changes. We use the following notations to investigate this case. We will analyze the continuity of N1 and M 1 in Sections 4. The second point of view is concerned with the stability of the solution sets in the case in which the parameter v varies; that is, in which the domination structure of the decision maker changes.
To investigate this case, we also use the following notations. We will analyze the continuity of N2 and M 2 in Sections 4. In order to simplify the notation, the fixed set of feasible solutions Y u and the fixed domination structure D v will be denoted by Y and D, respectively, hereafter in this chapter. The results in this chapter are mainly from Tanino and Sawaragi [TlO]. Naccache [N2] developed stability results in connection with scalarization, and Jurkiewicz [13] considered the stability of compromise solutions.
Namely, the stability of the solution set will be investigated when the set of feasible solutions specified by a parameter vector u is perturbed. It is a generalization of the study of the continuity of optimal value functions or perturbation functions in ordinary scalar optimization problems. The continuity, especially lower semicontinuity, of this function is closely related to the duality in scalar optimization. Lemma 4. Then Zk E Q yk except for a finite number of k's.
Proof We denote the closed unit ball in RP by B. Here we may assume that Q yk is a convex set, since Q y is convex for any y near y. Then the separation theorem of convex sets Theorem 2. The following theorem provides sufficient conditions under which the map N 1 is upper semicontinuous. First, note that y E Y u , since the map Y is upper semicontinuous at U. Hence we can apply Lemma 4.
Therefore, y E Nt u ; that is, N, is upper semicontinuous at U. We will give some examples to illustrate that each condition in the above theorem is essential for assuring the upper semicontinuity of the point-to-set map Nt at U. Namely, we will illustrate that the lack of only one of the conditions might violate the upper semicontinuity of Nt. For each example, the condition that does not hold is indicated in the parenthesis.
All the other conditions are satisfied. Example 4. Example 4,2. If we notice that y E Nl Ii , ji must coincide with y. Hence N 1 is u, upper semicontinuous at and the proof of the theorem is completed. Some examples illustrate the necessity of each condition in the above theorem. The condition in the parenthesis does not hold, but all the other conditions in the theorem are satisfied.
In other words, we will investigate the continuity of the point-to-set map N z , which is defined on the space V by parameter vectors v that specify the domination structure of the decision maker. This point of view is peculiar to multiobjective optimization problems. The aim of the decision maker in ordinary optimization problems is to maximize or minimize scalar objective functions and so is quite clear; hence, there is no room for such consideration.
On the other hand, in multiobjective optimization problems, the domination structure of the decision maker is a very important and essential factor. Hence, it is quite natural to consider that it is perturbed according to changes of environmental factors or of feelings of the decision maker.
Thus, the study of the stability of the solution set for perturbations of the domination structure seems to be necessary and interesting. As has been already noted in Section 4. Some sufficient conditions for the continuity of this point-to-set map N z will be investigated in this section. Proof Let and We must show that y E Nz v. Since Y is a closed set, then y E Y. Then, in view of Lemma 4. Therefore y must be contained in N 2 v.
Namely, N 2 is upper semicontinuous at v and the proof is completed. Some examples are given in the following to illustrate the necessity of each condition in the theorem. Y2 Y2 1 1 Proof If Yis empty, the theorem is obviously true. So we assume that Y is non empty. Therefore, the map N 2 is lower semicontinuous at v and the proof of the theorem is completed. First, as in Section 4. Then, the lemma is basic for the results in this section.
Since X is uniformly compact near ii, we may assume without loss of generality by taking a subsequence if necessary that x k converges to some point x. Hence, Y is upper semicontinuous at U. This completes the proof of the lemma. Then, we get the following lemma, which is also fundamental in this section.
This lemma provides the relationships between the continuity of point-to-set maps in the decision and objective spaces. We can combine these lemmas and the theorems obtained in Sections 4.
First, with regard to the continuity of the point-to-set map M I , the following theorems are obtained. The following theorems are readily obtained from Theorems 4. Remark 4. Every domination structure is assumed to be provided by a closed convex cone. N 1 u and Niv». In order for properly efficient sets to be well-defined, it is assumed throughout this section that D or D v is a pointed closed convex cone for every v E V If D is not pointed,.
We shall obtain sufficient conditions for the point-to-set maps Ql or Q2 to be upper or lower semicontinuous. Since Y is upper semicontinuous at it, y E Y it. Therefore y E Ql U. Indispensability of conditions i and ii in Theorem 4. Though the second condition is indispensable, it may be too restrictive. Since Y is upper semicontinuous at U, y E y u.
