Vector Optimization with Domination Structures: Variational Principles and Applications
Truong Q. Bao, Boris S. Mordukhovich, Antoine Soubeyran, Christiane Tammer
aa r X i v : . [ m a t h . O C ] F e b VECTOR OPTIMIZATION WITH DOMINATION STRUCTURES:VARIATIONAL PRINCIPLES AND APPLICATIONSTruong Q. Bao , Boris S. Mordukhovich , Antoine Soubeyran and Christiane Tammer Abstract . This paper addresses a large class of vector optimization problems in infinite-dimensionalspaces with respect to two important binary relations derived from domination structures. Motivated bytheoretical challenges as well as by applications to some models in behavioral sciences, we establish newvariational principles that can be viewed as far-going extensions of the Ekeland variational principle to coverdomination vector settings. Our approach combines advantages of both primal and dual techniques in varia-tional analysis with providing useful sufficient conditions for the existence of variational traps in behavioralscience models with variable domination structures.
Key words . Set-valued and variational analysis, vector optimization, domination structures, variable or-dering cones, variational rationality
Mathematics Subject Classification (2000)
It has been well recognized over the years that problems of vector and set/set-valued optimizationhave great many mathematical challenges and important intrinsic issues, which significantly differthem from the conventional areas of scalar optimization. Thus such problems require developingnovel tools of analysis to deal with their theory and applications. The spectrum of applications ofvector and set-valued optimization is indeed enormous: economics, finance, ecology, radiotherapytreatment in medicine, environmental and behavioral sciences to name just a few; see [1, 5, 20, 25,27, 29, 30, 32, 33, 36] for more information and references. Variational principles, together withvariational techniques and tools of generalized differentiation in set-valued and variational analysis,provide powerful machinery for the study and applications of vector and set optimality, particularlyrelated to Pareto-type optimal/efficient solutions. We refer the reader to, e.g., [3, 6, 25, 29, 30, 31,32, 33] and the vast bibliographies therein for various approaches, concepts, and results in theseand related directions.This paper is devoted to the study and applications of vector optimization problems with dom-ination structures that can be viewed as extensions of variable ordering cones. The nondomination concept for problems of multiobjective optimization (i.e., with finitely many scalar objectives) wasintroduced by Yu [44] and then was studied and developed in many publications; see, e.g., [8, 20]and the references therein. This concept is significantly more general than the conventional (Pareto)efficiency concept in vector optimization with fixed ordering cones; it has been realized as a crucialfactor for a variety of applications to decision making, games, etc. Department of Mathematics & Computer Science, Northern Michigan University, Marquette, Michigan 49855,USA ([email protected]). Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA ([email protected]).Research of this author was partly supported by the US National Science Foundation under grants DMS-1512846and DMS-1808978, by the US Air Force Office of Scientific Research under grant Aix-Marseille University (Aix-Marseille School of Economics), CNRS & EHESS, Marseille 13002, France ([email protected]). Martin-Luther-University Halle-Wittenberg, Faculty of Natural Sciences II, Institute of Mathematics, D-06099Halle (Saale), Germany ([email protected]). nondomination and efficiency . After revealing importantproperties of both nondominated and efficient solutions with respect to variable domination struc-ture in general linear space settings, we turn to the study of ordered-value mappings defined on quasimetric decision spaces. Besides mathematical novelty and interest, our motivation to involvequasimetric spaces (i.e., spaces with nonsymmetric distances) into consideration is due to unavoid-able appearing such spaces in models of behavioral sciences ; see the discussions in Section 5.The composition of the paper is as follows. In Section 2 we formulate vector optimizationproblems with preference relations induced by the notion of domination defined via a set-valuedmapping D : Y ⇒ Y from the image space Y to itself. This notion is a practically motivatedextension of ordering structures given by variable cones . We define here several notions of optimalsolutions with respect to two binary relations induced by a given domination structure, establishrelationships between them, and then reveal some of their basic properties.Section 3 overviews and discusses known achievements in the theory of variational principlesof the Ekeland-type for vector optimization problems with variable domination structures. For thereader’s convenience and making comparison with our new developments obtained below, simplifiedproofs and clarifications of the major known results are given in this section.Section 4 is the culmination of the paper, which presents new Ekeland-type variational principles in vector optimization with general domination structures and cost mappings defined on quasimetricspaces. We have two main motivations to develop these novel results. The first motivation comesfrom a strong mathematical call to obtain variational principles of the aforementioned type withtaking into account drawbacks of the known results and their proofs discussed in Section 3 as wellas significant challenges that intrinsically appear in the new framework under consideration. The second motivation comes from the aimed applications to models of behavioral sciences in the vein ofSoubeyran’s variational rationality approach, which unavoidably involves quasimetric spaces (evenin finite dimensions) and highly benefits from imposing domination structures on decision spaces;see [36, 39, 40] and the last section below. It is worth mentioning here that the progress achievedin this paper on variational principles and their applications is strongly based on the marriageof variational methods to Gerstewitz’s nonlinear scalarization functional [22, 23] and its recentdevelopments given [9, 26, 42].The concluding Section 5 is devoted to applications of the obtained variational results in thegeneral framework of vector optimization to some behavioral science models via developing thevariational rationality approach to human dynamics initiated and conceptionally described in [36,37, 38, 39, 40]. We first briefly review basic concepts of variational rationality that are closely relatedto quasimetric and domination structures. The major notions to analyze in these frameworks are variational traps of different types. Using our variational developments allows us to establish theexistence of the so-called ex ante (before moving) and ex post (after moving) traps. In this way,we formulate generalized efficiency and domination structures, which extend those in [44] to thesettings when resistance to move matters, and then derive the existence results in the ex ante andex post traps in the new settings. Observe to this end that the proofs of the variational principlesdeveloped in Section 4 provide constructive dynamic procedures to obtain such variational traps.
This section is devoted to introducing the main concepts of our study, establishing relationshipsbetween them, and revealing some of their important properties.2irst we recall the classical notion of vector optimization with respect to a fixed ordering cone.
Definition 2.1 (Pareto preorder)
Let Y be a linear space, let C be a convex cone in Y , and let y, v ∈ Y . The Pareto preorder on Y denoted by ≤ C is defined by v ≤ C y : ⇐⇒ v ∈ y − C ⇐⇒ y ∈ v + C. Given two vectors y and v in a decision linear space Y , we write v = y + d for some vector d ∈ Y . If y is preferred by the decision maker to v , then d can be viewed as a domination factor .The set of all the domination factors for y together with the zero vector 0 ∈ Y is denoted by D ( y ).Then, the multifunction D : Y ⇒ Y is called a domination structure . It is also called a variableordering structure in the majority of publications if D ( y ) is an ordering cone for each y ∈ Y .The concept of domination structures was introduced by Yu in [43], where the sets D ( y ) aresupposed to be cones. Yu defined a domination structure as a family of cones D ( y ), whereas Engau[21] considered it as a set-valued mapping. Domination factors were launched by Bergstresser,Charnes, and Yu [8] in a finite-dimensional setting with respect to convex domination sets.In contrast to vector optimization with a fixed ordering cone, we now define two binary relationsin Y with respect to the choice of domination sets. These binary relations will be used in theformulation of the variational principles in Sections 3 and 4 and also in the context of applicationsin behavioral sciences in Section 5. Definition 2.2 (binary relations)
Given a domination structure D : Y ⇒ Y in a linear space Y , and given vector y, v ∈ Y , we introduce the following binary relations: (i) The nondomination binary relation denoted by ≤ N, D is defined by v ≤ N, D y : ⇐⇒ y ∈ v + D ( v ) ⇐⇒ B ( y, v ) := y − v ∈ D ( v ) . (ii) The efficiency binary relation denoted by ≤ E, D is defined by v ≤ E, D y : ⇐⇒ v ∈ y − D ( y ) ⇐⇒ B ( y, v ) := y − v ∈ D ( y ) . The notation B ( y, v ) allows us to unify the aforementioned binary relations being also usefulin the application Section 5. It signifies the so-called “worthwhile balance without inconvenienceto move.” The unified relation B ( y, v ) ∈ D ( r ), where the chosen reference point r stands for N as r = v , for E as r = y , and for Θ as r = v = f ( x ). The worthwhile balance without inconvenienceto move is B ( f ( x ) , f ( u )) = f ( x ) − f ( u ), the worthwhile balance without inconvenience to moveand r = N is B ( f ( x ) − √ εq ( x, u ) , f ( u )) = f ( x ) − √ εq ( x, u ) − f ( u ), and the worthwhile balancewithout inconvenience to move and r = E is B ( f ( x ) , f ( u ) − √ εq ( x, u )) = f ( x ) − f ( u ) − √ εq ( x, u ).For simplicity, we drop the subscript D in the above binary notations if the context is clear. When D ( y ) ≡ C for some ordering cone of Y , both domination and efficiency binary relations reduce tothe classical Pareto binary relation generated by C , i.e., ≤ N = ≤ E = ≤ C .Given further a mapping f : X → Y acting from a nonempty set to a linear space, we considertwo solution concepts corresponding to both binary relations introduced in Definition 2.2. Thesenotions are important to deriving of the variational principles in Sections 3 and 4 with the subse-quent applications to the models of behavioral sciences given in Section 5. Denote in what followsdom f := { x ∈ X | f ( x ) = ∅} and rge f := f ( X ) with f ( X ) := ∪ x ∈ X f ( x ). Definition 2.3 (nondominated and efficient solutions with respect to domination struc-tures).
