Wetnam Joumal of Mathematics 26:'2(199D 95-110 V Ie trn a nn ] oiLrrrrrnLarlL o,lf MtA\If ]HtlENt7\T ]t C SS o Springer-Verlag1998
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GeneralizedConvexity and SomeApplications to Vector Optimization*
Dinh The Luc Institute of Mathematics, P.O. Box 631, Bo Ho 10000, Hanoi, Vietnam
Received January 25, 1998
Abstract. In this paper we present an overview of recent developmentson the characterizations of convex and generalized convex functions via the nonsmooth analysis approach. Generalized convex vector functions are also consideredtogether with their applicationsto vector optimization.
1. Introduction
Convex sets and convex functions have been studied for some time by Hdlder [19], Jensen[20], Minkowski [39] and many others. Due to the works of Fenchel, Moreau, Rockafellar in the 1960sand 1970s,convex analysis became one of the most beautiful and most developed branches of mathematics. It has a wide range of applications including optimization, operations resea.rch,economics, engineering, etc. However, severalpractical models involve functions which arenot exactly convex,but sharecertain properties of convex functions. These functions are a modification or generalizationof convex functions. The first generalization is probably due to de Finetti (1949) who introduced the notion of quasiconvexity.Other types of generalized convex functions were later developedin the works of Thy [45], Hanson [7], Mangasarian[37], Ponstein [41], Karamardianl2ll, Ortega-Rheinboldt [40], Avriel-Diewert-schaible-Zang[2]. This paper is an overview of recent developments on the study of convex and generalizedconvex functions. Our emphasisis on characterizationsofthese functions by the nonsmoothanalysis approach, namely, by subdifferential and directional derivatives. The second part of this overview is to study vector functions, mainly convex vector functions. We shall seethat presently,the study of generalizedconvex vector functions is quite far from being satisfactory.There still remain open problems in relation to the
* Invited lecture delivered at the XXI AMASES Congress,Rome, 10-13 September,1997. 96 Dinh Thel;uc
definitions ofgeneralized convex vector functions, their structureand characterizations. Nevertheless,many resultshave alreadybeen obtainedin the applicationsof generalized convex vector functions to multi-objective optimization. Becauseof the surveycharacter of the paper,proofs are not provided except for those of unpublishedresults.
2. Part A: Generalized Convex Functions
2.1. Most knportant GeneralizedConvex Functions
Let A be a subsetof a finite-dimensional spaceR". It is said to be convex if it.contains the whole interval linking any two points of A, that is, ),a I (l - ),)b e A whenever a, b e A and,l. e (0, 1). In this section, / is assumedto be a real function defined on a nonempty convex set A c Rn.
2.1.l. ConvexFunctions
Definition l. Thefunctionf issaidtobeconvexif,foreveryx,y € Aand), e (o, l), one has f ().x+ (1- r)y) < Lf (x)+ (l - L)f (y). (t)
If inequality (l) holds strictly for every x * y, we say that f is strictly convexon A.
Convex functions are a direct generalizationof affine functions because,for the latter, (1) becomes an equality. convex functions possessmany interesting properties. The readeris referred to Rockafellar l42l for a complete study of thesefunctions. Let us recall some properties that will be neededin understandinggeneralized convex functions. Rememberthat the epigraph of / is the set
