Set-Valued Maps

Home , Absolutely convex set, Algebraic interior, Balanced set

Appendix A Set-Valued Maps

Let X and Y be two nonempty sets. A set-valued map or multivalued map or point- to-set map or multifunction T W X ! 2Y from X to Y is a map that associates with any x 2 X a subset T.x/ of Y;thesetT.x/ is called the image of x under T.Theset Dom.T/ Dfx 2 X W T.x/ ¤;gis called the domain of T. Actually, a set-valued map T is characterized by its graph, the subset of X Y deﬁned by

Graph.T/ Df.x; y/ W y 2 T.x/g:

Indeed, if A is a nonempty subset of the product space X Y, then the graph of a set-valued map T is deﬁned by

y 2 T.x/ if and only if .x; y/ 2 A:

The domain of T is the projection of Graph.F/ on X.Theimage of T is a subset of Y deﬁned by [ [ Im.T/ D T.x/ D T.x/: x2X x2Dom.T/

It is the projection of Graph.T/ on Y. A set-valued map T from X to Y is called strict if Dom.T/ D X, that is, if the image T.x/ is nonempty for all x 2 X.LetK be a nonempty subset of X and T be a strict set-valued map from K to Y.Itmaybe useful to extend it to the set-valued map TK from X to Y deﬁned by T.x/; when x 2 K; T .x/ D K ;; when x … K; whose domain Dom.TK/ is K.

Let T be a set-valued map from X to Y and K Â X, then we denote by TjK the restriction of T to K. Deﬁnition A.1 Let T W X ! 2Y be a set-valued map. For a nonempty subset A of X, we write [ T.A/ D T.x/: x2A

If A D;, we write T.;/ D;.ThesetT.A/ is called the image of A under the set-valued map T. Y Theorem A.1 Let fA˛g˛2 be a family of nonempty subsets of X and T W X ! 2 be a set-valued map.

(a) If A1 Â A2!then T.A1/ Â T.A2/; [ [ (b) T A˛ D T .A˛/; ˛2 ! ˛2 \ \ (c) T A˛ T .A˛/; ˛2 ˛2 (d) T .X n A1/ Ã T.X/ n T .A1/.

Deﬁnition A.2 Let T1 and T2 be two set-valued maps from X to Y.

•Theunion of T1 and T2 is a set-valued map .T1 [ T2/ from X to Y deﬁned by

.T1 [ T2/.x/ D T1.x/ [ T2.x/; for all x 2 X:

•Theintersection of T1 and T2 is a set-valued map .T1 \ T2/ from X to Y deﬁned by

.T1 \ T2/.x/ D T1.x/ \ T2.x/; for all x 2 X:

•TheCartesian product of T1 and T2 is a set-valued map .T1 T2/ from X to Y Y deﬁned by

.T1 T2/.x/ D T1.x/ T2.x/; for all x 2 X:

•IfT1 is a set-valued map from X to Y and T2 is another set-valued map from Y to Z, then the composition product of T2 by T1 is a set-valued map .T2 ı T1/ from X to Z deﬁned by

.T2 ı T1/.x/ D T2.T1.x//; for all x 2 X: A Set-Valued Maps 489

Theorem A.2 Let T1 and T2 be set-valued maps from X to Y and A be a nonempty subset of X. Then

(a) .T1 [ T2/.A/ D T1.A/ [ T2.A/; (b) .T1 \ T2/.A/ Â T1.A/ \ T2.A/; (c) .T1 T2/.A/ Â T1.A/ T2.A/; (d) .T2 ı T1/.A/ D T2.T1.A//: Deﬁnition A.3 If T is a set-valued map from X to Y, then the inverse T1 of T is deﬁned by

T1. y/ Dfx 2 X W y 2 T.x/g; for all y 2 Y

Further, let B be a subset of Y.Theupper inverse image T1.B/ and lower inverse 1 image TC .B/ of B under F are deﬁned by

T1.B/ Dfx 2 X W T.x/ \ B ¤;g and

1 TC .B/ Dfx 2 X W T.x/ Â Bg:

1 1 We also write T .;/ D;and TC .;/ D;. It is clear from the deﬁnition of inverse of T that .T1/1 D T and y 2 T.x/ if and only if x 2 T1. y/. We have the following relations between domain, graph and image of T and T1.