Let f : X → Y be a mapping from a nonempty set to a linear space, and let D : Y ⇒ Y be a domination structure in the image space Y . Given ¯ x ∈ dom f , we say that: i) ¯ x is a conventional nondominated solution of f with respect to D , or a conventional D -nondominated solution, or a conventional ≤ N -minimal solution, if ∀ x ∈ dom f, f ( x ) ≤ N f (¯ x ) = ⇒ f (¯ x ) ≤ N f ( x ) . (ii) ¯ x is a D -nondominated solution of f with respect to D if ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) N f (¯ x ) . (iii) ¯ x is a conventional efficient solution of f with respect to D , or a conventional D -efficient solution, or a conventional ≤ E -minimal solution, if ∀ x ∈ dom f, f ( x ) ≤ E f (¯ x ) = ⇒ f (¯ x ) ≤ E f ( x ) . (iv) ¯ x is a D -efficient solution of f with respect to D if ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) E f (¯ x ) , which is equivalent to the condition rge f ∩ ( f (¯ x ) − D ( f (¯ x ))) = { f (¯ x ) } . Recall that the concept of D -nondominated solutions in Definition 2.3(ii) was initiated by Yu[43, 44] for conic domination structures. The concept of D -efficient solutions of f with respect to D in Definition 2.3(iv) was introduced by Chen, Huang, and Yang [10, Definition 1.13] under thename of “nondominated-like minimal points”; see also Chen and Yang [11, Definition 3.1]. We willdeal with (approximate) D -efficient and D -nondominated solutions in Sections 3, 4, and 5.Observe also that a D -efficient solution ¯ x of f is an element, which is not dominated by anotherpoint x with respect to the associated set D ( f ( x )) at the D -efficient solution x . However, given a D -nondominated solution x of f , a domination set D ( f ( x )) is a set associated with another point x . Important properties of these elements can be found in [11, 10, 20, 18, 17, 44].In order to combine these two solution concepts, we use in the following proposition the languageof ≤ ◦ -minimality , where ≤ ◦ stands for either the domination binary relation ≤ N , or for the efficientbinary relation ≤ E formulated in Definition 2.2. Proposition 2.4 (relationships between minimal solutions, I)
Let ‘ ≤ ◦ ’ stand for both thedomination binary relation ≤ N and the efficient binary relation ≤ E taken from Definition .Then, we have the relationships: (i) If ¯ x is a ≤ ◦ -minimal solution of f , then it is a conventional ≤ ◦ -minimal solution of f . (ii) Assume that the pointedness condition for { f, D } at ¯ x ∈ dom f ∀ x ∈ dom f, D ( f ( x )) ∩ ( −D ( f (¯ x ))) = { } (2.1) holds. If ¯ x is a conventional ≤ ◦ -minimal solution of f , then it is ≤ ◦ -minimal to f . Proof . Let us verify both conclusions in this proposition for the case of nondominated solutions;the proof for efficient solutions is similar.To justify (i), assume that ¯ x is a D -nondominated solution of f , i.e., ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) N f (¯ x ) , ∀ x ∈ dom f, f ( x ) ≤ N f (¯ x ) = ⇒ f ( x ) = f (¯ x ) . By f ( x ) = f (¯ x ), we have f (¯ x ) ≤ N f ( x ), and so ¯ x is a conventional D -nondominated solution of f .To prove (ii), assume that (2.1) is satisfied, and that ¯ x is a conventional D -nondominatedsolution of f . To check the D -nondomination of ¯ x to f , fix an arbitrary element x ∈ dom f satisfying f ( x ) = f (¯ x ). We claim that f ( x ) N f (¯ x ). Arguing by contraposition, suppose that f ( x ) ≤ N f (¯ x ). The conventional D -nondominatedness of ¯ x to f yields f (¯ x ) ≤ N f ( x ). Then,Definition 2.2(i) tells us that f (¯ x ) ∈ f ( x ) + D ( f ( x )) and f ( x ) ∈ f (¯ x ) + D ( f (¯ x )) , and therefore f (¯ x ) − f ( x ) ∈ D ( f ( x )) ∩ ( −D ( f (¯ x ))) = { } , where the last equality holds due to thepointedness condition for { f, D } at ¯ x . Thus we get f (¯ x ) = f ( x ), a contradiction, which shows that f ( x ) N f (¯ x ). Since x was chosen arbitrary in dom ( f ) while satisfying f ( x ) = f (¯ x ), we verify that¯ x is a D -nondominated solution of f and hence complete the proof of the proposition. △ Observe that when D ( y ) ≡ Θ is a fixed domination set or D ( y ) ≡ C is a fixed orderingcone, there is no difference between the two concepts of nondomination and efficiency defined inDefinition 2.3, and they both reduce to Pareto minimality . In such a situation, the pointednesscondition for D and f at ¯ x ∈ dom f is nothing but the pointedness property of the ordering set Θand of the ordering cone C , respectively.Next we establish relationships between minimal solutions with respect to a domination struc-ture D and Pareto minimal solutions with respect to a fixed domination set Θ. Denote the twodomination sets associated with { f, D } byΘ u D := [ (cid:8) D ( f ( x )) (cid:12)(cid:12) x ∈ dom f (cid:9) and Θ i D := \ (cid:8) D ( f ( x )) (cid:12)(cid:12) x ∈ dom f (cid:9) and call them the union (respectively, intersection ) domination set for f and D . We skip mentioning f in the above notations for simplicity. Proposition 2.5 (relationships between minimal solutions, II)
The following hold: (i) If ¯ x is a conventional D -efficient solution of f , then it is e D -efficient to f , where e D : Y ⇒ Y is defined by e D ( y ) := D (¯ y ) \ ( − Θ u D ) . (ii) If ¯ x is a D -efficient solution of f , then it is Θ -minimal with Θ = D ( f (¯ x )) . (iii) If ¯ x is a D -nondominated solution of f , then it is Θ i D -minimal to f . (iv) If ¯ x is a Θ u D -minimal solution of f , then it is D -nondominated to f . Proof . (i) Assume that ¯ x is a conventional D -efficient solution of f , i.e., ∀ x ∈ dom f, f ( x ) ≤ E, D f (¯ x ) = ⇒ f (¯ x ) ≤ E, D f ( x ) ⇐⇒ ∀ x ∈ dom f, f ( x ) ∈ f (¯ x ) − D ( f (¯ x )) = ⇒ f (¯ x ) ∈ f ( x ) − D ( f (¯ x )) . Arguing by contraposition, suppose that ¯ x is not a e D -efficient solution of f . Then, we could find x ∈ dom f satisfying f ( x ) = f (¯ x ) and such that f ( x ) ≤ E, e D f (¯ x ), i.e., f ( x ) ∈ f (¯ x ) − e D ( f (¯ x )) = f (¯ x ) − D ( f (¯ x )) \ ( − Θ u D ) ⊆ f (¯ x ) − D ( f (¯ x )) , (2.2)5hich clearly implies that f ( x ) − f (¯ x ) Θ u D and f ( x ) ≤ E, D f (¯ x ). Since ¯ x is a conventional D -efficient solution of f , we have f (¯ x ) ≤ E, D f ( x ), i.e., f (¯ x ) ∈ f ( x ) − D ( f ( x )) = ⇒ f ( x ) − f (¯ x ) ∈ D ( f ( x )) ⊆ Θ u D . The obtained contradiction verifies the implication in (i).(ii) This is straightforward from the definitions. Indeed, we have¯ x is a D -efficient solution of f ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) E f (¯ x ) ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) f (¯ x ) − D ( f (¯ x )) ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) D ( f (¯ x )) f (¯ x ) ⇐⇒ ¯ x is a ≤ D ( f (¯ x )) -minimal solution of f . (iii) This also follows from the definitions. Indeed, we have¯ x is a D -nondominated solution of f ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) N f (¯ x ) ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f (¯ x ) f ( x ) + D ( f ( x )) ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f (¯ x ) − f ( x ) ∈ Y \ D ( f ( x )) ⊆ Y \ Θ i D ⇐ = ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) Θ i D f (¯ x ) ⇐⇒ ¯ x is a (Pareto) ≤ Θ i D -minimal solution of f . (iv) Similarly to the above we get the equivalences¯ x is a Θ u D -minimal solution of f ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) Θ u D f (¯ x ) ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f (¯ x ) f ( x ) + Θ u D ⇐⇒ ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f (¯ x ) − f ( x ) ∈ Y \ Θ u D ⊆ Y \ D ( f ( x )) ⇐ = ∀ x ∈ dom f, f ( x ) = f (¯ x ) = ⇒ f ( x ) N f (¯ x ) ⇐⇒ ¯ x is a D -nondominated solution of f , which therefore complete the proof of the proposition. △ We illustrate the differences of solution notions above by the following example.
Example 2.6 (differences between solution notions)
Let X := { , , } , and let the valuesof a mapping f : X → R are given by f (1) := A = (0 , , f (2) := A = (4 , − , and f (3) := A = ( − , . Set e := (1 , e := (0 , e := (1 , e := ( − , −
1) and then consider a dominationstructure D on f ( X ) with the following values in R : D ( A ) := conv cone { e , e } , D ( A ) := conv cone { e , e , e } , and D ( A ) := conv cone {− e , e } . Then, we have the optimal solutions:a) 1 and 3 are D -efficient solutions of f .b) 3 is a D -nondominated solution of f .c) 3 is a Θ i D -minimal solution of f . 6 Overview and Elaborations of Known Results
It has been well recognized that the
Ekeland variational principle (EVP) plays a fundamental rolein variational analysis and in a vast variety of applications, including those to vector and set-valuedoptimization; see, e.g., the books [30, 31, 32] with the references and commentaries therein. Quiterecently, several extensions of the EVP have been developed for problems of vector optimizationwith domination structures . Let us recall and elaborate them in this section, which makes a bridgeto our new developments and applications in the subsequent sections.In the first part of this section we deal with approximate solutions of f as an extension of D -efficient solutions of f in the sense of Definition 2.3(iv). This concept of (weakly) approximateefficient solutions with respect to D is given in the next definition. Definition 3.1 (approximate efficient solutions)
Let f : X → Y , D : Y ⇒ Y , k ∈ Y \ { } ,and ε ≥ . Then, we have: (a) An element x ε ∈ X is called an εk -efficient solution of f with respect to D if f ( X ) ∩ ( f ( x ε ) − εk − ( D ( f ( x ε )) \ { } )) = ∅ . (b) An element x ε ∈ X is called a weakly εk -efficient solution of f with respect to D if int D ( f ( x ε )) = ∅ and f ( X ) ∩ ( f ( x ε ) − εk − int D ( f ( x ε ))) = ∅ . Note that for the special case where ε = 0, the notion of (weakly) εk -efficient solutions of f withrespect to D reduces to that of D -efficient solutions of f formulated to Definition 2.3(iv).The classical EVP concerns approximate solutions of scalar optimization problems with extended-real valued, lower semicontinuous, and bounded from below objectives in the setting of completemetric spaces. First we recall extensions of EVP to the case of εk -efficient solutions of f withrespect to a domination structure D : Y ⇒ Y in the sense of Definition 3.1.In [5, Theorem 3.1], Bao et al. established a version of EVP for set-valued mappings withordering structures in quasimetric spaces. They used a variational approach based on an extendedversion of the Dancs-Hegedu¸s-Medvegyev¸s fixed point theorem. A simplified version of this resultfor vector-valued mappings from a complete metric space to a normed space is given below. Theorem 3.2 (EVP for conic domination structures)
Let ( X, d ) be a complete metric space,let Y be a normed space, and let k ∈ Y \ { } . Given f : X → Y and a cone Θ ⊂ Y . Consider adomination structure D : Y ⇒ Y such that the sets D ( y ) are proper, pointed, and closed cones forall y ∈ rge f . Consider Θ i D = ∩{D ( f ( x )) | x ∈ dom f } . Picking ε > , take an εk -efficient solution x ε ∈ X of f with respect to D together with k ∈ Θ i D \ ( − Θ − D ( f ( x ))) and assume that: (A1) ( boundedness condition ) f is quasibounded with respect to Θ in the sense that there is abounded subset of Y such that f ( x ) ⊆ M + Θ for all x ∈ dom f . (A2) ( limiting monotonicity condition ) f satisfies the limiting decreasing continuity conditionover X with respect to D in the sense that for every sequence { x n } ⊆ X such that x n → x ∗ and f ( x n ) − f ( x n +1 ) ∈ D ( f ( x n )) we have f ( x n ) − f ( x ∗ ) ∈ D ( f ( x n )) for all n ∈ N . A3) (transitivity condition for ≤ E ) D enjoys the monotonicity property on rge f in thesense of the application ∀ x, u ∈ X, f ( u ) − f ( x ) ∈ D ( f ( u )) = ⇒ D ( f ( x )) ⊆ D ( f ( u )) . Then, there exists ¯ x ∈ dom f such that the following conditions hold: (i) ¯ x is an εk -efficient solution of f with respect to D . (ii) d ( x ε , ¯ x ) ≤ √ ε . (iii) ¯ x is an efficient solution of f ¯ x with respect to D , where f ¯ x := f + √ εd (¯ x, · ) k , i.e., ∀ x ∈ X \ { ¯ x } , f (¯ x ) − f ( x ) − √ εd (¯ x, x ) k
6∈ D ( f (¯ x )) . Comment 3.3 (a)
Conclusions (i) and (ii) can be formulated in the form f ( x ε ) − f (¯ x ) − √ εd ( x ε , ¯ x ) k ∈ D ( f ( x ε )) , which clearly implies that both (i) and (ii) hold. (b) The requirement k ∈ Θ i D is equivalent to that k ∈ D ( f ( x )) for all x ∈ dom f , and themonotonicity property of the variable ordering structure D is essential for the transitivity propertyof the efficiency binary relation ≤ E . Furthermore, both these assumptions are essential in thevariational approach in [3]. (c) The space Y should be a normed space in [5, Theorem 3.1] instead of a real topologicalspace, since the existence of a bounded set in condition (A1) is not defined in the latter case . In [35, Theorem 5.1], Soleimani established a version of EVP in vector optimization with adomination structure D : X ⇒ Y whose domination sets are not necessarily cones . He used the scalarization approach first developed in [41] for vector optimization with domination sets, i.e., inthe case of a constant domination structure. The result formulated in Banach spaces holds in thissetting. It was improved in [2, Theorem 3.8], while in [35, Theorem 5.1] and [2, Theorem 3.8] itwas supposed that X is a Banach space and that Y a topological linear space.The Ekeland-type variational principle given in the next theorem is derived for εk -efficientsolutions of f with respect to D : Y ⇒ Y in the sense of Definition 3.1 under the assumptionthat ( X, d ) is a complete metric space and that Y a topological linear space. It provides a certainimprovement of [35, Theorem 5.1] and [2, Theorem 3.8] with a simplified proof as presented below. Theorem 3.4 (EVP with relaxed conic domination)