epif ::{(x,a) e Rn x R:.x € A, f (x) lev(f;a) :- {x e A: f(x) < a}. Proposition 2. Thefunction f is convex if and only if one of thefollowing conditions holds: (i) epi/ is convex; (ii) lev(/ i€; a)isconvexforalla e Randall{ € R"wherethesumfunctionf -t€ is definedby(f + 6)(x) : f(x) + (€,x). It is worth noting that according to Proposition 2(ii), convex functions have convex level sets; however, the converseis not true. A generalized convexity will be defined to meet this converse.The next result of Fenchel [13] and Mangasarian I38l deals with differentiable functions. GeneralizedCowexity to VectorOptimization 97 Proposition 3. AssumeA is open and f is dffirentiable on A. Then f is convex(resp. strictly cowex) if andonly if,foreachx, ! € A, x # y, onehas ftf(x)-vf0),r-y)>0 (resp.(y/(x) - V,f0), r - y) > 0). For nondifferentiable functions, one uses the subdifferential to characteize convexitv. We recall that the subdifferential of a function f at x e A is defined by lf(x) :: {E e R" : f(y) - f(x) > (E, y -x) forally e A}. For convex functions, the subdifferential is nonvoid at every relative interior point of A. Proposition 4. Assume af @) is nonemptyfor x e A. Then f is convex (resp. strictly convex)ifandonlyifforeachx,y € A, x I y andg e 0f (x), n eAf 0), onehas (€-ry,r-y)>0, (resp.(€-n,x-y) >0). The above result is due to Rockafellar [42]. rt has been recently extended by correa-Jofr6-Thibaudt and by Luc (see [29]) for the case where / is defined on a Banach spaceand 0/ is defined in a more generalform, including Rockafellar-Clarke's subdifferential, Michel-Penot's subdifferential, etc. Now, we turn to a subclassof strictly convex functions which is frequently used in optimization and economics. Definition 5. f is said to be strongly cotvex if there existsot > 0 such that f Q,x(I -I)y) Equivalently, / is strongly convex if there is cv > 0 such that the difference function f (x) - allxll2 is convex. Using Proposition 4, we obtain the following characteization of strong convexity. Proposition 6. Assume0f (x) is nonemptyfor x e A. Then f is strongly convexif and only if thereexists a > 0 suchthat, for eachx, y e A, x I y and { e Of (x), n e Of (y), one has G - q, x - y) - allx -)ll2 > 0. 2.1.2. QuasiconvexFunctions Definition 7. f is saidtobequasicowexif,foreveryx,y € A, x I y and), € (0, 1), one has - f ().x + (1 r)y) < max{f (x), f (y)}. (2) If (2) holds strictly, then f is said to be strictly quasiconvex.If (2) hotds strictly for those x, y with f (x) * f (y), then f is said to be semistrictly quasiconvex. 98 Dinh The Luc It is evident that strict quasiconvexityimplies semistnct Qua-si6sa1-gxrq'andquasicon- vexity. Semistrict quasiconvexityimplies quasiconrerin'prorrded rhe funcuon is lower semicontinuous. Remember that the level surface of f at cy € R is the set surt-rr: at '.: 11 1 I '. f(x): q\. Proposition 8. Wehave thefollowing: (i) / ts quasiconvexif and only if lev (f , u) is com'ex_for all cr = R'. (11) Assumef is continuousand A is strictly com'er. Then .f is semisrricrk quasiconvex ifandonlyiflev(f;u)isconvexandeitherler-(/:al:surt'r.f:ratorsurft_f:a)ls in the boundaryoflev(f; a); (111)Assume f is continuous.Then f is strictly quasicomexi.f and onh if levt/:cv) ls strictly colrvexand surf(f ; a) consistsof exteme poinrs o.fletr ,f : q t onh. Proof of (iii). It is alreadyknown [2]that lev(/; a) is strictlv conve\ ii -f is continuous, strictly quasiconvex.If surf(/; a) containsa non-extremepoint of levr-f : a r. then there arex,y, z elev(f;cv)suchthata: f(z) >_mar{-fr.rr../rrr}.: € (.r._r').On the by strict quasiconvexity, < mar{-fr.rt..f,)'}. other hand, "f(r) a contradiction. Conversely,if lev(/; a) is strictly convex, cy € R. then it follou's rn panicular that / is quasiconvex.Now, if it is not strictly quasiconvex.,*'e can find .r. -r'= -{ and : e (x, y) such that f (z) : max{/(x),,f(y)}, say, equal to .f(.rt. Then br quasiconvexity, f(z): f(r): f(x)forallx e [2,"r].Thismeansthatsurfr_f:_fr.trrdoesnotonly consistofextremepointsof lev(f; f (x)). Theproof iscomplete. r Next is a characterizationof quasiconvexity in the differentiable case. Proposition 9.fll Assumef is dffirentiable on an open com ei set ,1,C Rn. Then f is (x) < (t) -l') < quasiconvexifandonlyif,foreveryx,y € A, "f f impliesT-f()')(.r 0. 