Dom.T1/ D Im.T/; Im.T1/ D Dom.T/ and

Graph.T1/ Df. y; x/ 2 Y X W .x; y/ 2 Graph.T/g:

Theorem A.3 Let fB˛g˛2 be a family of nonempty subsets of Y, A Â X and B Â Y. Let T W X ! 2Y be a set-valued map. 1 1 (a) If B1 Â B2,thenT .B1/ Â T .B2/; 1 (b) A TC .T.A//; . 1. // (c) B T TC B! ; [ [ 1 1 (d) TC B˛ TC .B˛/; ˛2 ! ˛2 \ \ 1 1 (e) TC B˛ D TC .B˛/; ˛2 ˛2 (f) T1.T.A// A; 490 A Set-Valued Maps ! \ \ 1 1 (g) T B˛ T .B˛/; ˛2 ! ˛2 [ [ 1 1 (h) T B˛ D T .B˛/. ˛2 ˛2

Y Theorem A.4 Let T1; T2 W X ! 2 be set-valued maps such that .T1 \ T2/.x/ ¤; for all x 2 X and B Â Y. Then 1 1 1 (a) .T1 [ T2/ .B/ D T1 .B/ [ T2 .B/; 1 1 1 (b) .T1 \ T2/ .B/ T1 .B/ \ T2 .B/; 1 1 1 (c) .T1 [ T2/C .B/ D T1C.B/ \ T2C.B/; 1 1 1 (d) .T1 \ T2/C .B/ T1C.B/ [ T2C.B/; Y Z Theorem A.5 Let T1 W X ! 2 and T2 W Y ! 2 be set-valued maps. Then for any set B Â Z, we have . /1 . / 1 1. / (a) T2 ı T1 C B D T1C T2C B ; 1 1 1 (b) .T2 ı T1/ .B/ D T1 T2 .B/ . Y Z Theorem A.6 Let T1 W X ! 2 and T2 W X ! 2 be set-valued maps. Then for any sets B Â Y and D Â Z, we have 1 1 1 (a) .T1 T2/C .B D/ D T1C.B/ \ T2C.D/; 1 1 1 (b) .T1 T2/ .B D/ D T1 .B/ \ T2 .D/. For further details and applications of set-valued maps, we refer to [1–8]andthe references therein. Appendix B Some Algebraic Concepts

Let K and D be nonempty subsets of a vector space X.Thealgebraic sum and algebraic difference of K and D are, respectively, deﬁned as

K C D Dfx C y W x 2 K and y 2 Dg and

K D Dfx y W x 2 K and y 2 Dg:

Let be any real number, then K is deﬁned as

K Dfx W x 2 Kg:

We pointed out that K C K D 2K is not true in general for any nonempty subset K of a vector space X. Deﬁnition B.1 Let X be a vector space. The space of all linear mappings from X to R is called algebraic dual of X and it is denoted by X0. Deﬁnition B.2 A subset K of a vector space X is called • balanced if it is nonempty and ˛K K for all ˛ 2 Œ1; 1; • absolutely convex if it is convex and balanced; • absorbing or absorbent if for each x 2 X, there exists >0such that x 2 K for all jjÄ. Note that an absorbing set contains the zero of X. Theorem B.1 Let X and Y be vector spaces and f W X ! Y be a linear map. (a) If K is a balanced subset of X, then f .K/ is balanced. (b) If K is absorbing and f is onto, then f .K/ is absorbing. (c) Inverse image under f of absorbing or balanced subsets of Y are absorbing or balanced, respectively, subset of X.

© Springer International Publishing AG 2018 491 Q.H. Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 492 B Some Algebraic Concepts

Deﬁnition B.3 Let K be a nonempty subset of a vector space X. • The set cor.K/ Dfy 2 K W for every x 2 X there is a >0N with y C x 2 C for all 2 Œ0; N g is called the algebraic interior of K (or the core of K). •ThesetK is called algebraic open if K D cor.K/. • The set of all elements of X which neither belong to cor.K/ nor to cor.X n K/ is called the algebraic boundary of K. •Anelementx 2 X is called linearly accessible from K if there exists y 2 K, y ¤ x, such that y C .1 /x 2 K for all 2 .0; 1. The union of K and the set of all linearly accessible elements from K is called the algebraic closure of K and it is denoted by

lin.K/ D K [fx 2 X W x is linearly accessible from Kg:

The set K is called algebraic closed if K D lin.K/. •ThesetK is called algebraic bounded if for every y 2 K and every x 2 K,there is a >0N such that y C x … K for all N . These algebraic notions have a special geometric meaning. Take the intersections of the set K with each straight line in the vector space X and consider these intersections as subsets of the real line R. Then the set K is algebraic open if these subsets are open; K is algebraic closed if these subsets are closed; and K is algebraic bounded if these subsets are bounded. Lemma B.1 ([9, Lemma 1.9]) Let K be a nonempty convex subset of a vector space X. (a) x 2 lin.K/,y2 cor.K/ )fy C .1 /x W 2 Œ0; 1gcor.K/; (b) cor.cor.K// D cor.K/; (c) cor.K/ and lin.K/ are convex; (c) If cor.K/ ¤;,thenlin .cor.K// D lin.K/ and cor .lin.K// D cor.K/ . Lemma B.2 ([9, Lemma 1.11]) Let K be a convex cone in a vector space X with a nonempty algebraic interior. Then (a) cor.K/ [f0g is a convex cone, (b) cor.K/ D K C cor.K/. Lemma B.3 A cone C in a vector space X is reproducing if cor.C/ ¤;. Proof Since cor.C/ is nonempty, we let x 2 cor.C/ and take any y 2 X. Then there is a >0N with x C N y 2 C implying 1 1 y 2 C x C C: N N

So, we get X C C and together with the trivial inclusion C C X, we obtain the assertion. ut B Some Algebraic Concepts 493

Given x and y in a vector space X, we denote by Œx; y and x; yŒ the closed and open line segments joining x and y, respectively. Deﬁnition B.4 Let K be a nonempty convex subset of a vector space X. A point x 2 K is said to be relative algebraic interior point of K if for any u 2 X such that x C u 2 K, there exists ">0such that x u; x C uŒ Â K. The set of all relative algebraic interior points of K is denoted by relint.K/. We note that if 0<˛<, then x ˛u; x C ˛uŒ Â x u; x C uŒ. Appendix C Topological Vector Spaces

Definition C.1 (Directed Set) Aset together with a reflexive and transitive ordering relation 4 such that every finite set of has an upper bound in (that is, for ˛; ˇ 2 ,thereisa 2 such that ˛ 4 and ˇ 4 ) is called a directed set. Definition C.2 Let X be a topological space and, and be any index sets.

(a)S A collection F DfO˛g˛2 of subsets of X is said to be a cover of X if ˛2 O˛ D X; If each member of F is an open set, then F is called an open cover of X; (b) A subcollection C of a cover F of X is said to be a subcover if C is itself a cover of X; If the number of members of C is finite, then C is called a finite subcover. (c) A collection C DfUˇgˇ2 of subsets of X is said to be a refinement of the cover F DfO˛g˛2 of X if C is an open cover of X and for each member Uˇ 2 C ,thereisO˛ 2 F such that Uˇ Â O˛. Note that an open subcover is a refinement, but a refinement is not necessarily an open cover.

Definition C.3 An open cover F DfO˛g˛2 of a topological space X is said to be locally finite if each point of X has a neighborhood which meets only finitely many O˛. Definition C.4 A Hausdorff space X is said to be paracompact if every open cover of X has a locally finite open refinement. Definition C.5 Let X be a topological space and f W X ! R.Thesupport of f is the set Supp. f / WD cl.fx 2 X W f .x/ ¤ 0g/. Since a finite cover is necessarily locally finite, it follows that every open cover of a compact space has locally finite open refinement.