Let ( X, d ) be a complete metric space,let Y be a topological linear space, and let k ∈ Y \ { } . Given a domination structure D : Y ⇒ Y , amapping f : X → Y , and a number ε ≥ , we consider an εk -efficient solution x ε of f with respectto D and denote y ε := f ( x ε ) . Assume the following conditions: (B1) (boundedness condition) f is bounded from below in the sense that there is y ∈ Y suchthat ∀ x ∈ X, f ( x ) ∈ y − D ( y ε ) . (B2) (lower semicontinuity condition) f is ( k, D ) -lsc in the sense that for every y ∈ f ( X ) and for every t ∈ R the sets M ( y, t ) := (cid:8) x ∈ X (cid:12)(cid:12) f ( x ) ∈ tk − cl ( D ( y )) (cid:9) are closed in X . B3) (scalarization conditions) (B3-a) ∈ D ( y ε ) , D ( y ε ) is a proper, pointed, closed, and solid set with D ( y ε ) + D ( y ε ) ⊆ D ( y ε ) and D ( y ε ) + (0 , ∞ ) k ⊆ int( D ( y ε )) . (B3-b) There is a cone-valued mapping C : Y ⇒ Y satisfying k ∈ int( C ( y ε )) and D ( y ε ) +( C ( y ε ) \ { } ) ⊆ int D ( y ε ) . (B3-c) D ( y ) ⊆ D ( y ε ) for all y ≤ D ( y ε ) y ε .Then, there exists ¯ x ∈ dom f such that we have the assertions: (i) ¯ x is an εk -efficient solution of f with respect to C , i.e., ∀ x ∈ X, f (¯ x ) − f ( x ) − √ εk
6∈ C (¯ y ) \ { } , where ¯ y := f (¯ x ) . (ii) d ( x ε , ¯ x ) ≤ √ ε . (iii) ¯ x is an efficient solution of the perturbed function f ¯ x with respect to C , where f ¯ x := f + √ εd (¯ x, · ) k , i.e., ∀ x ∈ X, f (¯ x ) − f ( x ) − √ εd (¯ x, x ) k
6∈ C (¯ y ) \ { } . Proof . Set Θ := D ( f ( x ε )), define S := (cid:8) x ∈ X (cid:12)(cid:12) f ( x ε ) − f ( x ) − √ εd ( x ε , x ) k ∈ D ( f ( x ε )) (cid:9) , and consider the scalarization function ϕ : Y → R given by ϕ ( y ) := ϕ Θ ,k ( y ) = (cid:8) t ∈ R (cid:12)(cid:12) y ∈ tk − Θ (cid:9) . By [25, Theorem 2.3.1], condition (B3-a) ensures that dom ϕ = Y and that ϕ is translation invariantalong k . Furthermore, condition (B3-b) implies that ϕ is strictly C ( y ε )-monotone in the sense that a ∈ b − ( C ( y ε ) \ { } ) and a = b = ⇒ ϕ ( a ) < ϕ ( b ) . It is easy to check that the εk -efficiency to f of x ε guarantees that x ε is an ε -minimal solution ofthe scalarized function ψ := ϕ ◦ f .By (B1) and (B2), the scalarized function ψ is bounded from below and lower semicontinuous.The classical EVP ensures the existence of ¯ x ∈ dom f such that(i’) ψ (¯ x ) + √ εd ( x ε , ¯ x ) ≤ ψ ( x ε ) and(ii’) ∀ x ∈ X \ { ¯ x } , ψ ( x ) + √ εd (¯ x, x ) > ψ (¯ x ).Since x ε is an ε -minimal solution of ψ , it follows from (i’) that d ( x ε , ¯ x ) ≤ √ ε . By [25, Theorem 2.3.1]we have that f (¯ x )+ √ εd ( x ε , ¯ x ) k ∈ y ε −D ( y ε ), and thus f (¯ x ) ∈ y ε − int D ( y ε ) due to (B3-a). Arguingby contraposition, now we verify the fulfillment of (i). Indeed, suppose that ¯ x is not an εk -efficientsolution of f with respect to C , i.e., there is some x ∈ X such that f ( x ) + εk ∈ f (¯ x ) − ( C (¯ y ) \ { } ) . f ( x ) + εk ∈ f ( x ε ) − int D ( y ε ) − ( C (¯ y ) \ { } ) ( B − b ) ⊆ f ( x ε ) − int D ( y ε ) − ( D (¯ y ) \ { } ) ( B − c ) ⊆ f ( x ε ) − int D ( y ε ) − ( D ( y ε ) \ { } ) ⊆ f ( x ε ) − ( D ( y ε ) \ { } ) , which clearly contradicts the εk -efficiency of x ε for f with respect to D .We complete the proof by verifying that (ii’) yields (iii). Indeed, it follows that( ii ′ ) ⇐⇒ ∀ x ∈ X \ { ¯ x } , ϕ ( f ( x ) + √ εd (¯ x, x ) k > ϕ ( f (¯ x ))= ⇒ ∀ x ∈ X \ { ¯ x } , f ( x ) + √ εd (¯ x, x ) k f (¯ x )) − D ( f ( x ε ))= ⇒ ∀ x ∈ X \ { ¯ x } , f ( x ) + √ εd (¯ x, x ) k f (¯ x )) − ( C ( f (¯ x )) \ { } ) , where the first implication holds due to [25, Theorem 2.3.1] and the last one holds due to (B3-c).Thus (iii) is satisfied, which ends the proof of the theorem. △ Comment 3.5 (a)
The result of [35, Theorem 5.1] was formulated for ordering structures D X : X ⇒ Y acting from the domain space to the image space of the mapping f . By using the sameline as in the proof of Theorem 3.4, a better result can be established since the hypotheses of [35,Theorem 5.1] are more restrictive. In particular, (B3) is assumed therein for all x ∈ dom f insteadof at the element x ε . The idea of the proof in [35, Theorem 5.1] is to scalarize the vector-valuedmapping by using the nonlinear scalarized function ϕ := ϕ D ( y ε ) ,k . Thus it is sufficient to imposethe assumptions on the domination set D ( y ε ) as in our elaborated proof. (b) The lower semicontinuity condition (B2) can be weaken to strictly decreasing lower semi-continuity as in [1]. (c)
The boundedness condition of the scalar function ψ is equivalent to the existence of a realnumber m such that ψ ( x ) = ϕ ( f ( x )) > m for all x ∈ dom f ⇐⇒ f ( x ) mk − D ( y ε ) . (3.3)It is easy to check that this condition is weaker than (B1). (d) Condition (B3) does not guarantee that the binary relation ≤ D ( y ε ) is transitive. (e) The result was formulated with respect to the cone-valued domination (i.e., ordering) struc-ture C . In the next section, we will establish a corresponding (even better) one in terms of a givendomination structure D , which may be nonconic.Next we recall Theorem 3.12 in [2], which is a version of EVP with nonsolid domination setsfor εk -efficient solutions of f with respect to D : Y ⇒ Y in the sense of Definition 3.1. It wasestablished by using the nonlinear scalarization approach. Theorem 3.6 (EVP with nonsolid domination sets)
Let ( X, d ) be a complete metric space,let Y be a Banach space, and let k ∈ Y \ { } . Given a domination structure D : Y ⇒ Y and amapping f : X → Y , for each ε ≥ and k ∈ Y \ { } consider an εk -efficient solution x ε of f withrespect to D and denote y ε := f ( x ε ) . Impose the following assumptions: (C1) (quasiboundedness condition) f is quasibounded from below with respect to D ( y ε ) in thesense that there is a bounded set M ⊆ Y such that f ( x ) ∈ M − D ( y ε ) for all x ∈ dom f . C2) (lower continuity condition) f is D ( y ε ) -lower semicontinuous over dom f in the sensethat the level sets lev( y ; f ) := (cid:8) x ∈ dom f (cid:12)(cid:12) f ( x ) ∈ y − D ( y ε ) (cid:9) are closed in X for all y ∈ Y . (C3) (scalarization conditions) (C1-a) D ( y ε ) is a proper, closed, convex, and pointed cone. (C2-b) D ( f ( x )) ⊆ D ( y ε ) for all x ∈ dom f with d ( x ε , x ) ≤ √ ε .Then, there exists an element ¯ x ∈ dom f for which we have: (i) f ( x ε ) − f (¯ x ) ∈ D ( y ε ) , and thus ¯ x is an εk -efficient solution of f with respect to D . (ii) d ( x ε , ¯ x ) ≤ √ ε . (iii) ¯ x is an efficient solution of f ¯ x with respect to D , where f ¯ x := f + √ εd (¯ x, · ) k , i.e., f (¯ x ) − f ( x ) − √ εd (¯ x, x ) k
6∈ D ( f (¯ x )) for all x ∈ dom f. Comment 3.7 (a)
Theorem 3.12 in [2] was formulated for εk -efficient solutions of constrainedproblems, where the cost mapping f acted between two Banach spaces. However, the proof of thetheorem holds true when the domain space is a complete metric one. (b) The quasiboundedness condition (C1) is more restrictive then condition (3.3), which isequivalent to the boundedness from below of the scalarized function of f . (c) Condition (C2) implies that the composite function ψ = ϕ ◦ f defined in the proof ofTheorem 3.4 is lower semicontinuous. Therefore, we can weaken it to the requirement that ψ isstrictly decreasingly lower semicontinuous.In contrast to the aforementioned developments of EVP for εk -efficient solutions of vector-valuedmappings with respect to domination structures in the sense of Definition 3.1 as an extension of D -efficient solutions of f in the sense of Definition 2.3(iv), there have been almost no results for εk -nondominated solutions as an extension of D -nondominated solutions of f in the sense ofDefinition 2.3(ii). To the best of our knowledge, the only result in this direction has been obtained byBao et al. [2, Theorem 4.7], where the conclusions are formulated via a certain auxiliary scalarizedfunction, but not in terms of the given vector-valued mapping. Namely, the scalarized functionemployed in [2] is an extended version of the Gerstewitz scalarization function s : Y → R ∪ {±∞} being defined by the formula s ( y ) := inf (cid:8) t ∈ R (cid:12)(cid:12) y ∈ a + tk − D ( y ) (cid:9) . (3.4)To formulate the aforementioned result, we need to recall the following notion, which is alsoused in the subsequent developments of Section 4. Definition 3.8 (approximate nondominated solutions)
Let f : X → Y , D : Y ⇒ Y , k ∈ Y \ { } , and ε ≥ . An element x ε ∈ X is said to be an εk -nondominated solution of f withrespect to D if we have ∀ x ∈ X, f ( x ε ) − εk f ( x ) + ( D ( f ( x )) \ { } ) . ε = 0, the concept of εk -nondominated solutions of f with respect to D reduces to D -nondominated solutions of f in the sense of Definition 2.3(ii).The next theorem is an extension of [2, Theorem 4.7], where it is obtained under the assumptionthat X is a Banach space. Theorem 3.9 (EVP for approximate nondominated solutions)
Let ( X, d ) be a complete met-ric space while Y is a Banach space, let k ∈ Y \ { } , let D : Y ⇒ Y be a domination structure, andlet f : X → Y be a vector-valued mapping. Given ε ≥ , consider an εk -nondominated solution x ε of f with respect to D and denote y ε := f ( x ε ) . Impose the following assumptions: (D1) (boundedness condition) f is bounded from below with respect to the element y ∈ Y andthe set Θ := D ( y ) , i.e., f ( x ) ∈ y + Θ for all x ∈ dom f . Furthermore, D ( y ) + D ( y ) ⊆ D ( y ) for all y ∈ rge f , k ∈ int(Θ) , and D ( y ) + int(Θ) ⊆ D ( y ) . (D2) (continuity conditions) f is continuous over dom f , and the domination mapping D is ofclosed graph over rge f in the sense that for every sequence of pairs { ( y n , v n ) } with y n ∈ f (Ø) and v n ∈ D ( y n ) for all n ∈ N the convergence ( y n , v n ) → ( y ∗ , v ∗ ) as n → ∞ yields theexistence of x ∗ ∈ X such that y ∗ = f ( x ∗ ) and v ∗ ∈ D ( y ∗ ) . (D3) (scalarization conditions) (D3-a) ∀ y ∈ rge f we have ∈ D ( y ) and D ( y ) is closed in Y . (D3-b) ∀ y ∈ rge f we have D ( y ) + (0 , + ∞ ) k ⊆ D ( y ) \ { } and ( −∞ , k ∩ D ( y ) = ∅ .Then there exists an element ¯ x ∈ dom f such that (i) s ( f (¯ x )) + √ εd (¯ x, x ε ) ≤ s ( f ( x ε )) . (ii) d (¯ x, x ε ) ≤ √ ε . (iii) ¯ x is an exact solution of the scalarized function defined by f ¯ x := s ◦ f + √ εd (¯ x, · ) . Comment 3.10
It is worth mentioning that the conclusions of Theorem 3.9(i,iii) are formulatedin a scalarized form via the scalarization function (3.4). It is important to find appropriate as-sumptions on the given data such that the scalarized function under consideration is bounded frombelow in condition (D1), and that the continuity condition (D2) is satisfied; cf. Lemmas 4.2 and 4.6in [2] for more details. Natural questions arise on whether it is possible to weaken the assumptionsof this result and/or to obtain conclusions in terms of the nondomination to the given vector-valuedmapping by using either a nonlinear scalarization approach, or a nonscalarization approach, or amixed approach. We develop new results in this direction in the next section.
This section provides new versions of EVP, which are significantly better than those discussed aboveand are obtained under weaker assumptions. These new versions of EVP seem to be important fortheir own sake while having interesting applications to behavioral sciences presented in Section 5.In particular, the results below address εk -efficient solutions of f with respect to D in the senseof Definition 3.1 as well as εk -nondominated solutions of f with respect to D in the sense ofDefinition 3.8 under the assumptions that the underlying space X is a quasimetric space (which is12ssential for applications in Section 5) and that Y is a real linear space. In our approach we usenew developments for the Gerstewitz scalarization functions of type (3.4) given in [26, 42].First we recall the definition of vectorial closedness with respect to a direction and the definitionof the Gerstewitz scalarization function. Given a real linear space Y and a nonempty subset A ⊆ Y ,the vectorial closure of A in the direction k ∈ Y is defined byvcl k A := (cid:8) y ∈ Y (cid:12)(cid:12) ∀ λ > , ∃ t ∈ [0 , λ ] , y + tk ∈ A (cid:9) . We refer the reader to [26, 34, 42] for more results and discussions on the directionally vectorclosedness and its relationships with vector closedness and topological closedness.