2.1.3. PseudoconvexFunctions We use Ortega-Rheinbold's definitions where no differentiabi-liq'assumptionis made. Definition 10. / ls said to be pseudoconvexif, for eve^' x. y e A and /(.r) < /(y), there exists B > O such that f().x* (1-r)y) =f(y)-x(l-L)p (3) for all.r, e (0, 1). f issaidtobestrictlypseudoconvexif,foreveryx,y,eA. .x f t and f (x) = f (y), there exists B as above. Observethat convexity implies pseudoconvexity,which implies semistrict quasicon- vexity, and strict convexity implies strict pseudoconvexity. Proposition ll. For dffirentiable functions, the above definition is equivalent to the classical one of Mangasarian [37]: f ispseudocowexif,forx,ye A, f (x) < f (y)impliesVf 0)(x -y) < O,and f isstrictpseudocorNexif,forx,ye A, x +y, f (x) S f $)impliesVf 0)@-y) < 0. Generalized.Convexity to VectorOptimization 99 For pseudoconvexfunctions, no known criteria exist using epigraph or level sets. 2.2. Extrema of GeneralizedConvex Functions For convex functions, it is known [13] that any local minimum is global. This property is very important in optimization since most existing theoretical and computational methods allow us to find local minima only. The concQptof quasiconvexityis developed under the influence of this property, namely, we have the following proposition. Proposition 12. Let f be a real function on A c R". Then (i) If f is quasicorwex,then every strict local minimum is strictly global minimum; (11) If f is continuous, then f is semistrictly quasicowex if and only if every local minimum is global minimum; (iii) If f is strictly quasiconvex,then it cannot have a local maximum and it may have a uniqueminimum. Actually, we can characterize quasiconvexity using extrema properties of the function on line segments. In the next proposition, "strict maximum" on [x, y] meansthat the point is maximum on [x, y] and unique in some neighborhood on [x, y]. Proposition 13. Let f be a real function on a convexset A C R". Then thefollowing assertionshold true: (i) / ls quasicorwex if and only if its restriction on any closed segment attains its maximum at a segmentend; (ii) Assume f is lower semicontinuouson A, f is semistrictly quasiconvexif and only if its restriction on any closed segmentattains a strict maximum at a segmentend wheneverit is not constant at that point. (iii) / ,s strictly quasicowex if and only if its restriction on any closed segmentattains a strict maximum at a segmentend. Proof. The first and third assertions are directly obtained by using definitions. We show the second assertion. Assume / is semistrictly quasiconvex.Let [x, y] be any segmentin A with x f y. Assume further that / is not constanton this segment,that is, f (c) * /(y) for somec e [r, y]. lt f (x) * f (y),say /(x) < f (y),then by definition, f Q"x 1- (1 - f)y) < f (y) for ). e (0, 1) which shows that y is a sfrict maximum of / on [x, y]. Because / is lower semicontinuousand semistrictly quasiconvex,/ is quasiconvex.Hence, lf f (x) : f (y), onehas f (c) < /(y). Again, by definition, fQ"x+ (1-.r.)c) < f(x), f(Ly+ (1-^)c) Another important property of convex functions is that a point x e A is minimum if (x) and only 1t v f @) : 0 when / is differentiable, or o e 3f in a more general case. The concept ofpseudoconvexity refers to this property' Proposition 14.111,37l Let f be dffirentiable on the open convexset A C R". If f is (strictly) pseudocorwexand a f @) : o, then x is a (strict) global minimum of f on (strictly) A. Moreover if f is continuously dffirentiable on A, then f is pseudoconvex if and only if v f @) : o implies x is a (strict) local minimum of f ' 2.3. Nonsmooth GeneralizedConvex Functions 2.3.1. Subdifferential Operator and Support Operator Denote by S(m,n ) the set of set-valuedmaps from R* to Rn and by B(m x n) the set of single-valued functions from R- x Ru to the extendedreal line R U {*oo}, which are positively homogeneousin the secondvariable. Let us define an operator denotedby o and called support operator from 5(m, n) to B(m x n) and another