© Springer International Publishing AG 2018 495 Q.H. Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 496 C Topological Vector Spaces

Deﬁnition C.6 (Partition of Unity) Let X be a topological space. A family fˇgi2I of continuous functions deﬁned from X into Œ0; 1/ is called a partition of unity associated to an open cover fUigi2I of X if

(i)X for each x 2 X, ˇi.x/ ¤;for only finitely many ˇi; (ii) ˇi.x/ D 1 for all x 2 X. i2I Theorem C.1 ([10, pp. 68]) A Hausdorff space X is paracompact if and only if every open cover of X has a continuous locally finite partition of unity. We note that every compact Hausdorff space is paracompact and every metrizable space is paracompact. Definition C.7 A subset in a topological space is precompact (or relatively compact) if its closure is compact. Note that every element in Rn has a precompact neighborhood. Definition C.8 ([11, 12]) Let X be a Hausdorff topological vector space and L be a lattice with least one minimal element, denoted by 0. A mapping ˚ W 2X ! L is said to be a measure of noncompactness provided that the following conditions hold for all M; N 2 2X: (i) ˚.M/ D 0 if and only if M is precompact; (ii) ˚.cl.M// D ˚.M/; (iii) ˚.M [ N/ D max f˚.M/; ˚.N/g : It follows from condition (iii) that if M Â N,then˚.M/ Ä ˚.N/.

Deﬁnition C.9 A net in a topological space X is a mapping ˛ 7! x˛ from a directed set into X; we often write fx˛g˛2, fx˛ W ˛ 2 g,orsimplyfx˛g. We say that a net fx˛g˛2 converges to x 2 X if for any neighborhood V of x, there exists an ˛ (an index ˛) such that xˇ 2 V for all ˇ < ˛. The point x is called a limit of the net fx˛g. When a net fx˛g converges to a point x, we denote it by x˛ ! x. We say that x is a cluster point of the net fx˛g if for any neighborhood V of x and for any index ˛, there exists ˇ < ˛ such that xˇ 2 V. Deﬁnition C.10 A vector space X with a topology T under which the mappings

.x; y/ 7! x C y from X X ! X and

.˛; x/ 7! ˛x from R X ! X are continuous, is called a topological vector space. C Topological Vector Spaces 497

Theorem C.2 Let X be a vector space and B be a family of subsets of X such that the following conditions hold. (i) Each U 2 B is balanced and absorbing; (ii) For any given U1; U2 2 B, there exists U 2 B such that U U1 \ U2; (iii) For any given U 2 B, there exists V 2 B such that V C V U. Then there is a unique topology T on X such that .X; T / is a topological vector space and B is a neighborhood base (or local base) at 0. Theorem C.3 Let X be a topological vector space. (a) The closure of a balanced set A Â X is balanced. (b) The closure of a convex set A Â X is convex. (c) The interior of a convex set A Â X is convex. Theorem C.4 Let X be a topological vector space and K be a subset of X. (a) If K is balanced and 0 2 int.K/, then its interior is also balanced. (b) Any superset of an absorbing set is absorbing. So if K is absorbing then so is its closure. The interior of an absorbing set is not generally absorbing; (c) If K is open, then so is its convex hull. Theorem C.5 Let X be a topological vector space and U be a neighborhood of its zero element 0.ThenU is a neighborhood of 0 for all nonzero real . Deﬁnition C.11 A topological vector space X is said to be locally bounded if there is a bounded neighborhood of 0. Trivially, every normed space is locally bounded. A well-known example of a locally bounded topological vector space is Lp for 0

cl.K/ DfK C U W U is a neighborhood of 0g:

In particular, cl.K/ K C U for any neighborhood U of 0. 1;2;:::; Proposition C.1 ([14, Proposition 15]) For each i D m, let KSi be a com- m pact convex subset of a Hausdorff topological vector space X. Then co iD1 Ai is compact. Corollary C.1 ([14, Corollary 1]) In a Hausdorff topological vector space, the convex hull of a ﬁnite set is compact. 498 C Topological Vector Spaces

Definition C.12 A topological vector space X with its topology T is said to be locally convex if there is a neighborhood base (local base) at zero consisting of convex sets. The topology T is called locally convex. Theorem C.8 Let T be a locally convex topology on X. Then there exists a local base B whose members have the following properties: (i) Every member of B is absolutely convex and absorbing; (ii) If U 2 B and >0,thenU 2 B. Conversely, if B is a filter base on X which satisfies conditions (i) and (ii), then there exists a unique locally convex topology T on X such that B is a local base at zero for T . Definition C.13 ([15, pp. 188]) Let X and Y be vector spaces. A bilinear functional or bilinear form B W X Y ! R, .x; y/ 7! B.x; y/ is a map which is linear in either argument when the other is held fixed. A pairing or pair is an ordered pair .X; Y/ of linear spaces together with a fixed bilinear functional B. Usually, B.x; y/ will be denoted by hx; yi. If X is a linear space and X0 its algebraic dual then the natural pairing of X and X0 is that arising from the (natural or canonical) bilinear functional on X X0,which sends .x; x0/ into x0.x/,thatis,hx; x0iDx0.x/. If X and Y are any two paired vector spaces, then we also have the natural pairing in the following sense: If y isanyfixedelementofY, then the map y0 W X ! R, x 7!hx; yi is obviously a linear functional on X,thatis,y0 2 X0. It is clear that