Definition 4.1 (nonlinear scalarization functions with domination sets)
Let Y be a linearspace, let A be a nonempty subset of Y , and let k be a nonzero direction in Y . The function ϕ A,k : Y → R ∪ {±∞} defined by ϕ A,k ( y ) := inf (cid:8) t ∈ R (cid:12)(cid:12) y ∈ tk − A (cid:9) with inf ∅ = + ∞ (4.5) is called the Gerstewitz nonlinear scalarization function generated by the set A and thescalarization direction k . By setting B := A + R + k , we get the equalities ∀ y ∈ Y, ϕ vcl k ( B ) ,k ( y ) = ϕ B,k ( y ) = ϕ A,k ( y ) . Comment 4.2
The the scalarization function ϕ A,k was defined in [25, Theorem 2.3.1] for closedsets A satisfying A + R + k ⊆ A in real topological vector spaces, where ϕ A,k was called a scalarizationfunction with uniform level sets due to the description of its level sets by ∀ t ∈ R , Lev( t ; ϕ A,k ) = tk − vcl k ( B ) = tk − vcl k ( A + R + k ) . Furthermore, ϕ A,k is translation invariant along the direction k in the sense that ∀ y ∈ Y, ∀ t ∈ R , ϕ A,k ( y + tk ) = t + ϕ A,k ( y ) . Given a subset B of Y , ϕ A,k is B -monotone in the sense that a ∈ b − B = ⇒ ϕ A,k ( a ) ≤ ϕ A,k ( b )if and only if A + B ⊆ A . For other properties of the Gerstewitz scalarization functions; see [25,Theorem 2.3.1], [26, Theorem 4], and the references therein.Next we recall some important concepts of quasimetric spaces theory taken from [13]. Definition 4.3 (quasimetric spaces)
A quasimetric space is a pair ( X, q ) consisting of a set X and a function q : X × X R + := [0 , ∞ ) on X × X having the following properties: (i) q ( x, x ′ ) ≥ for all x, x ′ ∈ X and q ( x, x ) = 0 for all x ∈ X ( positivity ) . (ii) q ( x, x ′′ ) ≤ q ( x, x ′ ) + q ( x ′ , x ′′ ) for all x, x ′ , x ′′ ∈ X ( triangle inequality ) . Note that quasimetric spaces may be finite-dimensional , which is the case of our applicationsto behavioral science models given in Section 5. 13 efinition 4.4 (convergence and completeness in quasimetric spaces)
Let ( X, q ) be a quasi-metric space, and let { x n } be a sequence in X . (i) The sequence { x n } is said to be forward-Cauchy if for every ε > there exists some N ε ∈ N such that whenever n ≥ N ε and m ∈ N we have q ( x n , x n + m ) < ε . (ii) The sequence { x n } is said to be forward-convergent to x ∞ if q ( x n , x ∞ ) → as n → ∞ . (iii) The space ( X, q ) is said to be forward Hausdorff if every forward-convergent sequencehas a unique forward-limit. (iv) The space ( X, q ) is forward-forward-complete if every forward-Cauchy sequence is forward-convergent. Since a quasimetric in not symmetric , there are the corresponding backward concepts, whichcan be found in [13].The next concept of generalized Picard sequences of set-valued mappings is taken from [14].
Definition 4.5 (generalized Picard sequences)
A sequence { x n } in a topological space X iscalled generalized Picard for a set-valued mapping S : X ⇒ X if we have ∀ n ∈ N , x n +1 ∈ S ( x n ) . Our approach in this paper is to scalarize a vector-valued mapping f : X → Y by using theGerstewitz scalarization function ϕ A,k defined in (4.5) to construct a generalized Picard sequenceof a certain set-valued mapping that converges to the desired element giving us new versions ofEVP. Note that the scalarized function ϕ A,k ◦ f might not be lower semicontinuous.Recall that an Ekeland-type variational principle for set-valued mappings F acting between acomplete Hausdorff quasimetric space X and a vector space Y is formulated in [6, Theorem 4.2].The binary relation in [6, Theorem 4.2] is called post-less ordering relation , which agrees withthe efficiency binary relation introduced in Definition 2.2(ii). We derive our first result in thissection for εk -efficient solutions in the sense of Definition 3.1 under weaker assumptions than in[6, Theorem 4.2], especially those concerning the variable domination structure, monotonicity, andboundedness. It is supposed in [6, Theorem 4.2] that the set-valued objective mapping F : X ⇒ Y with values in a linear space Y is quasibounded in the following sense. Definition 4.6 (quasiboundedness)
A set-valued mapping F : X ⇒ Y with values in a linearspace Y is quasibounded if there exist a bounded set M in Y such that ∀ x ∈ X, F ( x ) ⊆ M + Θ , where Θ is the given ordering cone of the image space Y . The following major theorem significantly extends the one in [6, Theorem 4.2] with an essentiallydifferent proof. The main tool of our analysis here is a scalarization technique based on the nonlinearscalarization function introduced in Definition 4.1.
Theorem 4.7 (variational principle for efficient solutions under variable domination)
Let ( X, q ) be a quasimetric space, let Y be a linear space equipped with a variable domination structure D : Y ⇒ Y , and let f : X → Y be a vector-valued mapping. Given k ∈ Y \ { } , x ∈ X , y := f ( x ) , Θ := D ( y ) , and ε ≥ , we consider the set-valued mapping W : X ⇒ X defined by W ( x ) := (cid:8) u ∈ X (cid:12)(cid:12) f ( x ) − f ( u ) − √ εq ( x, u ) k ∈ D ( f ( x )) (cid:9) (4.6)14 nd the extended-real-valued function ψ : X → R ∪ {±∞} defined by ψ ( x ) := ϕ Θ ,k ( f ( x ) − f ( x )) (cid:0) = ϕ Θ − f ( x ) ,k ( f ( x )) (cid:1) , (4.7) where ϕ Θ ,k is taken from (4.5) . Impose the following assumptions: (E1) (boundedness condition) The function ψ from (4.7) is bounded from below over W ( x ) . (E2) (limiting monotonicity condition) For every infinite nonconstant generalized Picardsequence { x n } of the set-valued mapping W from (4.6) the convergence of the series P ∞ n =0 q ( x n , x n +1 ) yields the existence of x ∗ such that ∀ n ∈ N , x ∗ ∈ W ( x n ) . (4.8) (E3) (scalarization condition) Θ is k -vectorial closed with ∈ Θ , Θ + Θ ⊆ Θ , Θ + cone ( k ) ⊆ Θ , and Θ ∩ ( − cone ( k )) = { } .Then, there exists x ∗ ∈ W ( x ) satisfying the inclusion W ( x ∗ ) ⊆ { x ∗ } := (cid:8) u ∈ X (cid:12)(cid:12) q ( x ∗ , u ) = 0 (cid:9) . (4.9) If in addition the condition (E4) ( X, q ) is forward-Hausdorff is satisfied, then the conclusions of this theorem reduce to (i) f ( x ) − f ( x ∗ ) − √ εq ( x , x ∗ ) k ∈ D ( f ( x )) and (ii) ∀ x ∈ X \ { x ∗ } , f ( x ∗ ) − f ( x ) − √ εq ( x ∗ , x ) k
6∈ D ( f ( x )) . That is, x ∗ is a Θ -efficient solutionof the perturbed function f x ∗ , where f x ∗ : X → Y is defined by f x ∗ ( x ) := f ( x ) + √ εq ( x ∗ , x ) k. Furthermore, imposing the domination inclusion (E5) D ( f ( x ∗ )) ⊆ D ( f ( x )) ensures that x ∗ is a D -efficient solution for the perturbed function f x ∗ , i.e., ∀ x ∈ X \ { x ∗ } , f ( x ∗ ) − f ( x ) − √ εq ( x ∗ , x ) k
6∈ D ( f ( x ∗ )) . (4.10) If finally the starting point x is an εk -efficient solution of f with respect to D , then x ∗ canbe chosen so that in addition to (i) and (ii) we have (iii) q ( x , x ∗ ) ≤ √ ε . Proof . It is easy to observe from the construction of W in (4.6) that f ( x ) − f ( x ) ∈ −√ εq ( x , x ) k − Θ ∈ R k − Θ for all u ∈ W ( x ) , and thus f ( x ) − f ( x ) ∈ dom ϕ Θ ,k . This yields the inclusion W ( x ) ⊆ dom ψ and allows us toconstruct inductively a generalized Picard sequence satisfying the conclusions of the theorem.Starting with x , we assume that x n is given. Then, choose x n +1 ∈ W ( x n ) satisfying ψ ( x n +1 ) ≤ inf u ∈ W ( x n ) ψ ( u ) + 12 n +1 . (4.11)15t is obvious that such an element x n +1 exists due to the boundedness from below of the function ψ assumed in (E1). We aim at verifying that the generalized Picard sequence { x n } forward-convergesto the desired element by splitting the proof into several steps. Claim 0 : If u ∈ W ( x ) , then f ( u ) ≤ Θ f ( x ) and W ( u ) ⊆ W ( x ). A proof is straightforward. Thistells us therefore that ∀ n ∈ N , f ( x n ) − f ( x n +1 ) ∈ Θ and W ( x n +1 ) ⊆ W ( x n ) . Claim 1 : For every u ∈ W ( x n ) we have the estimate ∀ n ∈ N , ∀ x ∈ W ( x n ) , √ εq ( x n , u ) ≤ n . Indeed, it follows from Θ + Θ ⊆ Θ assumed in (E3) that the scalarization function ϕ Θ ,k is Θ-monotone, i.e., if v ≤ Θ y , then ϕ Θ ,k ( v ) ≤ ϕ Θ ,k ( y ). Fixing an arbitrary number n ∈ N and anarbitrary element x ∈ W ( x n ) yields x ∈ W ( x n ) ⇐⇒ f ( x n ) − f ( x ) − √ εq ( x n , x ) k ∈ Θ ⇐⇒ f ( x ) − f ( x ) + √ εq ( x n , x ) k ≤ Θ f ( x n ) − f ( x ) . Since the scalarization function ϕ Θ ,k given by (4.5) is Θ-monotone and translation invariant alongthe direction k (see Comment 4.2), we have ϕ Θ ,k ( f ( x ) − f ( x ) + √ εq ( x n , x ) k ) ≤ ϕ Θ ,k ( f ( x n ) − f ( x ))= ⇒ ϕ Θ ,k ( f ( x ) − f ( x )) + √ εq ( x n , x ) = ψ ( x ) + √ εq ( x n , x ) ≤ ψ ( x n ) . This readily implies the inequalities √ εq ( x n , x ) ≤ ψ ( x n ) − ψ ( x ) ≤ ψ ( x n ) − inf u ∈ W ( x n ) ψ ( u ) ≤ ψ ( x n ) − inf u ∈ W ( x n − ) ψ ( u ) ≤ n , where the last two estimates hold due to W ( x n ) ⊆ W ( x n − ) and (4.11), respectively. Claim 2 : The series P ∞ n =1 q ( x n , x n +1 ) is convergent . To show this, for every n ∈ N we have x n +1 ∈ W ( x n ) ⇐⇒ f ( x n ) − f ( x n +1 ) − √ εq ( x n , x n +1 ) k ∈ Θ . (4.12)Summing up these inequalities for n = 0 , . . . , i gives us the inclusion f ( x ) − f ( x i +1 ) − √ ε ( i X n =1 q ( x n , x n +1 )) k ∈ Θ , which can be rewritten in the form f ( x i +1 ) − f ( x ) + √ ε ( i X n =1 q ( x n , x n +1 )) k ∈ − Θ . Taking into account the Θ-monotonicity and transitivity properties of ϕ Θ ,k , we get ϕ Θ ,k ( f ( x i +1 − f ( x ) + √ ε ( i X n =1 q ( x n , x n +1 )) k ) ≤ ⇒ √ ε ( i X n =1 q ( x n , x n +1 )) ≤ − ϕ Θ ,k ( f ( x i +1 ) − f ( x ) = ψ ( x i +1 ) ≤ − inf u ∈ W ( x ) ψ ( u ) < ∞ , i was chosen arbitrary, we arrive at the claimedseries convergence √ ε ( ∞ X n =1 q ( x n , x n +1 )) < ∞ . Claim 3 : The inclusion in (4.9) is satisfied . Using the assertion of Claim 2 and the limitingmonotonicity condition (E2) ensures the existence of x ∗ satisfying (4.8). Since we obviously have x ∗ ∈ W ( x n ) ⊆ W ( x ), to get (4.9) it is sufficient to prove that ∀ u ∗ ∈ W ( x ∗ ) , q ( x ∗ , u ∗ ) = 0 . (4.13)To this end, it is easy to check that u ∈ W ( x ) = ⇒ f ( x ) − f ( u ) ∈ Θ and W ( u ) ⊆ W ( x ) . Fixing now an arbitrary element u ∗ ∈ W ( x ∗ ) gives us by (4.8) that ∀ n ∈ N , u ∗ ∈ W ( x n ) . It follows from Claim 1 that q ( x n , u ∗ ) → n → ∞ , i.e., u ∗ is a forward-limit of the sequence { x n } . Denote further α := lim n →∞ ψ ( x n ) and show that ψ ( x ∗ ) = α . Indeed, the