operator denoted by 0 and called subdifferential operator from B(m x n) to S(m,n) asfollows:For M e 3(m, n), x e R^, u e Rn oM(x, r) : )) ,suP ,(r, andforheB(mxn),xeR^ \h(x) : {u e R' : h(x,y) > (u,y) for all y € Rn}. It can be seenthat M(x) c D(oM)(x), in addition, equality holds provided M(x) is closed convex and h(x , y) >- o (?h)(x, Y), in additon, equality holds if h is sublinearlower semicontinuousin y. Examples. o Let f be a differentiable function on R'. Then the directional derivative of / can be seen as a function of two variables f'(x,u): (Yf(x),u). Then0//(.x)-{u: lu,u) < (V"f(x),u)fotallue R"} : {v"f(x)}' that is. the subdifferential of the directional derivative f'(*, u) at r only consistsof the (n gradient V f @) .On the other hand, x r-> {V f @)} is an element of S , n) . o(yf)(x,u): sup (u,Y) ye{V.f (x)} : {af@),u) GeneralizedConvexity to VectorOptimization 101 that is, the support of y/ is the directional derivative of /. o Let f be a convex function on R'. The directional derivative is defined by f'(x:u) : y,t-;- f(x*tu)-f(x) Then 0/' is exactly convex analysis subdifferential of 0, f: - - 0"f(x):- {u e R" : (u,y x) < f(y) f(x) forally € R'}. cLet f be a locally Lipschitz function on R". The Clarke directional derivativeis defined by n f6 * tu) - f(x) fu(-r:u\-lim"" t J1" Then the subdifferential 3/0 is exactly Clarke's subdifferential (denotedby 0a fl of f . 2.3.2. GeneralizedMonotonicity Throughout this subsection,F is a set-valuedmap from Rn to Rn. We say that F is (i) monotone(resp. strictlymonotone) if, forx, y € Dom F,x # y,xx e F(x),y* e F(y), onehas (x",y-x)*()*,x-y) <0; (4) (i') (resp.(x*,! -r) * ()*,x - y) < 0); (ii) quasimonotoneif, for.r, y € Dom F, x*, y* as above, min{(x*,y-xil,ly*,* -y)}<0; (5) (iii) semistrictlyquasimonotoneifitisquasimonotone,andif(x*,y-xl > 0ande > 0, therecan be found z e (y - e(y - x), !), z* e F(z) suchthat (z*, y - x\ # 0; (iv) strictly quasimonotoneif it is quasimonotone,and lf x I y in Dom F, therc are - z € (x , l), z* e F(z) suchthat (z*, y x\ * 0; (v) pseudomonotoneif (5) is strict whenever (x*, y - xl # 0; (vi) stictly pseudomonotoneif (5) is strict wheneverx + y. Observethat (i') + (vi) + (v) + (ii); (vi) + (iv) + (iii) + (ii); (i) + (iD. The inverse implications are generally not true. The conceptof monotone set-valuedmaps is known in convex and functional analysis. Quasimonotonicity was introduced in [3 1] . Under a more complicatedform, it was given in [12] for Clarke's subdi-fferential. Now, we tum to the concept of generalized monotonicity for bifunctions. I'rut f e B(n, n). We saythat / rs (i) monotone if, for every x, y e Rn, one has f(x,y-x)+ f(y,x-y)<0; (6) (i') strictly monotone if (6) is strict wheneverx t' y; t02 DinhThe Luc (ii) quasimonotoneif, for every x, y e Rn , one has rnn{f(x,y-x),f(y,x- y)} <0; (7) (iii) semistrictlyquasimonotoneifitisquasimonotone,andif/(x,y-x) > O, f (y,x- y) > -oo, e > 0 implies the existenceof z e (y - t(y - x), y) such that f(z,y-x)>o; (iv) strictly quasimonotone if it is quasimonotone, and if x, y € X, x I y with - -oo f(x,y -r) > -oo and f(y, x y) > implies the existenceof z e (x, y) - suchthateither f (z,y - x) > 0or f (z,x y) > 0; (v) pseudomonotoneif the inequality in (7) is strict whenever f (x, y * x) I 0; (vi) strictly pseudomonotoneif the inequality in (7) is strict wheneverx f y. As in the caseof set-valuedmaps, we have the following implications (i') + (vi) =f (v) + (ii); (vi) + (iv) + (iii) + (ii); (i) + (ii). Monotonicity of bifunctions was usedin [35]. Generalizedmonotone bifunctions were developedin 127,3I1. Below we present a relation between generalizedmonotonicity of a set-valuedmap and that of its support bifunction. Theorem 15. Let F e S(n,n).Then (i) F is G-monotone if and only if o F is G-monotone,where G may take the prefix Q, quasi, semistrictly quasi, and strictly quau; (ii) if o F is strictly monotone (resp.pseudornonotone, strictly pseudomonotone),then