In view of Examples 4. Y, int D v. Indispensability of each condition can be understood by Examples 4. Then, in view of Theorem 3. Then, E D v , since D is upper semicontinuous at v. Examples 4. However, the condition in Theorem 4. Therefore, the results in this subsection are valid for the stability of the efficient set in linear problems.
As was seen in the previous chapters, efficient solutions correspond to minimal maximal solutions in ordinary mathemati- cal programming. The next section develops duality in nonlinear multiobjective optimization in parallel with ordinary convex programming, which was originally developed by Tanino and Sawaragi [T7, T9].
Some of modified versions are due to Nakayama [N5]. The third section considers duality from a geometric viewpoint. A geometric meaning of the vector-valued Lagrangian function will be given.
Other duality theorems will be derived from geometric consideration. The main parts of this section are due to Nakayama [N5, N7]. Earlier results along a similar line can be found in Nieuwenhuis [N15] and Jahn [11].
They considered the following matrix optimization problem. Furthermore, we shall identify the set of all m x n matrices with R'"":".
This notation is also valid for matrices with other dimensions. The following problem is a simple extension of that given by Gale et al. The dual problem associated with the problem PKGT then becomes Maximize k subject to Before proceeding to duality for the stated problem, we will prove the theorem of alternative in a generalized form.
Then Proof Easy. Lemma 5. Proof The given A condition is equivalent to Furthermore, since M and Q are convex polyhedral cones, from Proposition 2.
The following two lemmas are extensions of those by Gale et al. Proof We shall prove here the first part of the lemma, since the second one follows readily in a similar fashion. Choose a matrix fo.. Theorem 5. Now with a help of Lemma 5. On the other hand, statements ii and iii follow immediately from Lemma 5. Proof See Kornbluth [K8]. Since iii is dual to i and ii , we shall prove ii here. Hence, it follows immediately from iii of Theorem 5.
On the other hand, Isermann [I6] has given a more attractive formulation without auxiliary parameters such as y and u. In the following, we shall consider it in an extended form. The following duality properties hold for these problems.
Then there exists A. E 2 0 such that A. In a similar fashion, we can conclude that x is a D-minimal solution to the primal problem. Let be aD-minimal x solution to the primal problem p. Then it is readily seen from Theorem 3. Let B denote the submatrix of [A, - I] that consists of m columns cor- responding to the basic variables. In view of the result ii , the last relation implies that A is aD-maximal solution to dual problem D. Next, we shall prove the reverse inclusion.
Suppose that A is aD-maximal solution to the dual problem D 1. Then it is clear that for every j. In fact, suppose to the contrary, that there exist some A' E QO and j. Rewriting Eqs. Thus from Lemma 5. The following lemma, which was used in the proof of Theorem 5. Since efficient solutions correspond to maximal solutions in ordinary mathematical programming, it is hard to derive duality for efficient solutions in the absence of the compactness condition. Throughout this section, we impose the following assumptions: i X' is a nonempty compact convex set, ii D and Q are pointed closed convex cones with nonempty interiors of RP and R"', respectively, iii f is continuous and D-convex, iv g is continuous and Q-convex.
Although these conditions might seem too strong, something like these conditions would be more or less inevitable as long as we consider efficient solutions. Several authors have been recently attempting to relax the compactness condition by using the notion of inf instead of min, which will be restated later. It is known that the set r is convex see, for example, Luenberger [L14]. Definition 5. In the following we shall investigate the properties of W.
Proposition 5. On the other hand, since Y u is D-compact, Theorem 3. Proof Immediate from Proposition 5. Proof In view of Proposition 5. Remark 5. This linear vector-valued functional leads geometrically to a supporting conical variety i. Such a geometric meaning will be discussed in more detail in the next subsection. We now have a Lagrange multiplier theorem for multiobjective optimization. In fact, due to the D- convexity off, since for any x!
Then clearly A. All results in this subsection are also valid for such a particular form of Ag. In this subsection, we will state an equivalent algebraic form of the theorem. Hereafter, we shall denote by :t' a family of all p x m matrices A such that AQcD.
Such matrices are said to be positive in some literature Ritter [R6], Corley [C16]. This fact will be often used later. Proof necessity Condition i is the same as a part of the definition of saddle point for L x, A. This result and condition i imply that the pair x, A is a saddle point for L x, A.