hx; yiD0; for all x 2 X implies y D 0 equivalently,

y ¤ 0 implies that there is some x 2 X such that hx; yi¤0:

Definition C.14 Let X be a vector space. A semi-norm on X is a function p W X ! R such that the following conditions hold: (i) p.x/ 0 for all x 2 X; (ii) p.x/ Djjp.x/ for all x 2 X and 2 R; (iii) p.x C y/ Ä p.x/ C p. y/ for all x; y 2 X. Theorem C.9 ([14, Corollary, p. TVS II.24]) Let X be a topological vector space with its topology T .ThenT is defined by a set of semi-norms if and only if T is locally convex. Definition C.15 ([15, pp. 189]) Let X and Y be vector spaces. The map x 7! jhx; yij D py.x/ determines a semi-norm on X for all y 2 Y. The topology generated by the family of semi-norms fpy W y 2 Yg is the weakest topology on X and it C Topological Vector Spaces 499 is called weak topology on X determined by the pair .X; Y/, and it is denoted by .X; Y/. Remark C.1 Clearly, .X; Y/ is a locally convex topology on X and also it is a Hausdorff topology. Let X and Y be Hausdorff topological vector spaces and L.X; Y/ denote the family of continuous linear functions from X to Y.Let be the familyS of bounded subsets of X whose union is total in X, that is, the linear hull of fU W U 2 g is dense in X.LetB be a neighborhood base of 0 in Y,where0 is the zero element of Y.WhenU runs through , V through B, the family ( ) [ M.U; V/ D 2 L.X; Y/ W h ;xiÂV x2U is a neighborhood base of 0 in L.X; Y/ for a unique translation-invariant topology, called the topology of uniform convergence on the sets U 2 , or, briefly, the - topology (see [16, pp. 79–80]). Lemma C.1 ([16]) Let X and Y be Hausdorff topological vector spaces and L.X; Y/ be the topological vector space under the -topology. Then the bilinear mapping h:; :iWL.X; Y/ X ! Y is continuous on L.X; Y/ X. We now present an open mapping theorem due to Brezis [17]. Theorem C.10 ([17, Théorème II.5]) Let X and Y be two Banach spaces and T W X ! Y be a continuous linear surjective mapping. Then there exists a constant c >0such that BcŒ0 T.B1Œ0/,whereBcŒ0 denotes the closed ball of radius c around 0 in Y and B1Œ0 is the closed unit ball in X. Remark C.2 From Theorem C.10, we see that T W X ! Y is an open mapping. Indeed, let U be an open subset of X. We show that T.U/ is open. Let y 2 T.U/, where y D T.x/ for some x 2 U.Letr >0such that BrŒx U,i.e.,x C BrŒ0 U. Then we have y C T.BrŒ0/ T.U/.DuetoTheoremC.10, we obtain BrcŒ0 T.BrŒ0/, and consequently, BrcŒy T.U/. Theorem C.11 ([18, Chapter 6, Theorem 1.1]) If X and Y are Banach spaces and T W X ! Y is a linear operator, then T is bounded if and only if it is continuous from the weak topology of X to the weak topology of Y. Definition C.16 Let .X; d/ be a metric space and A be a nonempty subset of X.The diameter of A, denoted by ı.A/,isdefinedas

ı.A/ D supfd.x; y/ W x; y 2 Ag:

We close this subsection by presenting the following Cantor’s intersection theorem. Theorem C.12 (Cantor’s Intersection Theorem) Let .X; d/ be a complete metric space and fAmg be a decreasing sequence (that is, AmC1 Â Am) of nonempty closed 500 C Topological Vector Spaces subsets of X such that the diameter of Am ı.Am/ ! 0 as m !1. Then the T1 intersection Am contains exactly one point. mD1