so is F. The converseis also true if, in addition, F is compact-valued. Similarly, a link between generalizedmonotonicity of a bifunction and its subdiffer- ential is given in the next theorem. Theorem 16. Let f e B(n,n) be sublinear lower semicontinuousin the second variable. Then (i) / ,r G-monotone if and only if 3f is G-monotone, where G may be Q, quasi, sernistrictly quasi, and strictly quasi; (1i) if f is strictly monotone (resp.pseudornonotone, strictly pseudomonotone),then so is 0f . The corf,verseis also true if 0f is compact-valued. The merit ofthe aboveresults is resignedin the fact thatin orderto establishgeneralized monotonicity of a set-valuedmap, one can verify it for its supportfunction or vice versa. 2.3. 3. Characterizations of Generalized Conv ex Functions In this subsection, we charucteize generalized convex functions by means of Clarke- Rockafellar's subdifferential.Other types of subdifferentialscan also be usedin a similar , way. We recall that Clarke-Rockafellar's subdifferential a1/ is defined as in Subsec.2.3.l by using the following directional derivative: f0 -l tu) - a /t(x; a) : 5uP limsuP ,n ' e>o(y,a)+rxiIgueB(u'e) t GeneralizedConvexity to VectorOptimization 103 --> -+ where / is alowersemicontinuousfunctionon Rn, 0,a) If -r meansy x, a - -ull f (x)andot >_f (y): B(u,e) {u e Rn : llu < e}.Theaboveformulatakesa simple form when / is locally Lipschitz: - f (v-rtu)-f (v) fr(x;u) limsup y+x;t 1,0 In Sec. 1 (Proposition 4), we have already formulated a criterion for convex functions. A fundamental result in characterizing generalized convexity via subdifferential is the following theorem of [30]. Theorem 17. Let f be a lower semicontinuousfunction from Rn (even a Banach space) to R U {+m1. Then f is quasiconvexif and only if AI f is quasimonotone. With the aid of this theorem,it is possibleto establishsubdifferential characterizations of other types of generaTizedconvexity. We shall assume/ is locally Lipschitz and dom / is open and convex. Proposition 18. Thefunction f is strictly (resp. semistrictly) quasicorwexif and only if thefollowing conditions hold: (i) atl is quasimonotone; (11)For x + y and f(x) : f(y), thereexist a,b e (x,y) and a* e 0I f(x), b* < Ar f 0) suchthat (a*,y - x) . (b*,y-,r) > 0. (resp.) - -x),y) (ii/) For f(x) <,f(y) and e > 0, thereexist z e (y e(y and z* e 0r 7171 suchthat (z*, ! - z) + O). In particular, if / is strictly quasiconvex,then 3T/ is strictly quasimonotone,and if 6rl is semistrictly quasimonotone,then / is semistrictly quasiconvex. Note that the converseof the last assertionin the preceding proposition is also true when n : 1, and not true generally in a higher dimension. Proposition 19. If thefunction f is pseudoconvex(resp. strictly pseudocorwex),then (i) atf is quasimonotone; (11)for f(x) < f(y) (resp.f(x) = f(y) and x * y), thereexists y* e 0r f(y) such that{y*,r-y) <0. Conversely, if AI f is pseudomonotone (resp. strictly pseudomonotone),then f is p seudoc onv ex ( resp. strictly p seudoc onv ex ). Observe that conditions for generalized convexity presentedin Propositions 18 and 19 involve both the values of the function and that of the subdifferential. Subdifferential alone is not sufficient or necessary to characterize generalized convexity except for the quasiconvexity. Because generalized convex functions are determined exclusively by their restrictions on line segments,the classicaldirectional derivatives,which only make use of the values of functions on direction, seem to be more appropriate to produce generalizedconvexity characterizations. lO4 DinhTheLuc Proposition 20. Let f be a lower semicontinuousfunction on an open convex set A c R". Then f is G-convex if and only if its directional upper derivative f+ is a G-monotone bifunction on A, where G may be Q, strictly, quasi, semistrictly quasi, strictly quasi, pseudo, and strictly pseudo. - The directional lower derivative f can also be used in the previous proposition. However, for (strict) pseudoconvexity,the sufficient part may fail. Let us specify the aboveresult for the caseof differentiable functions. Rememberthat in thiscase, + f (*, r): (yl.