Corollary 5. Then, there exists a p x m matrix A E! Proof Immediate from Theorems 5. Thus, we have verified that a properly efficient solution to the problem P together with a matrix give a saddle point for the vector-valued Lagrangian function under the convexity assumptions and the appropriate regularity conditions.
Conversely, the saddle point provides a sufficient condition for optimality for the problem P. To begin with, we define a point-to-set map corresponding to the dual function. Proof Since the maps f and g are D-convex and Q-convex, respectively, the map L. Proof Note that Theorem 3.
Hence, in view of Proposition 3. Hence, we may see from Proposition 5. This is an extension of a property of the primal function w in ordinary convex optimization stated in Subsection 5. Furthermore, suppose thatf x 5. Q c D and for all x E X' under an appropriate regularity condition such as Slater's constraint quali- fication.
The theorem in ordinary scalar convex optimization that cor- responds to this theorem asserts the existence of a supporting hyperplane for epi w at O,j x. We shall discuss this in more detail. All assumptions on f, g, D, and Q of the previous section are inherited in this section.
Hereafter we shall assume that the pointed closed convex cone D is polyhedral. Since D is pointed, M 1 has full rank p. We now have the following lemma. Proof Immediate from Lemma 2. Furthermore, let I K denote the lineality space of K, i.
We now can establish with the help of Lemma 5. Therefore, we can conclude that Theorem 5. Geometric interpretation of vector-valued Lagrangian. This accords with the well-known result. We now investigate a relationship between supporting hyperplanes and supporting conical varieties. Let H A. Define h A. Then, associated with the hyperplane H A.
In particular, let H A. The following lemma clarifies a relation between supporting conical varieties and supporting hyperplanes. If, to the contrary, a conical variety of K given by relation 5.
We now summarize several properties for the support property of conical varieties, saddle points of the vector-valued Lagrangian function, duality, and unconstrained D-minimization of the vector-valued Lagrangian function as the following mutual equivalence theorem. Proof of Theorem 5. X E X is immediate. As in the previous section, the convexity assumption on f and g will be also imposed here, but X' is not necessarily compact. Associated with this primal problem, we shall consider the following two kinds of dual problems: D-maximize U AE!
The following results assert weak duality theorems for these problems, Theorem 5. Y iDf x. Proof Part i follows in a similar fashion to Theorem 5. The following lemma is essential for deriving strong duality. AEQ' Proof Since G is a closed convex set, it is well known that G is also represented by the intersection of the closed halfspaces including it.
Since G is closed, we can now apply Lemma 5. Hence, it follows from Lemma 5. In other words, for a matrix A E. P such that ATp. Thus, We now show the last inclusion in the proposition. Then, it follows immediately from the weak duality Theorem 5. The following lemma will be used in deriving strong duality. Since D is a pointed convex cone, according to Proposition 3. We can now establish a strong duality. The first part of the following theorem is due to Nakayama [N5], and the second part is due to Jahn [Jl].
Proof It can be shown in a similar fashion to Theorem 5. We prove that each efficient point of the dual problem is also an efficient point of the primal problem. Then, according to the weak duality Theorem 5. The proof of ii is quite similar, and thus it is omitted here. Instead of these conditions, in his original work Jahn [11] assumed that YG is closed and some normality condition.
We shall discuss this condition in more detail. In this event, note that, in general, e. The normality condition plays a role to exclude the so-called duality gap here. Notice that Slater's constraint qualification yields the normality.
In Jahn's formulation for multiobjective optimization, the stated normality condition can be extended as follows. For a given u, let G f. The convexity of G is assumed throughout this subsection. On the other hand, Nieuwenhuis [N15] suggested extending the normality condition in a more natural way to multiobjective optimization. We arrive at the desired result via Rockafellar [R7, Corollary 6.
On the other hand, a direct proof is as follows. E [0, 1 IY. O, y' E G, which implies lY. Notice that under the condition, by the same reason of Lemma 5. Then for an appropriate subsequence is a direction of recession of cl G. Finally, it follows from the assumption that Y E D. We can now relax some of conditions of Proposition 5. It should be noted that N-normality is assured by the existence of a proper D-minimal solution to the primal problem and by J-normality Theorem 5.
Notice that, in deriving the relation 5. By considering cl G instead of G itself and using the N-normality condition, the first inclusion follows immediately. The proof of the second inclusion is the same as that of Proposition 5. The third inclusion can be obtained similarly by replacing G by cl G and by using N-normality. Proof Immediate from Theorem 5. Convexity D-convexity or Q-convexity assumptions off, g, and X' are also assumed here, but X' is not necessarily compact nor closed.