choice of x n +1 readily implies that ψ ( x n +1 ) ≤ inf u ∈ W ( x n ) ψ ( u ) + 12 n +1 ≤ ψ ( u ∗ ) + 12 n +1 , where the passage to the limit as n → ∞ yields α ≤ ψ ( u ∗ ). On the other hand, we have from thechoice of u ∗ ∈ W ( x n ) that f ( u ∗ ) − f ( x ) + √ εq ( x n , u ∗ ) k ∈ f ( x n ) − f ( x ) − Θ . Taking into account the Θ-monotonicity and the transitivity along the direction k of the scalariza-tion function ϕ Θ ,k ensures that ψ ( u ∗ ) + √ εq ( x n , u ∗ ) ≤ ψ ( x n ) , and thus we get by passing to the limit as n → ∞ that ψ ( u ∗ ) ≤ α . Hence ψ ( u ∗ ) = α . Since x ∗ ∈ W ( x ∗ ), we have ψ ( x ∗ ) = α as claimed. It now follows from u ∗ ∈ W ( x ∗ ) that f ( u ∗ ) − f ( x ) + √ εq ( x ∗ , u ∗ ) k ∈ f ( x ∗ ) − f ( x ) − Θ . The aforementioned properties of the scalarization function ϕ Θ ,k lead us to ψ ( u ∗ ) + √ εq ( x ∗ , u ∗ ) ≤ ψ ( x ∗ ) . Substituting ψ ( u ∗ ) = ψ ( x ∗ ) = α into the last inequality, we have q ( x ∗ , u ∗ ) ≤ q ( x ∗ , u ∗ ) =0, which verifies (4.13) Since u ∗ was chosen arbitrarily in W ( x ∗ ), we have x ∗ ∈ W ( x ) and W ( x ∗ ) ⊆ { x ∗ } . (4.14)This ensures the fulfillment of (4.9) and thus completes the proof of Claim 3. Claim 4 : Imposing (E4) gives us assertions (i) and (ii) of the theorem . Assumption (E4) tells usthat the forward-limit is unique if exists. Thus { x ∗ } = { x ∗ } , and the two inclusions in (4.14) reduceto assertions (i) and (ii), respectively, by the construction of the sets W ( x ) in (4.6).17 laim 5 : The domination inclusion in (E5) yields (4.10) . Arguing by contraposition, supposethat x ∗ is not a D -efficient solution of the perturbed function f x ∗ . Then, we find x = x ∗ such that f ( x ) + √ εq ( x ∗ , x ) k ∈ f ( x ∗ ) − D ( x ∗ ) ( E ⊆ f ( x ∗ ) − Θ , which clearly contradicts (ii). Claim 6 : If x is an εk -efficient solution of f with respect to D , then we have (iii) . We againargue by contraposition and suppose that (iii) fails, i.e., q ( x , x ∗ ) > √ ε . Then, (i) yields f ( x ∗ ) ∈ f ( x ) − √ εq ( x , x ∗ ) k − D ( f ( x ))= f ( x ) − εk − √ ε ( q ( x , x ε ) − √ ε ) k − D ( f ( x )) ( E ⊆ f ( x ) − εk − D ( f ( x )) . This readily contradicts the assumption that x is an εk -efficient solution of f with respect to D and thus completes the proof of the theorem. △ It is clear that the obtained Theorem 4.7 weakens and/or drops many assumptions in Theo-rem 3.2 and Theorem 3.4. Let us comment on the major assumptions imposed in Theorem 4.7.
Comment 4.8 (a)
Condition (E1) allows us to extend EVP to the class of vector-valued mappingshaving their image spaces as arbitrary linear spaces. When the image space happens to be a normedone, (E1) is weaker than the quasiboundedness condition (A1) and the boundedness condition (B1).In fact, (E1) can be equivalently written as ∃ e ∈ Y, ∀ x ∈ W ( x ) , f ( x ) − f ( x ) ∈ e + Y \ ( − Θ) . (b) Condition (E2) is better than conditions (A2) and (B2) as shown in the next two propositions.The following simple example shows that the boundedness from below condition (E1) imposedin Theorem 4.7 is weaker than the quasiboundedness assumption (A1) imposed in Theorem 3.2.
Example 4.9 (boundedness from below condition of Theorem 4.7 versus quasibound-edness)
Consider a vector-valued mapping f : R ⇒ R defined by f ( x ) := (cid:26) (0 , − x ) if x ≥ , ( x,
0) if x < , and consider a fixed domination structure D : R ⇒ R with D ( y ) ≡ R . In this case we haveΘ = R . Take k := (1 , ∈ Θ \ − Θ and q ( x, u ) := | x − u | . Then, W ( x ) = { x } for all x ∈ R . Themapping f clearly satisfies the boundedness from below condition (E1), but it is not quasiboundedin the sense of (A1) as required in Theorem 3.2. Indeed, we haverge f = cone { (0 , − , ( − , } . Next we present an illustrative example for other assumptions of Theorem 4.7.
Example 4.10 (illustrating the assumptions of Theorem 4.7)
Let X := R and Y := R ,and let f : X → Y be defined by f ( x ) := ( ( x, x −
1) if x < , ( x,
1) if x ≥ . D : R ⇒ R is given by D ( y ) := ( conv cone (cid:8) (1 , , ( | y | , | y | ) (cid:9) if y < y < , R otherwise.Take ε = 1, x = 0, k = (1 , d ( x, u ) = 1 / | x − u | . In this case we have ψ ( x ) = ϕ R ,k . It iseasy to check that: a) W (0) = [ − . ,
0] and W ( − .
5) = {− . } . b) f is ϕ R ,k -bounded from below. c) f is not R -lower semicontinuous sincelev( f, R ) = (cid:8) x ∈ X (cid:12)(cid:12) f ( x ) ∈ − R (cid:9) = ( −∞ , R . d) f ( − .
5) = ( − . , − / √ f (0) = (0 , D ( f ( − . { (1 , , (0 . , − / √ } ,and D ( f (0)) = R . It is obvious that f ( − . ≤ D ( f ( x )) f (0) and D ( f ( − . ⊆ D ( f (0)),and hence condition (E5) is satisfied. e) Condition (E2) holds since for any nonconstant generalized Picard sequences in W (withoutloss of generality it can assumed that x n <
0) we have that W ( x n ) is closed whenever n ∈ N .Then, the existence of x ∗ follows from the classical Cantor theorem.Our next goal is to derive efficient conditions expressed entirely via the given problem dataensuring the fulfillment of the limiting monotonicity assumption (E2) of Theorem 4.7. First weintroduce the following new notion. Definition 4.11 (decreasing lower semicontinuity with respect to domination sets)
Avector-valued mapping f : X → Y is said to be Θ -decreasing lower semicontinuous over aset A if for every forward-convergent sequence { x n } ⊆ A with a forward-limit x ∗ , the Θ -decreasingmonotonicity of { f ( x n ) } with respect to the domination set Θ (i.e., f ( x n +1 ) ≤ Θ f ( x n ) for all n ∈ N ) implies that ∀ n ∈ N , f ( x ∗ ) ≤ Θ f ( x n ) . Proposition 4.12 (sufficient conditions for limiting monotonicity)
Let ( X, q ) be a forward-complete quasimetric space, and let the vector-valued mapping f : X → Y be Θ -decreasing lowersemicontinuous over W ( x ) . Then the set-valued mapping W from (4.9) satisfies the limitingmonotonicity condition (E2) of Theorem . Proof . Take an arbitrary generalized Picard sequence { x n } ⊆ W ( x ) of the set-valued mapping W satisfying P ∞ n =0 q ( x n , x n +1 ) = ℓ < ∞ . Then, for each ε > N ε ∈ N such that N ε − X n =0 q ( x n , x n +1 ) ≥ ℓ − ε and ∞ X n = N ε q ( x n , x n +1 ) ≤ ε. For every i, j ≥ N ε with j > i we have q ( x i , x j ) ≤ j − X n = i q ( x n , x n +1 ) ≤ ∞ X n = N ε q ( x n , x n +1 ) ≤ ε. { x n } is a forward-Cauchy sequence. Since ( X, q ) is forward-forward-complete, it forward-converges to some forward-limit x ∗ ∈ X . Thus { x n } is a generalized Picard sequence of W , i.e., ∀ n ∈ N , x n +1 ∈ W ( x n ) , which implies that f ( x n +1 ) ≤ Θ f ( x n ) for all n ∈ N by (E4). The imposed Θ-decreasing lowersemicontinuity of f ensures that ∀ n ∈ N , f ( x ∗ ) ≤ Θ f ( x n ) . Since { x n } is a generalized Picard sequence of W , it follows that ∀ n ∈ N , f ( x n +1 ) + √ εq ( x n , x n +1 ) k ≤ Θ f ( x n ) . Summing up these relations from n = i to i + j while taking into account (E4) and the triangleinequality for the quasimetric, we have f ( x i + j ) + √ εq ( x i , x i + j ) k ≤ Θ f ( x i ) . Adding this to f ( x ∗ ) ≤ Θ f ( x i + j ) yields f ( x ∗ ) + √ εq ( x i , x i + j ) k ≤ Θ f ( x i ) . Taking again into account the triangle inequality for the quasimetric and (E4) gives us the estimates f ( x ∗ ) + √ εq ( x i , x ∗ ) k + √ q ( x i + j , x ∗ ) k ≤ Θ f ( x ∗ ) + √ εq ( x i , x i + j ) k ≤ Θ f ( x i ) . Since j was chosen arbitrarily, Θ is k -vectorial closed, and since lim n →∞ q ( x n , x ∗ ) = 0, the passageabove to the limit as j → ∞ yields f ( x ∗ ) − √ εq ( x i , x ∗ ) k ∈ f ( x i ) − Θ , i.e., x ∗ ∈ W ( x i ). The latter verifies the fulfillment of (E2), since i was also chosen arbitrarily. △ Proposition 4.13 (other sufficient conditions for the fulfillment of (E2))
Let Θ satisfy con-dition (E3) , and let f be (Θ , k ) -lower semicontinuous in the sense of Soleimani [35] : ∀ x ∈ W ( x ) , ∀ t ∈ R , L ( x, t ) := (cid:8) u ∈ X (cid:12)(cid:12) f ( u ) ∈ f ( x ) + tk − Θ (cid:9) is closed in X. Then, f satisfies condition (E2) of Theorem . Proof . Pick an arbitrary generalized Picard sequence { x n } ⊆ W ( x ) of the set-valued mapping W satisfying P ∞ n =0 q ( x n , x n +1 ) = ℓ < ∞ , and then fix an arbitrary number n ∈ N . For every i ∈ N , itis not difficult to check that f ( x n + i ) + √ εq ( x n , x n + i ) k ≤ Θ f ( x n )by using (E3) and the triangle inequality for the quasimetric. We can further proceed as follow: f ( x n + i ) ∈ f ( x n ) − √ εq ( x n , x n + i ) k − Θ= f ( x n ) − √ εq ( x n , x ∗ ) + √ εq ( x n + i , x ∗ ) k −√ ε (cid:0) q ( x n + i , x ∗ ) k − q ( x n , x ∗ ) + q ( x n , x n + i ) (cid:1) k − Θ ⊆ f ( x n ) − √ εq ( x n , x ∗ ) k + √ εq ( x n + i , x ∗ ) k − Θ . i →∞ q ( x n + i , x ∗ ) = 0, for every δ > N δ ∈ N such that f ( x n + i ) ∈ f ( x n ) − √ εq ( x n , x ∗ ) k + √ ε ( δ ) k − Θ . Since f is (Θ , k )-lower semicontinuous, we have f ( x ∗ ) ∈ f ( x n ) − √ εq ( x n , x ∗ ) k + √ εδk − Θ . Remembering that δ > k -vectorially closed, it follows that f ( x ∗ ) ∈ f ( x ) + f ( x n ) − √ εq ( x n , x ∗ ) k − Θ , i.e., x ∗ ∈ W ( x n ). Since n was also chosen arbitrarily, condition (E2) is verified. △ Our second major result is a new variational principle for εk - nondominated solutions of f withrespect to D in the sense of Definition 3.8. Note to this end that in [6, Theorem 4.5] an Ekeland-type variational principle is obtained for set-valued objective mappings F : X ⇒ Y between acomplete Hausdorff quasimetric space X and a linear space Y . The binary relation considered in[6, Theorem 4.5] is called pre-less ordering relation , which agrees with the nondomination binaryrelation introduced in Definition 2.2(i).Now we establish our variational principle in quasimetric spaces for nondominated solutions under weaker assumptions than in [6, Theorem 4.5], especially those concerning variable dominationstructures, monotonicity and boundedness. Furthermore, as in the case of Theorem 4.7, the proofof the following theorem is based on the nonlinear scalarization function from Definition 4.1, whichis significantly different from the proof of [6, Theorem 4.5]. Theorem 4.14 (variational principle for nondominated solutions under variable domi-nation)