(x), u). The following conditions are necessaryand sufficient for f to be respectively quasi- convex, semistrictly quasiconvex, strictly quasiconvex, pseudoconvex and strictly pseudoconvex.Forevery x,y e A, x * y, (i) min{(V/@),y - xl, (v fj),r - y)} < 0; (8) (ii) (y/(x), y - x) > 0implies (V/0), x - y) < 0, andin anyneighborhoodofy on [x, y], thereis z suchthat (V f k), y - x) > 0; (iii) (8) inequality holds and thereis z e (x, y) with (V f k), y - x) # 0; (iv) inequality (8) is strict if one of two terms under min is nonzero: (v) inequality (8) is strict. The result of (iv) is shown by [21], the results of (i) and (v) were given by 116,22, 441. 3. Vector Functions and Vector Optimization 3.1. Efficient Points Let c be a convex closed pointed cone in R'. It producesa partial order in Rn by a>b if a-beC. Let A be a nonempty subsetof Rn. A point a e A is said to be efficient (or minimal with respectto c) if thereis no b e A such thata > b, a # b.The setof minimalpoints of A is denotedby Min A. When int C is nonempty, one is also interestedin the set of weakly efficient points: lTMinA :- {a e A : thereis no b E A with a - b eintC}. Similarly, one defines the set of maximal points and the set of weakly maximal points Max A, I4zMin A. A particular case is when c is the positive orthant R! of Rn, i.e., R\: {a: (ar,...,an)€ Rn : ai > O, i: 1,...,n}.Bfficientpointswithrespectto Rf are called Pareto-minimal points. Occasionally,one also considersideal points. Recall that a e A is said to be ideal if a < b fot all b e A. Generally, ideal points exist under very restrictive conditions. 3.2. Convex Vector Functions Let X be a convex subset of R- and / a function from X to Rn. As in the previous section, Rn is ordered by a convex closed pointed cone C. Generalized Convexity to Vector Optimization r05 We say that / is convex if, for.r, y e X, l. e (0, 1), f (),x-r(1- r)y) < Lf (x)+ (l - x)f (y). - In the case int C I 0, by writing a )) b, we mean a b eint C. Then / is strictly convexif,for x * y,L as above,one has f ()"x* (1- r)y) < Lemma 21. f is convex(resp. strictly convex)if and only if I o f is convex(resp. strictly corwex)for every l, € C' \ {01, where C' is the dual cone of C, i.e., C' : {t € Rn : (1,x)>0,reC). Let us define subdifferential for a convex vector function in a standardway. - - \f(x) :: {A e L(m,n) : f(y) f (x) > A(y x) for all y e X}, where L(m, n) denote the spaceof m x n-matrices. One can prove severalproperties of subdifferential of convex vector functions similar to those of scalarconvex functions. For instance,0/(x) is a convexclosed set. If / is differentiable,then {y/(x)} : 0f (x). Moreover, using Lemma 21, one can establishthe following relation: 0(6 o /)(x) > EAf @) for every € e C' and equality holds when x e riX (the relative interior of X). Now, assume/ is locally Lipschitz. We recall the generalizedJacobian --> J f (x) :: co{lim V f @i) i x; € X, / is differentiableat xi andx; x}, where co denotesthe closed convex hull. It is known that if f is scalar-valued,then J f (x) is the Clarke subdifferential and coincides with the convex analysis subdifferential Af @). This fact is no longer true for vector functions. Nevertheless,we have the following relation establishedby [36]. Proposition 22. Let x e int X. Then Jf (x) c 0f (x) and EJf (x) : t1f (x) for all Eec'. TheequalityisobtainedbythefactthatJ(fof)(x): lJf (x) and0(f o/) : l1f .The composition{ o/ isaconvex scalarfunctionwhenE e C'.TheinclusioninProposition 22 is strict.For instance,with /(x, y) : (lxl, lxl + y) from R2 to R2 and C : R2+,one has0/(0, o) : {(31), P € [0,1]], while "r/(0, 0) : *{(l !),(-l ill To study convex vector", functions, we also need the concept of monotone maps. Let F be a set-valuedmap from X c R^ to L(m, n). Remember that R' is ordered by the coneC.WesaythatFismonotoneif,forx,y € X, A e F(x), B e F(y), onehas (A-B)(x-y)>0. It is easy to see that F is monotone if and only if f o F is monotone in the classical sensefor every | e C'. Now, let f be a vector function, not necessarilyconvex. One defines subdifferential of / as if / were convex, that is, - - 0f (x) :: {A e L(m,n) : f (y) f (x) > A(y x) for all y e X}. The following result was partly proved in [36]. 