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B-pseudomonotone map C-semicompact set, 98 weighted, 272 C-strictly convex set, 160 C-h-convex function, 287 C-subconvex-like function, 104 strictly, 288 C-upper hemicontinuous function, 124 C-h-pseudoconvex function, 288 C-upper semicontinuous function, 117 strongly, 288 C-upper sign continuous function, 278, 352 weakly, 288 Cx-convex function, 178, 254 C-bi-pseudomonotone function, 373 strictly, 178, 254 C-bounded set, 96 Cx-monotone operator, 233 C-closed set, 96 Cx-pseudoconvex function, 178, 254 C-compact set, 98 strongly, 178, 254 C-complete set, 101 weakly, 178, 254 C-continuous function, 117 Cx-pseudomonotone operator, 233 C-convex function, 102 strongly, 233 C-convex-like function, 104 weakly, 233 C-diagonally concave function, 362 Cx-pseudomonotoneC operator, 236 C-diagonally convex function, 362 Cx-upper sign continuous map, 228 C-hemicontinuous function, 124 strongly, 227 C-lower hemicontinuous function, 124 weakly, 228 C-lower semicontinuous function, 117 ˚-condensing map, 47 C-monotone bifunction, 343 -transfer lower semicontinuous function, 71 C-properly subodd function, 278 -transfer upper semicontinuous function, 71 C-pseudocontinuous function, 117 H -continuous set-valued map, 42 C-pseudoconvex function, 110 H -hemicontinuous map, 305 C-pseudomonotone bifunction, 343, 348 "-constraint method, 195 maximal, 344 "-equilibrium point, 382 strictly, 343 "-generalized strong vector equilibrium C-pseudomonotone function, 277 problem, 473 strongly, 277 "-weak vector equilibrium problem, 390 weakly, 278 h-vector variational inequality problem, 276 C-pseudomonotoneC function, 284 Minty, 276 C-quasi-strictly convex set, 160 Minty strong, 276 C-quasiconcave-like function, 112 Minty weak, 276 C-quasiconvex function, 103 strong, 276 C-quasimonotone bifunction, 344 weak, 276

© Springer International Publishing AG 2018 505 Q.H. Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 506 Index v-coercive condition, 241 convex set, 2 v-coercive condition C1, 241 core of a set, 492 v-coercive operator, 245 correct cone, 9 weakly, 245 v-coercive set-valued map, 442 v-hemicontinuous operator, 228 Daniell cone, 100 decision (variable) vector, 79 decision variable space, 79 absolutely convex set, 491 diagonally quasiconvex function, 70 absorbing set, 491 -generalized, 70 acute cone, 9 Dini directional derivative, 34 additive dual vector equilibrium problem, 453 lower, 34 additive duality, 453 upper, 34 affine combination, 3 directed set, 495 affine function, 20 direction of recession, 13 affine hull, 5 directional derivative, 33 affine set, 2 domination factor, 86 algebraic boundary, 492 domination structure, 86 algebraic bounded set, 492 downward directed function, 113 algebraic closed set, 492 downward directed set, 113 algebraic closure, 492 dual cone, 12 algebraic interior, 492 dual generalized weak vector equilibrium algebraic open set, 492 problem, 449 approximating net, 417 dual implicit weak vector variational problem, arc-concave-like function, 114 365 asymptotic cone, 14 duality operator, 450 balanced set, 491 efficient element, 144 base, 10 efficient solution, 155 bilinear functional, 498 dominated, 175 binary relation, 82 dominated properly in the sense of Henig, Bouligand tangent cone, 17 175 boundedly order complete space, 100 dominated strong, 175 dominated weakly, 175 global properly in the sense of Henig, 166 Cantor’s intersection theorem, 499 local properly in the sense of Henig, 166 Clarke directional derivative, 37 properly in the sense of Benson, 163 Clarke generalized subdifferential, 37 properly in the sense of Borwein, 162 closed cone, 8 properly in the sense of Geoffrion, 167 coercive operator, 245 efficient solution set, 155 weakly, 245 Ekeland variational principle, 378 completely continuous map, 239, 326 vectorial form, 378 concave function, 21 epigraph, 21, 109 cone, 2 equilibrium problem, 65 cone combination, 3 dual, 67 conic hull, 5 Minty, 67 contingent cone, 17 equivalence relation, 83 continuous set-valued map, 38 explicitly C-quasiconvex function, 104 convex combination, 3 externally stable function, 140 convex cone, 2 convex function, 20 strictly, 21 feasible objective region, 79 convex hull, 5 feasible region, 79 Index 507