Let ( X, q ) be a quasimetric space, let Y be a linear space, let D : Y ⇒ Y be a dominationstructure on Y with the nondomination relation ≤ N , let k ∈ Y \ { } , and let Θ := D ( f ( x )) . Given x ∈ X and ε ≥ , define the set-valued mapping W = W f,q,ε : X ⇒ X by W ( x ) := (cid:8) u ∈ X (cid:12)(cid:12) f ( x ) − √ εq ( x, u ) k − f ( u ) ∈ D ( f ( u )) (cid:9) ; (4.15) where we drop the parameters f, q , and ε in the notation of W for simplicity. Consider also theextended-real-valued function ψ : X → R ∪ {±∞} given by ψ ( x ) := ϕ Θ ,k ( f ( x ) − f ( x )) (cid:0) = ϕ Θ − f ( x ) ,k ( f ( x )) (cid:1) , (4.16) where ϕ Θ ,k was introduced in (4.5) . Impose the following assumptions: (F1) (boundedness condition) ψ is bounded from below on W ( x ) ; i.e., there exists τ ∈ R suchthat ψ ( x ) ≥ τ for all x ∈ W ( x ) . (F2) (limiting monotonicity condition) for every generalized Picard sequence { x n } of theset-valued mapping W , the convergence of the series P ∞ n =0 q ( x n , x n +1 ) yields the existence of x ∗ satisfying x ∗ ∈ W ( x n ) for all n ∈ N . (F3) (scalarization conditions) (F3-a) Θ is k -vectorially closed, Θ + Θ ⊆ Θ , Θ + cone ( k ) ⊆ Θ , and Θ ∩ − cone ( k ) = { } . (F3-b) ∀ x ∈ W ( x ) , D ( f ( x )) + cone ( k ) ⊆ D ( f ( x )) . (F3-c) ∀ f ( u ) − f ( w ) ∈ D ( f ( w )) , D ( f ( w )) + D ( f ( u )) ⊆ D ( f ( w )) . F3-d) Θ u ⊆ Θ , where Θ u := ∪{D ( f ( x )) : x ∈ W ( x ) } .Then, there exists x ∗ ∈ W ( x ) such that W ( x ∗ ) ⊆ { x ∗ } , where { x ∗ } := (cid:8) u ∈ X (cid:12)(cid:12) q ( x ∗ , u ) = 0 (cid:9) . Assuming furthermore that (F4) the forward-limit of a forward-convergent sequence in the quasimetric space ( X, q ) is unique ,the conclusions of the theorem can be written in the equivalent form (i) f ( x ) − √ εq ( x , x ∗ ) − f ( x ∗ ) ∈ D ( f ( x ∗ )) , (ii) ∀ x = x ∗ , f ( x ∗ ) − √ εq ( x ∗ , x ) k − f ( x )
6∈ D ( f ( x )) .If finally the starting point x is an εk –nondominated solution of f with respect to D , i.e., ∀ x ∈ X, f ( x ) − εk − f ( x )
6∈ D ( f ( x )) , (4.17) then we have in addition to (i) and (ii) that x ∗ satisfies the localization condition (iii) q ( x , x ∗ ) ≤ √ ε . Proof . Let us construct inductively a generalized Picard sequence that satisfies all the requirementsof the theorem. Observe that for every u ∈ W ( x ) we have f ( u ) ∈ f ( x ) − √ εq ( x , u ) k − D ( f ( u )) ( F − a ) ⊆ f ( x ) − D ( f ( u )) ( F − d ) ⊆ f ( x ) − Θ , which clearly implies that ϕ Θ ,k ( f ( u ) − f ( x )) is finite due to (F1). This gives us W ( x ) ⊆ dom ψ .Starting with x , suppose that x n is defined, and then choose x n +1 ∈ W ( x n ) such that ψ ( x n +1 ) ≤ inf u ∈ W ( x n ) ψ ( u ) + 12 n +1 . (4.18) Claim 1 : If u ∈ W ( x ) , then W ( u ) ⊆ W ( x ). To verify this, fix an arbitrary element w ∈ W ( u ) andby using the construction of W get the inclusions f ( x ) − √ εq ( x, u ) k − f ( u ) ∈ D ( f ( u )) and f ( u ) − √ εq ( u, w ) k − f ( w ) ∈ D ( f ( w )) . (4.19)By (F3-b), the second inclusion in (4.19) yields f ( u ) − f ( w ) ∈ D ( f ( w )), and thus D ( f ( u )) + D ( f ( w )) ⊆ D ( f ( w )) due to (F3-c). Combining both inclusions in (4.19) tells us that f ( x ) − √ εq ( x, u ) k − √ εq ( u, w ) k − f ( w ) ∈ D ( f ( w )) . Employing (F3-b) and the triangle inequality of the quasimetric q , we obtain f ( x ) − √ εq ( x, u ) k − √ εq ( u, w ) k − f ( w ) ∈ D ( f ( w )) ⇐⇒ f ( x ) − √ εq ( x, w ) k − f ( w ) ∈ D ( f ( w )) + √ ε ( q ( x, u ) + q ( u, w ) − q ( x, w )) k = ⇒ f ( x ) − √ εq ( x, w ) k − f ( w ) ∈ D ( f ( w )) , which readily implies that w ∈ W ( x ). Since w was chosen arbitrarily in W ( u ), it gives us W ( u ) ⊆ W ( x ) as asserted in Claim 1. 22 laim 2 : For all n ∈ N and for every u ∈ W ( x n ) we have √ εq ( x n , u ) ≤ n . Picking an arbitraryelement u ∈ W ( x n ), observe that f ( x n ) − √ εq ( x n , u ) k − f ( u ) ∈ D ( f ( u )) ( F − d ) = ⇒ f ( x n ) − √ εq ( x n , u ) k − f ( u ) ∈ Θ ⇐⇒ f ( u ) − f ( x ) ∈ f ( x n ) − f ( x ) − √ εq ( x , x n +1 ) k − Θ . Since Θ + Θ ⊆ Θ by (F3-a), the scalarization function ϕ Θ ,k is Θ-monotone. This leads us to ϕ Θ ,k ( f ( u ) − f ( x )) ≤ ϕ Θ ,k ( f ( x n ) − f ( x ) − √ εq ( x n , u ) k )= ϕ Θ ,k ( f ( x n ) − f ( x )) − √ εq ( x n , u ) . Using the assertion of Claim 1, we have W ( x n ) ⊆ W ( x n − ) and thus arrive at the estimates √ εq ( x n , u ) ≤ ψ ( x n ) − ψ ( u ) ≤ ψ ( x n ) − inf u ∈ W ( x n ) ψ ( u ) ≤ ψ ( x n ) − inf u ∈ W ( x n − ) ψ ( u ) ≤ n , where the two last inequalities hold due to W ( x n ) ⊆ W ( x n − ) and (4.11). This verifies the claim. Claim 3 : The series P ∞ n =1 q ( x n , x n +1 ) is convergent . For every n ∈ N we have x n +1 ∈ W ( x n ),which is equivalent to the inclusion f ( x n ) − √ εq ( x n , x n +1 ) k − f ( x n +1 ) ∈ D ( f ( x n +1 )) . (4.20)It follows from (F2-b) that f ( x n ) − f ( x n +1 ) ∈ D ( f ( x n +1 )). Then, using (F3-c) gives us D ( f ( x n +1 ))+ D ( f ( x n )) ⊆ D ( f ( x n +1 )). Summing up the inclusions in (4.20) for n = 0 , . . . , i , we get that f ( x ) − √ ε ( i X n =1 q ( x n , x n +1 )) k − f ( x i +1 ) ∈ D ( f ( x i +1 )) . Taking further (F3-d) into account ensures that f ( x i +1 ) − f ( x ) + √ ε ( i X n =1 q ( x n , x n +1 )) k ∈ − Θ . Since ϕ Θ ,k is Θ-monotone and translation invariant along the direction k as discussed in Com-ment 4.2, we obtain the inequality ϕ Θ ,k ( f ( x i +1 ) − f ( x )) ≤ −√ ε ( i X n =1 ) q ( x n , x n +1 )) . Then, the boundedness condition (F1) tells us that √ ε ( i X n =1 q ( x n , x n +1 )) ≤ − ψ ( x i +1 ) ≤ − inf u ∈ W ( x ) ψ ( u ) < ∞ . Since i was chosen arbitrarily, we arrive at √ ε ( ∞ X n =1 q ( x n , x n +1 )) < ∞ , Claim 4 : We have the inclusion W ( x ∗ ) ⊆ { x ∗ } . Since (F2) yields the existence of x ∗ with x ∗ ∈ W ( x n ) for all n ∈ N , to justify this claim it is sufficient to show that ∀ u ∗ ∈ W ( x ∗ ) , q ( x ∗ , u ∗ ) = 0 . Fix an arbitrary element u ∗ ∈ W ( x ∗ ). It follows from Claim 3 with u ∗ ∈ W ( x n ) and the assertionof Claim 2 that √ εq ( x n , u ∗ ) ≤ n for all n ∈ N , i.e., u ∗ is a forward-limit of the sequence { x n } .Denoting α := lim n →∞ ψ ( x n ), we intend to prove that ψ ( u ∗ ) = α . Indeed, it follows from thechoice of x n +1 that ψ ( x n +1 ) ≤ inf u ∈ W ( x n ) ψ ( u ) − n +1 ≤ ψ ( u ∗ ) − n +1 , where the passage to the limit as n → ∞ yields α ≤ ψ ( u ∗ ). To verify the opposite inequality,deduce from u ∗ ∈ W ( x n ) that f ( u ∗ ) − f ( x ) ∈ f ( x n ) − f ( x ) − √ εq ( x n , u ∗ ) k − D ( f ( u ∗ )) ( F − d ) ⊆ f ( x n ) − f ( x ) − √ εq ( x n , u ∗ ) k − Θ . Since the scalarization function ϕ Θ ,k is Θ-monotone and translation invariant along the direction k (see Comment 4.2), we get ψ ( u ∗ ) ≤ ψ ( x n ) − √ εq ( x n , u ∗ ) . Passing there to the limit as n → ∞ yields ψ ( u ∗ ) ≤ α , and thus ψ ( u ∗ ) = α . Furthermore, weobtain ψ ( x ∗ ) = α since x ∗ ∈ W ( x ∗ ).Deduce now from u ∗ ∈ W ( x ∗ ) the inclusion f ( u ∗ ) − f ( x ) ∈ f ( x ∗ ) − f ( x ) − √ εq ( x ∗ , u ∗ ) k − Θ . Employing again the aforementioned properties of the scalarization function ϕ Θ ,k implies that ψ ( u ∗ ) + √ εq ( x ∗ , u ∗ ) ≤ ψ ( x ∗ ) . Substituting ψ ( u ∗ ) = ψ ( x ∗ ) = α into the last inequality, we arrive at q ( x ∗ , u ∗ ) ≤ q ( x ∗ , u ∗ ) = 0. This readily verifies Claim 4 due to the construction of the set W ( x ∗ ) in (4.15). Claim 5 : Assertions (i) and (ii) under assumption (F4). Assuming (F4) ensures that { x ∗ } = { x ∗ } . This yields by the constructions above that x ∗ ∈ W ( x ) and W ( x ∗ ) = { x ∗ } , which can beequivalently written as forms (i) and (ii) of the theorem. Claim 6 : Completing the proof . It remains to estimate the quasidistance between x and x ε when x is an εk –nondominated solution of f with respect to D . Indeed, if (iii) fails, we have q ( x , x ε ) > √ ε . Then, it follows from (i) and (F3-b) that f ( x ε ) ∈ f ( x ) − √ εq ( x , x ε ) − D ( f ( x ε )) = f ( x ) − εk − √ ε ( q ( x , x ε ) − √ ε ) k − D ( f ( x ε )) ⊆ f ( x ) − εk − D ( f ( x ε )) , which clearly contradicts the choice of x and thus completes the proof of the theorem. △ Comment 4.15 (a)
In the proof of Theorem 4.14 we provide a new way to construct a generalizedPicard sequence of W by using the scalarization function ϕ Θ ,k . This seems to be more effectivethan the procedure in [3, Theorem 3.4]: find x n +1 ∈ W ( x n ) such that q ( x n , x n +1 ) ≥ sup u ∈ W ( x n ) q ( x n , u ) − n +1 . b) The boundedness condition (F1) is less restrictive than the requirement that f is quasi-bounded with respect to Θ (see Definition 4.6) as supposed in (H3) of [6, Theorem 4.5]. (c) The limiting monotonicity condition (F2) is related to the level-decreasing-closedness con-dition (H4’) used in [6, Theorem 4.5] under the name of “pre-less preorder.” (d)
Since there is only one domination set Θ = D ( f ( x )) satisfying condition (F3-a), theadditional condition (F3-d) is essential in comparison with (D3). Condition (F3-c) guarantees thatthe domination binary relation ≤ N is transitive.The next proposition provides efficient conditions in terms of the given data of the problemthat ensure the fulfillment of the boundedness assumption (F1) in Theorem 4.14 expressed thereinvia the auxiliary function (4.16). Proposition 4.16 (sufficient conditions for the boundedness assumption of Theorem 4.14)
Let Y be a normed space, and let a set Θ be topologically closed in Y with k ∈ Θ \ ( − Θ) . Assume inaddition that (F3-a) holds, and that the mapping f in Theorem is quasibounded over W ( x ) .Then, the boundedness condition (F1) is satisfied. Proof . Since f is quasibounded over W ( x ), there exists a bounded set M such that ∀ x ∈ W ( x ) , f ( x ) − f ( x ) ∈ M + Θ . (4.21)Arguing by contraposition, assume that the function ψ from (4.16) is not bounded from below over W ( x ). Then, there is a sequence { x n } ⊆ W ( x ) such that ψ ( x n ) ≤ − n , and thus ∀ n ∈ N , f ( x n ) − f ( x ) ∈ − nk − Θ . Fixing an arbitrary number n ∈ N , we find θ k ∈ Θ such that f ( x n ) − f ( x ) = − nk − θ k . It followsfrom (4.21) that − nk − θ k ∈ M + Θ, and then by (F3-a) we have k ∈ n M − Θ . Since n was chosen arbitrarily in N while Θ is a closed set, the passage to limit as n → ∞ yields k ∈ − Θ. This clearly contradicts the choice of k ∈ Θ \ ( − Θ) and thus completes the proof. △ Now we present an example illustrating the fulfillment of all the scalarization conditions inassumption (F3) of Theorem 4.14.