106 Dinh TheLuc Proposition 23. Let be f avectorfunctionfrom X c Rm to Rn.If 0f (x) is nonvoidfor every x e x, then is convex. f Moreove4 if f is corwex,then 0f is iaximat monotone. Proof. If 0f (x) is nonvoid for all ; e X, then given x, y e X, ). e (0, 1), one has _ f (x) f (),x 1_(1 _ r)y) > (1 _ L)A(x _ y), _ f(y) f(),x + (1 _ r)y) > ).A(y _ x), where A is any element - of 0f e'x + (1 ),)y). It fonows from the above inequalities that f ()"x + (1 - l)y) < Lf (x) (1 - x) (y).Hence, + f / is convex.The secondpart is proyen in a standard way. For a locally Lipschitz function, the generalized Jacobianis always--J- nonvoid. we can use it to charactenze convexity. I Proposition 24. Let be a locaily Lipschitzfunctionfrom f an open convexset x C Rm to Rn. Then f is convexif and only if J f is monotone. As a consequenceof propositions 23 and,24, we see that generarizedJacobian of a convex vector function is not maximal monotone in general. we study optimality conditions -Now, of convex functions. Recall that the tangentcone of a convex set X c R at xs e X is the cone T(X; xd :- {u e R- : xs + tv eXforsome / > 0}. we say that x6 e X is an efficient point of f if f (xd is an efficient point of the set f (x). rdeal points and weaklyefficient points of / are defined in a similar way. For a set0 c R^,wedenote e Rn: (x,q) e#:{* >0forsomeS € Ot. Proposition 25. Let beaconvexvectorfunctiononaconvexsetX f C R .Then (i) xs e X is ideat if and onty if 0 e 0f (xs); (ii) ;6 e X is weakly.efficient if and onty if there is g e C, \ {0} such that T(X; xd c [t\f (xdl#. Proof' The first assertionis immediate from definitions. Let x6 e x be weakly efficient. Then thereexists e c/ suchthatx. f \ {0} is aminimum of I o fover x. since o is convex,we have,for every f / u e T(X;xs) , 2 0. Equivalently, ,.rlylr,,r(-r,i) (€ e A, u) > 0 for someA e Ef (xs). Thus,Z(X; xs) c lf 0f (xe)l#.Conversely, if xs is not weaklyefficient, then there "i(^' is u e T (X;"0) ,u"t tt ut + rj _ ("0) O. Hence,for every .f << E e c' \ t0), 6 o f(xo* u) - t 0. consequenrly, (ry,u) < " f@d"< € u)- €" f(xo) < 0for everyn€ 0f o/(.rs),whichmeans thatuf[tAf@d]#."f(xo-f ) r\--v/ r A particularcase of Propos ition25 is the situationwhere x6 is a relativeinterior point of X,i.e.,xse riX.Thenr6isweaklyefficienrif andonlyifIii" il]itaf @s) Afor some e C'\ wherelLin I f {0}, Xla is theorthogonal complement oithe tinearsubspace spannedon X - x6. For X : R^, the aboverelation is equivalentto 0 e f0l(x6). GeneralizedConvexity to VectorOptimization 107 Another particular case is obtained when / is differentiable at xs. We then have Df (xd: {f'(xd}andProposition25(ii)becomesasfollows:.r0 € Xisweaklyeffrcient if and only if there is f e C' \ {0} such that \€f' (xd,u) Z 0 forall u e T(X; xd. Again, if in additionxs is arelativeinteriorpointof X, thenthe aboverelationis equivalent to '@il Ef e lLin xlL and,forX: R^, itmeansthat0: Ef'@d. 3.3: QuasiconvexVector Functions As in the previous section, / is a vector function from a convex set X S R- to R" and the spaceRn is partially ordered by a closed convex pointed cone C. Definition 26. We say that f is (strict) quasiconvexif, for x,y e X, x + y andfor ). e (0, l), one has - < (x), (y)l+ f Q.x-r(1 r)y) lf "f (resp.f Q"x+ (1- r)y) < This definition of quasiconvexity has the advantage in that quasiconvex vector functions can completely be characteized by level sets. We recall that, for a € Rn, lev(f ; a) consistsof x e X suchthat f (x) < a. Proposition 27. f is quasiconvexif and only if level sets of f &re convex' The definition given aboveis one way to generalizethe relation (2) to vector functions. That relation can also be interpreted in other ways, giving different definitions of quasiconvexvector functions. Let us mention some of them: (a) implies (),x-r (1 -r)y) < f (y). "f(x) s f 0) f o) /(r.r+(1 -r)y) < flf(x)+(t- fl)f(v)forsome fl e (0,1).