finite concave-like function, 113 ideal minimal element, 144 fixed point, 43 implicit variational problem, 68 fixed point problem, 51 implicit weak vector variational problem, 342 Fréchet derivative, 34 improperly efficient solution in the sense of Geoffrion, 167 Gâteaux derivative, 34 gap function, 367, 368 Jameson lemma, 12 for vector equilibrium problems, 369 Jensen’s inequality, 22 generalized Cx-monotone set-valued map, 312 strongly, 312 KKM-map, 43 generalized C -pseudomonotone set-valued x Kneser minimax theorem, 58 map, 313 strongly, 312 generalized Cx-quasimonotone set-valued map, L-condition, 239 325 line segment, 2 generalized Cx-upper sign continuous map, linear combination, 3 303 linear function, 20 strongly, 302 linear hull, 5 weakly, 303 linear order, 83 generalized d-coercive condition, 318 linear scalarization, 182 weakly, 318 linearly accessible element, 492 generalized v-coercive condition, 317, 323 Lipschitz continuous function, 26 weakly, 317, 323 locally bounded set, 497 generalized v-hemicontinuous map, 304 Locally Lipschitz function, 25 generalized complementarity problem, 60, 248 lower bound, 87 vector, 248, 324 lower semicontinuous function, 28 generalized duality, 450 lower semicontinuous set-valued map, 38 generalized KKM map, 44 generalized Minty variational inequality problem, 61 maximal element, 87, 144 weak, 61 measure of noncompactness, 496 generalized variational inequality problem, 58 minimal element, 87, 144 generalized vector equilibrium problem, 429, minimal solution, 155 431 properly in the sense of Geoffrion, 167 abstract, 431 minimization problem, 66 strong, 431 Minty lemma, 53 weak, 430 Minty variational inequality, 51 generalized vector variational inequality Minty vector equilibrium problem, 340 problem strong, 340 Minty, 300 weak, 340 Stampacchia, 300 monotone operator, 30 strictly, 30 strongly, 30 Hausdorff metric, 42 weighted, 269 hemicontinuous function, 53 multicriteria optimization problem, 79 lower, 52 multiobjective optimization problem, 79 upper, 52 multiplicative dual problem, 454 weighted, 269 multiplicative duality, 454 hybrid method, 199 multivalued map, 487 ideal efficient element, 144 Nadler theorem, 42 ideal efficient solution, 160 Nash equilibrium point, 66 ideal maximal element, 144 Nash equilibrium problem, 66 508 Index natural C-quasiconvex function, 103 pseudomonotone bifunction, 36 natural pairing, 498 pseudomonotone map, 31 net, 496 strictly, 31 nondominated set, 155, 159 nonlinear complementarity problem, 50 nonlinear scalarization, 192 quasi-order, 83 nonlinear scalarization function, 128, 133 quasiconvex function, 26 nonsmooth variational inequalities, 54 semistrictly, 26 nontrivial cone, 8 strictly, 26 quasimonotone map, 31 semistrictly, 32 objective function, 79 strictly, 32 objective space, 79 open lower section of a set, 46 ordered structure, 83 radially semicontinuous function, 53 lower, 34, 52 upper, 34, 52 pairing, 498 regular saddle point problem, 341 paracompact space, 495 relative algebraic interior point, 493 parametric strong vector equilibrium problem, relative boundary, 7 411 relative interior, 6 parametric weak vector equilibrium problem, relatively compact set, 496 401 relatively open set, 7 parametrically well-posed problems, 417 reproducing cone, 8 unique, 418 Pareto efficient solution, 155 Pareto optimal solution, 155 saddle point problem, 66 properly in the sense of Geoffrion, 167 selection, 325 Pareto solution continuous, 325 dominated, 175 set-valued bifunction dominated properly in the sense of Henig, .H/-C-pseudomonotone, 454 175 C-pseudomonotone, 440 dominated strong, 175 C-quasimonotone, 445 dominated weakly, 175 G-C-pseudomonotone, 451 partial order, 83 m.H/-C-pseudomonotone, 455 partition of unity, 496 maximal .H/-C-pseudomonotone, 454 pointed cone, 8 maximal C-pseudomonotone, 440 polar cone maximal m.H/-C-pseudomonotone, 455 strict, 12 set-valued fixed point problem, 60 positively homogeneous function, 20 set-valued inclusion problem, 59 precompact set, 496 set-valued Lipschitz map, 43 preorder, 83 set-valued map, 487 proper C-quasimonotone mapping, 286 C-continuous, 125 strongly, 286 C-convex, 115 weakly, 286 C-lower semicontinuous, 125 proper cone, 8 C-proper quasiconcave, 115 properly C-quasiconvex function, 103 C-quasiconcave, 114 properly efficient element, 147 C-quasiconvex, 115 properly efficient solution in the sense of C-quasiconvex-like, 115 Kuhn-Tucker, 169 C-upper semicontinuous, 125 properly maximal element, 147 H -hemicontinuous, 42 properly minimal element, 147 u-hemicontinuous, 42 pseudoconvex function, 29 completely semicontinuous, 327 strictly, 29 explicitly ı-C-quasiconvex, 115 pseudolinear function, 29 explicitly C-quasiconvex-like, 116 Index 509