Example 4.17 (scalarization conditions of Theorem 4.14)
In the setting of Y = R with k = (1 , D : R ⇒ R defined by D ( y ) := D ( y , y ) if y < y , D ( y , y ) if y > y , conv cone n (1 , − a | a | +1 ) , ( − a | a | +1 , o if y = y = a. It is easy to check that D ( y ) ⊆ R , that ∀ a, b ∈ R with a ≥ b, D ( a, a ) ⊆ D ( b, b ) , and that condition (F3-a) is satisfied due to the convexity of the cones D ( y ) for every y ∈ R . Tocheck the fulfillment of (F3-c), fix two arbitrary elements y, v ∈ R such that v ≤ N y . It followsfrom the definition of D that y ∈ v + D ( v ) ⊆ v + R , y ≥ v and y ≥ v . Denoting y := min { y , y } and v := min { v , v } ,we have y ≥ v and D ( y, y ) ⊆ D ( v, v ). Hence D ( y , y ) = D ( y, y ) ⊆ D ( v, v ) = D ( v , v ) . Since D ( y , y ) and D ( v , v ) are convex cones, we get D ( v , v ) + D ( y , y ) ⊆ D ( v , v ). Remem-bering that y and v were chosen arbitrarily in R allows us to conclude that D satisfies condition(F3-c), which completes our considerations in this example.The last result of this section provides efficient conditions via the problem data that ensurethe fulfillment of the limiting monotonicity assumption of Theorem 4.14. First we need to definethe following notion of decreasing lower semicontinuity with respect to nondomination relations ;cf. Definition 4.11 in a different setting. Definition 4.18 (decreasing lower semicontinuity with respect to nondomination)
Avector-valued mapping f : X → Y is said to be ≤ N -decreasing lower semicontinuous withrespect to nondomination relations over a subset A if for every forward-convergent sequence { x n } ⊆ A with a forward-limit x ∗ , we have that the decreasing monotonicity of { f ( x n ) } with respectto the nondomination relation ≤ N , i.e., f ( x n +1 ) ≤ N f ( x n ) for all n ∈ N , implies that ∀ n ∈ N , f ( x ∗ ) ≤ N f ( x n ) . Proposition 4.19 (sufficient conditions for limiting monotonicity in Theorem 4.14)
Let ( X, q ) be a forward-forward-complete quasimetric space, and let f : X → Y be a ≤ N -decreasinglower semicontinuous mapping with respect to the nondomination relation from Theorem . Thenthe mapping W defined by (4.15) satisfies the limiting monotonicity condition (F2) . Proof . Take an arbitrary generalized Picard sequence { x n } of the set-valued mapping W satisfying P ∞ n =0 q ( x n , x n +1 ) = L < ∞ . For each ε > N ε ∈ N such that ∞ X n = N ε q ( x n , x n +1 ) > L − ε/ , and thus for every i, j ≥ N ε with j > i we have the estimates q ( x i , x j ) ≤ j − X n = i q ( x n , x n +1 ) ≤ ∞ X n = N ε q ( x n , x n +1 ) < ε, which show that { x n } is a forward-Cauchy sequence. Since X is forward-forward complete, thissequence forward-converges to some x ∗ ∈ X .Remembering that { x n } is a generalized Picard sequence of W , we deduce from condition (F3-b)that f ( x n +1 ) ≤ N f ( x n ) for all n ∈ N . The imposed decreasing lower semicontinuity of f gives us ∀ n ∈ N , f ( x ∗ ) ≤ N f ( x n ) . Using now (F3-c) yields the inclusions ∀ n ∈ N , D ( f ( x n +1 )) ⊆ D ( f ( x n )) and D ( f ( x ∗ )) ⊆ D ( f ( x n )) . (4.22)Since { x n } is a generalized Picard sequence of W , we get f ( x n +1 ) ∈ f ( x n ) − √ εq ( x n , x n +1 ) k − D ( f ( x n +1 )) . n = i to j while taking into account (F3-b) and the triangleinequality of the quasimetric ensure that f ( x j +1 ) ∈ f ( i ) − √ εq ( x i , x j +1 ) k − D ( f ( x j +1 )) . Adding the latter to the inclusion f ( x ∗ ) ∈ f ( x j +1 ) − D ( f ( x j +1 )) yields f ( x ∗ ) ∈ f ( x i ) − √ εq ( x i , x j +1 ) k − D ( f ( x ∗ ))= f ( x i ) − √ q ( x i , x ∗ ) k + √ q ( x j +1 , x ∗ ) k −√ ε ( q ( x i , x ∗ ) + q ( x j +1 , x ∗ ) − q ( x i , x j +1 ) k − D ( f ( x ∗ )) ( F − b ) ⊆ f ( x i ) − √ q ( x i , x ∗ ) k + √ q ( x j +1 , x ∗ ) k − D ( f ( x ∗ )) . Since the index j was chosen arbitrarily and since D ( f ( x ∗ )) is k -vectorial closed, the passage tolimit above as j → ∞ gives us the inclusion f ( x ∗ ) ∈ f ( x i ) − √ q ( x i , x ∗ ) k − D ( f ( x ∗ )) , i.e., x ∗ ∈ W ( x i ). This verifies (F2) by taking into account that i was also chosen arbitrarily. △ This section is devoted to developing a variational rationality approach to human dynamics in thevein of [36, 37, 38, 39, 40], with taking now into account the new results on variational principlesin vector optimization with variable domination structures that were established above.After presenting the basic concepts of the variational rationality modeling of human dynamics,we mainly concentrate on the following issues: • Introducing generalized efficiency and domination structures of the type formalized in Definition2.2, while being adjusted to the variational rationality approach to human dynamics. These notionsextend those from [44] to the settings where the resistance to move matters . • Applying the obtained variational principles in vector optimization with variable dominationstructures to establish the existence of ex ante (before moving) and ex post (after moving) variationaltraps with showing that possible regrets can matter much .To highlight the major topics of the presentation, we split this section into several subsections.
In this subsection we discuss some basic notions in the modeling of human dynamics and con-ventional approaches to behavioral models based on efficiency and domination structures in theclassical sense of Pareto and more recent ones introduced by Yu.
In the finite-dimensional space Y = R m , consider a list of different pains J = { , . . . , m } and alist of vector amounts of pains v = ( v , . . . , v m ) ∈ R m + ⊆ Y associated with each pain j ∈ J . Theamount of pain v j ∈ R + represents a quantity of a “to be decreased” payoff , e.g., some degree ofunsatisfaction, loss, cost, lack of given things (size of needs), etc. In this context where the agentwants less of each pain and tries to minimize the amounts of pain v j as j ∈ J , the space Y = R m is the space of amounts associated with pains and pleasures. The first problem of the agent is tocompare the lists of different amounts of pains v = ( v , . . . , v m ) ∈ R m + and y = ( y , . . . , y m ) ∈ R m + .As usual, a vector of pains v is Pareto smaller (resp.
Pareto larger ) than another vector of pains y if and only if we have v j ≤ y j (resp. v j ≥ y j ) for all j ∈ J .27aving in mind the above descriptions of “Pareto smaller” and “Pareto larger” vectors, we neednow to clarify the related meanings of the expressions that a given vector of pains is “better than”or is “worse than” another vector of pains in all their aspects; cf. Definition 2.1. (Pareto-i) The vector of pains v is Pareto better than the vector of pains y (in all their aspects)if v = y − D ∗ , where D ∗ := R m + . In this setting, v = y − d with d ∈ D ∗ means that eachamount of pain v j of the list v is smaller than or equal to each amount of pains y j of the list y , i.e., v j ≤ y j for all j ∈ J . This tells us that “less of each pain is better.” (Pareto-ii) The vector of pains y is Pareto worse than the vector of pains v (in all their aspects)if y ∈ v + D ∗ , where D ∗ := R m + . In this case we have y = v + d with d ∈ D ∗ meaning thateach amount of pain y j of the list y is higher or equal to each amount of pains v j of the list v , i.e., y j ≥ v j for all j ∈ J . This tells us that “more of each pain is worse.”Note that in the above case of D ∗ := R m + we have the equivalencies (cf. Section 2) y − v ∈ D ∗ ⇐⇒ v ≤ D ∗ y ⇐⇒ v − y ≤ D ∗ . Thus defining the “better than” sets B ( y ) = y − D ∗ and the “worse than” sets W ( y ) = y + D ∗ , weget that v is Pareto better (resp. worse) than y if and only if v ∈ B ( y ) (resp. y ∈ W ( y )). (cf. Definitions 2.2 and 2.3). In[43, 44], Yu generalized the above concepts of Pareto efficiency and nondomination by consideringthe following conic domination structures : (Yu-i) An arbitrary fixed convex cone D ⊆ Y instead of the Pareto constant cone D ∗ = R m . (Yu-ii) Variable conic structures D ( y ) for all y ∈ Y .To discuss these concepts, consider first the “worse” and “better” relations with respect toconstant cones D ⊆ Y . The main emphasis here is that the same amount of two different painscan be more important or less important for an individual. In this setting, the agent may acceptto trade off a lower amount of a more important pain 1 to a higher amount of a less importantpain 2. For simplicity, take two pains 1 and 2 with the corresponding amounts of these pains v = ( v , v ) ∈ R and y = ( y , y ) ∈ R . The new meaning that Yu gave to the “better than”relation is: v is better than y if and only if v = y − D , when the cone D may be larger than thePareto cone D ∗ = R m + . In this simple situation where the agent compares the new short list ofpains v with the old short list of pains y , he/she prefers the new list of pains v to the old one y if accepting to trade off the lower amount v ≤ y of the most important pain 1 against a higheramount v ≥ y of the less important pain 2. This means that, moving from the list y to the list v ∈ y − D , the agent trades off the diminution v − y ≤ v − y ≥ y − v = d ≥ v − y = − d ≥ ≤ v − y = − d ≤ α ( y − v ) = α d giving us d + α d ≥
0. Then, α > ≤ v − y ≤ α if y − v = 1 . D of acceptable augmentations of less important pains relative to given diminutions of more important pains: D = (cid:8) d = ( d , d ) ∈ R (cid:12)(cid:12) d ≥ , α d + d ≥ (cid:9) . We also refer the reader to [28] for related discussions showing how tradeoffs modelize the relativeimportance of criteria in the case where D ⊇ R m + is not a too obtuse cone.The same meaning can be given to the relation “to be worse than” with the same dominatedcone D . In this case we say that y is worse than v if y ∈ v + D , i.e., if y = v + d as d ∈ D . Thistells us that the agent finds y worse than v if a given augmentation of the more important pain 1is not compensated by a large enough diminution of the less important pain 2. In this subsection we describe the variational rationality (VR) approach to human dynamics, whichbenefits from the variational principles developed in Section 4. In fact, one of the major motivationsfor developing our new research on variational principles in vector optimization problems withvariable domination structures came from the needs of the VR approach described below.