(Definitionbv Tanakaand Helbig.) It is easily seenthat / is quasiconvexin the senseof (a) if and only if lev(/; /('r)) is star-shapedwith center at x. Cambini andMartein [6] go evenfurther in generalizingrelation (2) by using separately the orders generatedby C, C \ {0}, int C in the inequalities in (a). The interestedreader is referred to [4] for details ofthose generalizations. As noticed before, a vector function is convex if and only if the composition f o / is convex for every E e C' .This property makes it possible to treat a family of scalar functions : e Ct} instead of the vector function /. This shows particular {€ " f $ importance in scalarization of vector problems. A similar result does not hold for quasiconvexfunctions; composition of a quasiconvexfunction with a functional { e C' is not necessarily quasiconvex.When C is the nonnegative orthant of the space R', wehavethefollowingresult: / : (fr,...,fr)is (strictly)quasiconvexifandonlyif fi, ..., f, are (strictly) quasiconvex. 108 DinhTheLuc Quasiconvexity is used in vector optimization to establish local-global properties of efficient solutions. It is also used to prove topological propertie, ,oJh u, connectedness and contractibility of the set of efficient solutions. Let us consider the case c - Ri. with this order, the spaceRn is a Banach lattice. In particular, for every a, b e Rn, one can find aunique element u € Rn such that u > a, u > b and,for any other u satisfying the above inequalities, one has u > a. Such an element a is denotedby sup{a, b}. Let us introduce the concept of semistrict quasiconvexity (called explicitly quasiconvexity in [3a]). we say that is semistrictly quasiconvexif / it is quasiconvex,*a ir /(") * fii, one has ()"x -l (1 - f r)v) < sup(/(-r), /(v)). Berow is the local-globalproperty in vector optimizatton. Proposition 28. Assume is semistrictry quasicorwex. f Then every local fficient solution of f over X is its globat fficient solution. Note that if the componentsfi, ..., f, of the function / are semistrictly quasiconvex and quasiconvex,then so is /. The converseis not true in general.The componentwise semistrict quasiconvexity can be used to characteize local-global property for weakly efficient solutions. Proposition 29. Assume f is componentwise semistrictquasiconvex (and quasiconvex). Then every local weakly solution of fficient f over x it tt, global weakly fficient solution. The local-global property is also satisfiedunder pseudoconvexity,a notion introduced by t6l. Assume is directionally g. / differentiable ar x and intc I f is said to be pseudoconvex ,(*, if f (y) < f (x) implies .f y - ,r) << 0. Proposition 30. If is pseudoconvex,then t6l f every local fficient solution of f over X is a global fficient solution. It is known and easy to see that if .r e X (weakly) is a efficient solution of / on x, then -intC f'(x,u) f for every v e T(x;x). It follows directly from the definition of pseudoconvexitythat the above condition is also sufficient foi efficiency when is pseudoconvex.Second order efficiency / conditions were developed by t4l using generalizedconvexity. References 1. K.J. Arrow and A.c. Enthoven, euasi-concave programming, Econometrica 29 (1961) 779-800. 2. M. Avriel, w.E. Diewert, s. Schaible, andr.Zang, Generalized concavity,plenumpress, New York-London.1988. 3' H-Brezis, OpdrateursMaximaw Monotones et Semi-Groupesde Contraction dans lesEspaces de Hilbert, North-Holland, 1973. 4' R. 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