generalized hemicontinuous, 62 variational inequality generalized pseudomonotone, 63 generalized, 58 generalized weakly pseudomonotone, 63 weak generalized, 58 inverse, 489 variational inequality problem, 48 properly C-quasiconvex, 115 variational principle, 367, 370, 371 strictly C-proper quasiconcave, 115 for Minty weak vector equilibrium strictly C-quasiconcave, 114 problems, 373 strongly semicontinuous, 327 for weak vector equilibrium problems, 370 weakly lower semicontinuous, 436 for WVEP. f ; h/, 375 solid cone, 8 vector biconjugate function, 139, 365 starshaped set, 18 vector conjugate function, 139, 365 strict order, 83 vector equilibrium problem, 339, 340 strict partial order, 83 dual, 340 strict properly C-quasiconvex function, 103 Minty, 340 strictly C-convex function, 102 strong, 340 strictly C-quasiconcave-like function, 112 weak, 340 strictly C-quasiconvex function, 103 vector optimization problem, 79 strictly convex body, 225 vector saddle point problem, 341 strong C-pseudomonotone bifunction, 352 vector variational inequality problem, 223 strong C-quasimonotone bifunction, 352 Minty, 226 strong efficient element, 144 Minty strong, 226 strong maximal element, 144 Minty weak, 226 strong minimal element, 144 perturbed, 229 strong order, 83 perturbed strong, 229 strong properly C-quasimonotone bifunction, perurbed weak, 229 353 strong, 223 strongly C-complete set, 101 weak, 224 strongly C-diagonally concave function, 363 strongly C-diagonally convex function, 362 weak order, 83 strongly v-coercive condition C1, 240 weak subdifferential of a vector-valued strongly v-coercive condition C2, 241 function, 139 strongly efficient set, 160 weak subgradient of a vector-valued function, strongly efficient solution, 155, 160 139 strongly nondominated set, 160 weakly v-coercive condition C1, 241 strongly nonlinear variational inequality weakly v-coercive condition C2, 241 problem, 68 weakly efficient element, 147, 148 strongly nonlinear weak vector variational weakly efficient set, 159 inequality problem, 343 weakly efficient solution, 155, 159 subodd function, 20 weakly maximal element, 148 subspace, 2 weakly minimal element, 147, 148 support of a function, 495 weakly minimal solution, 159 weakly Pareto efficient solution, 155 tangent, 17 weakly Pareto optimal solution, 159 topological vector space, 496 Weierstrass theorem, 96 locally convex, 498 vectorial form, 384 total preorder, 84 weighted pseudomonotone operator, 269 transfer closed-valued map, 45 maximal, 269 transfer open-valued map, 45 weighted variational inequality problem, 266 Minty, 268 well-ordered, 83 upper bound, 87 upper semicontinuous function, 28 upper semicontinuous set-valued map, 38 Zermelo’s theorem, 83 upper sign continuous bifunction, 54 Zorn’s lemma, 87