This implies Hence, which proves cl A c cl B 1. Next suppose that y' E cl B 1. We then have the following duality theorem. Hence, we shall prove the reverse inclusion. Hence, for a proof of 5. Now turn to the proof of part iii. According to parts i and ii , and Eq. As for w-Infj. As yet, unfortunately, there has been no development of inf playing an effective role in duality for strong efficiency.
The theory has not yet been brought to completion, but is now under development. There remain several unsolved problems. Hence, we present three different kinds of results owing to different authors, namely, Tanino and Sawaragi [Tll], Kawasaki [K2, K3] and Brumelle [B21].
They are based, respectively, on efficiency, weak efficiency, and strong optimality. Some well-known concepts in ordinary conjugate duality, such as conjugate functions, subgradients and stability, will be extended to the case of multiobjective optimization.
When f is an extended real-valued function on W i. Unfortunately, the concept of supremum of a set Yin RP or RP in the sense of efficiency has not yet been established, though some definitions have been proposed. Hence, we use Max and Min as an extension of usual sup and inf, respectively.
In order to consider the vector-valued case, we must define a paired space of W with respect to RP. A most natural paired space is the set of all p x n matrices, which is denoted by RP x ", However its dimension p x n is often too large.
Hence, we mainly discuss the case of the matrix variable, though most results in this section are valid in both cases. Definition 6. Proposition 6. Proof Left to the reader. Corollary 6. Lemma 6. Therefore, y2 E Max F2 x. Remark 6.
This drawback is probably due to the fact that we are using Max instead of Sup. The definition can be formally extended to a nonconvex function, though it essentially requires convexity at least locally. A subgradient has the geometrical interpretation as the normal vector to a supporting hyperplane of epi f at x,j x Remark 2. If we notice the separability of epi f and its supporting hyperplane, we may extend the definition of subgradients to the vector-valued case, which is also an intuitively direct extension of Definition 2.
If oJ x is not empty, thenJ is said to besubdifferentiable at X. The set of all subgradients of F at x; y is called the subdifferential of F at x;y and is denoted by of x; y.
If of x; y is not empty for every y E F x , then F is said to be subdifferentiable at X. Thus, F need not be convex in order for F to be subdifferentiable.
On Remark 6. However, 0, O? Thus af O is not closed. The following proposition provides a characterization of minimal so- lutions by the subgradient 0, that is, the stationary condition in an extended sense.
The following propositions show the relationships between conjugate or biconjugate maps and subgradients. Proof This proposition is obvious from the definitions of conjugate maps and subgradients.
Proof We prove ii only because the proof of i is similar. From Proposition 6. As is pointed out in Remark 6. However, it is often sufficient as is shown in the next proposition.
Then 0, y is clearly a boundary point of epi F, which is a convex set from the assumption. Hence, by Theorem 2. Hence, T E 8F 0; y from Proposition 6. E RP: J. Conversely, if TTJ. ELi J. Proof If T E 8f x , then from Proposition 6. Since f x - Tx is a convex function, there exists J. E SP such that J. See Theorem 3. Hence, P oE L J. Proof This result is obvious.
See x Proposition 6. In view of Proposition 6. From Corollary 6. See Proposition 6. From the definition of T, Y T.
Thus, from Theorem 3. Hence, assume that domf - 0. Since f is assumed to be subdifferentiable at x, there exists T E of x. Let tJ. Here tJ. We analyze Problem P by embedding it in a family of perturbed problems. The space of perturbation is assumed to be B" for simplicity. The first result is the so-called weak duality theorem. Then, from Proposition 6. Proof Immediate from Proposition 6.
We now define the perturbation map for P , which is an extension of the perturbation function or the optimal value function in ordinary scalar optimization. Throughout this chapter we assume that W u is externally stable for each u E R'". Proof Immediate from Lemma 6. Therefore, the discussion on the duality [i. Here we should note that we may take a vector subgradient A as pointed out in Remark 6. Noting that W u c Y u for every u, we can prove the proposition in a manner quite similar to the proof of Proposition 6.
Thus we have the following theorem, which is a generalization of the strong duality theorem in scalar optimization. Theorem 6. Proof These results are obvious from the previous discussions. A x Proof In the former case, in view of Corollary 6. A A Hence, we can directly apply Propositions 6. Proof This theorem is obvious from Propositions 6. Results similar to the above theorems for convex problems can be obtained with the vector dual variable cf.
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