According to Definition 2.2, a variabledomination and efficiency structures gives us, for each position v ∈ Y of a space of pains (positions) Y , a set of possible worse (dominated) positions v + D ( v ) ⊆ Y and a set of better (preferred)positions v − D ( v ) ⊆ Y . The variable cone D ( v ) ⊆ Y represents a set of pains, which can be addedto the vector of pains v to make worse the new vector of pains v + d with d ∈ D ( v ). On the otherhand, the variable cone P ( v ) = −D ( v ) ⊆ Y defines a set of pains that can be dropped from thevector of pains v while making better the vector of pains v − d with d ∈ D ( v ).The variable nondomination and efficiency binary relations under consideration are given by: • “ y is worse than v ” if and only if y ∈ v + D ( v ); this is the nondomination binary relationtaken from Definition 2.2(i). • “ v is better than y ” if and only if v ∈ y − D ( y ); this is the efficiency binary relation takenfrom Definition 2.2(ii).Note that these variable relations are not generally equivalent, while they becomes equivalentin the case of constant structures D ( y ) = D ( v ) = D for all v, y ∈ Y .The VR approach to human dynamics focuses the major attention on a short list of main con-cepts for modeling human behaviors: activities, payoffs (utilities and disutilities as satisfactions andunsatisfactions to move), moves, costs to move, advantages/disadvantages to move, inconveniencesto move, motivation and resistance to move, worthwhile balances, worthwhile moves, aspirationpoints, desires, and stationary or variational traps. This approach is well adapted to: (a) give anew interpretation of the Yu’s approach in the context of variable cones when there are no resis-tance to move (i.e., change rather than stay), and (b) generalize the nondomination and efficiencybinary relations in the vein of Definition 2.2 when resistance to move matters. Without resistanceto move, the VR approach provides the following: (i) Starts by focusing the attention on moves in a space of positions. (ii)
Makes a distinction between disadvantageous (utility deteriorating) moves d ∈ D ( v ) and ad-vantageous (utility improving) moves − d with d ∈ D ( v ) in the space of positions.29 iii) Makes an essential distinction between an ex ante perception to move and an ex post percep-tion to move. In this context, D ( v ) is a set of disadvantageous moves, while P ( v ) = −D ( v )is a set of advantageous moves starting from the initial position v in the payoff space Y .We refer the reader to [5, 6] for the first attempts to investigate adaptive aspects of the varia-tional rationality approach when resistance to move matters. Now we can do more. The corresponding logic of the efficiency and dominationstructures become very clear in the context of the variational rationality approach to human dy-namics. We start here with the VR discussions concerning the space of “to be decreased” payoffs Y = R m . Let y ∈ Y and v ∈ Y be the amounts of pains that the agent endorses in the previousand the current periods. Then, within the current period, a simplified definition of a move thatis well adapted to the present paper starts with “having suffered of the amounts of pains y ∈ Y in the previous period” and ends with “suffering of the amount of pains v ” in the current period.This move is ( y, v ) ∈ Y × Y . It is a change if v = y and a stay if v = y . Note that the move ( y, v )in the payoff space corresponds to some move ( x, u ) ∈ X × X in the activity space X . With thetwo given bundles of activities x ∈ X and u ∈ X , we have the amount of pains y = f ( x ) ∈ Y inthe previous period and the amounts of pains v = f ( u ) ∈ Y in the current period. The most basicquestion driving the VR approach is the following: “should I stay or should I move?” That is, atthe beginning of the current period (ex ante, i.e., before moving) the main alternative is: (a) Either to stay , i.e., doing the same bundle of activities x in the current period as before. Inthis case, the agent would suffer from the same amounts of pains y = f ( x ) as before. (b) Or to change , i.e., doing a different bundle of activities u = x in the current period as before.In this case, the agent will suffer from new amounts of pains v = f ( u ) in the current period.Let us discuss the aforementioned major alternative from both viewpoints of the efficiency and nondomination binary relations introduced in Definition 2.2. Efficiency binary relation (Definition 2.2 (ii)): should I change?
Yes, if ex ante v isbetter than y . In this case the advantages to move from y to v (change rather than stay) in thepayoff space is A ( y, v ) := y − v = f ( x ) − f ( u ) = A ( x, u ) ∈ Y . Consider an ex ante perception of amove. In this setting, the agent prefers to change before moving from x to u , rather than to stayat x , if the new amount of pains v = f ( u ) is lower than the old one y = f ( x ). This means that, exante, a given diminution of the most important pains compensates a not too large augmentationof the less important pains. The latter is equivalent to saying that A ( y, v ) = y − v ∈ D ( y ) ⇐⇒ A ( x, u ) = f ( x ) − f ( u ) ∈ D ( f ( x )), which means that there are ex ante advantages to move from y to v , i.e., from x to u . Appealing to Definition 2.2(ii)), this can be written as v ∈ y − D ( y ). Nondomination binary relation (Definition 2.2(i)): should I regret to have changed?
No, if ex post y is worse than v . Indeed, consider, the agent’s ex post perception of the samemove ( y, v ). In this new setting, the agent would prefer to change from y to v after moving, i.e.,to go from x to u rather than to stay at y provided that the new amount of pains v = f ( u ) isperceived ex post as lower than the old amount of pains y = f ( x ). This means that after movinga given diminution of the most important pains compensates a not too large augmentation of theless important pains. The latter is equivalent to saying that A ( y, v ) := y − v ∈ D ( v ) ⇐⇒ A ( x, u ) = f ( x ) − f ( u ) ∈ D ( f ( u )), which tells us that there are ex post advantages to move from y to v , i.e.,from x to u . Coming back to Definition 2.2(i), this can be written y ∈ v + D ( v ) meaning that theagent does not regret ex post to move from y to v . We refer the reader to [40] for more discussionsof the possible origins of regrets in the variational rationality context, where ex post regrets comefrom wrong ex ante evaluations of utility and costs of different moves.30 .2.3: When the resistance to move matters much . First let us offer an appropriate extensionof variable domination and efficiency structures in the variational rationality approach when the resistance to move matters . With respect to the binary relations in Definition 2.2, the VR approachcompares advantages to move to inconveniences/resistance to move and defines ex ante and ex post worthwhile moves that generalize ex ante and ex post advantageous moves , respectively. Inconveniences to move.
When the resistance to move (change rather than stay) matters ,the resistance to move generates the inconvenience to move rather than to stay defined by I ( y, v ) := C ( y, v ) − C ( y, y ) = C ( x, u ) − C ( x, x ) = I ( x, u ). In this formula, the amount C ( y, v ) = C ( x, u ) ∈ R m + ⊆ Y represents vectorial costs to move from “having done the bundle of activities x in theprevious period” to “being able to do and do the bundle of activities u in the current period”. Theamount C ( y, y ) = C ( x, x ) ∈ R m + ⊆ Y defines vectorial costs to stay at x . Observe that costs tomove are not symmetric, i.e., C ( v, y ) = C ( y, v ). This requires to use in modeling the framework of quasimetric spaces , which has been done in the variational theory developed in Section 4. In ourVR behavioral model we consider a specific vectorial case, where I ( x, u ) = √ εq ( x, u ) k with q ( x, u )being a given quasidistance; see [5, 6] for more discussions of such issues. Advantageous moves . When resistance to move does not matter , we define in our terminology(following an implicit construction of [40]) an advantageous move ( y, v ) from the viewpoint of r ∈ { y, v } by A ( y, v ) := y − v ∈ D ( r ) ⇐⇒ A ( x, u ) = f ( x ) − f ( u ) ∈ D ( r ). Worthwhile moves . Now we are ready to define, based on the binary relations from Def-inition 2.2, the new notion of worthwhile moves when the resistance to move matters . Namely,the worthwhile move ( y, v ) from the viewpoint of r ∈ { y = f ( x ) , v = f ( u ) , y = f ( x ) } is given by B ξ ( y, v ) := A ( y, v ) − ξ I ( y, v ) ∈ D ( r ). In our specific context of efficient and domination structures,the worthwhile balance is defined by B ξ ( y, v ) := A ( y, v ) − ξ I ( y, v ) = y − v − ξ I ( y, v ) , with ξ > . In [36, 37, 38], the reader can find some discussions on motivation of the resistance to move inother behavioral science settings. Note that the concepts from Definition 2.2 correspond to thecase where ξ := √ ε = 0. In what follows we consider even a more specific balance situation with I ( y, v ) = √ εq ( x, u ) k , i.e., B ξ ( y, v ) = f ( x ) − f ( u ) − √ εq ( x, u ) k, where ξ = √ ε ≥
0. Then, a move ( y, v ), which starts from the position y and goes to the position v , is worthwhile in the following senses: • Ex ante if we have B ξ ( y, v ) ∈ D ( y ) before moving, while choosing the viewpoint of the startingposition r = y = f ( x ) and the viewpoint of x (efficiency binary relation as in Definition 2.2(ii)). • Ex post if we have B ξ ( y, v ) ∈ D ( v ) after moving, while choosing the viewpoint of the final position r = v = f ( u ) and the viewpoint of u (nondomination binary relation as in Definition 2.2(i)).Observe finally that the move ( y = f ( x ) , v = f ( u )) is ex ante worthwhile if B ξ ( y , v ) = A ( y , v ) − ξ I ( y , v ) = f ( x ) − f ( u ) − √ εq ( x , u ) k ∈ D ( v ) , while choosing the initial viewpoint of r = y = f ( x ) and from viewpoint of x . The above discussions show that the behavioral model of human dynamics, which is describedin terms of the variational rationality approach, can be enclosed into the variational frameworkof vector optimization with variable domination structures in quasimetric spaces. Then, the new31ariational principles of Section 4 obtained in this general framework leads us to behavioral conclu-sions that can be interpreted as the existence of ex ante and ex post variational traps . The resultspresented below are direct consequences of the obtained variational principles, which are derivedin Theorems 4.7 and 4.14. Note that, besides the statements of the these theorems, their very proofs based on constructive generalized Picard sequences provide efficient dynamic procedures toapproach such traps, not only to establish their existence.To this end, we define in the framework of the VR approach the concept of variational trapsas follows. A given position x ∗ is a variational trap if this position is worthwhile to reach, but notworthwhile to leave. Using the definition of a worthwhile balance formulated in this section andthe results obtained in Theorems 4.7 and 4.14 allows us to arrive at the following conclusions: • Ex ante variational traps . It follows from the results of Theorem 4.7 with ξ := √ ε > (i) B ξ ( y = f ( x ) , v ∗ = f ( x ∗ )) = f ( x ) − f ( x ∗ ) − √ εq ( x , x ∗ ) k ∈ D ( f ( x )). (ii) B ξ ( v ∗ = f ( x ∗ ) , y = f ( x )) = f ( x ∗ ) − f ( x ) − √ εq ( x ∗ , x ) k ∈ D ( f ( x )).Note that condition (i) means that it is worthwhile to move from x to x ∗ , while condition(ii) tells us that it is not worthwhile to move away from x ∗ . The point of view that determinespreferences is the initial position ( x , r = f ( x )). This defines an ex ante variational trap as anefficiency binary relation from Definition 2.2(ii)), which gives us the ex ante motivation to movefrom x to x ∗ and then to stay at x ∗ . • Ex post variational traps . It follows from the results of Theorem 4.14 with ξ := √ ε > (i) B ξ ( y = f ( x ) , v ∗ = f ( x ∗ )) = f ( x ) − f ( x ∗ ) − √ εq ( x , x ∗ ) k ∈ D ( f ( x ∗ )). (ii) B ξ ( v ∗ = f ( x ∗ ) , y = f ( x )) = f ( x ∗ ) − f ( x ) − √ εq ( x ∗ , x ) k ∈ D ( f ( x )).As seen, condition (i) tells us that it is worthwhile to move from x to x ∗ , while condition (ii) meansthat it is not worthwhile to move away from x ∗ . The point of view that determines preferences inthis case for condition (i) is the final position ( x ∗ , r = f ( x ∗ )). On the other hand, for condition (ii)it is the position x with r = f ( x ) for each x away from x ∗ . This defines an ex post variational trapcorresponding to the nondomination binary relation from Definition 2.2(i), which excludes ex postregrets to move from x to x ∗ and then to stay at x ∗ .Thus Theorems 4.7 and 4.14 provide efficient conditions ensuring the existence of ex ante andex post variational traps in the variational rationality model of human dynamics. Acknowledgements . The research of the first author was initially conducted during his stayat the Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam. He wouldlike to thank the institute for hospitality and support.
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