Luận văn Quasiconvexity, quasimonotonicity and applicati0ns in variati0nal inequalities - Optimization

Chapter 2 APPLICATIONS IN VARIATIONAL INEQU AL- ITIES - OPTIMIZATION 2.1. SOLUTION EXISTENCE OF QUASIMONOTONE VARIA- TIONAL INEQUALITIES The purposeof the sectionis to provethe existenceof solutionsof the Stam- pachiavariationalinequalityfor a quasimonotonemultivaluedoperator. Let X bea Banachspace,K bea nonempty,convexsubsetof X andthemulti- valuedopertorT : K -t 2x*. The StampachiaVariational InequalitiesProblem (SVI) Find x E K :Vy E K,3x* E T(x) : (x*,y - x)2:00 The Minty Variational InequalitiesProblem (MVI) Find x E K :Vy E K, Vy*E T(y) : (y*,y - x)2:o. We denoteby S(T,K) thesetof thesolutionsof theStampachiavariationalin- equality x E 5(T, K) {:} or E K and Vy E K,3x* E T(x) : (x*,y - x) 2: 0 and by 5str(T, K) the set of strongsolutionsof the sameinequality orE 5str(T,K) {:} or E K and3x*E T(x),Vy E K: (x*,y - x) 2:o. Also,wedenotebyM(T, K) thesetofsolutionsoftheMintyvariationalinequality x E M(T,K) {:} x E K and Vy E K,Vy* E T(y): (x*,y - x) 2:00 Finally, wecall x E K a local solutionof the Minty variationalinequalityif there existsaneighborhoodU ofx suchthatx E M(T, KnU). WedenotebyLM(T, K) thesetof theselocalsolutions.ClearlyM(T, K) c LM(T, K). 23 Definition 27. Given a convexsetK c X and an operatorT : K + 2x* with nonemptyvalues,T is calleduppersign-continuousonK if, for everyx,y E K, thefollowingimplicationholds: (Vt E]O,1[,infx*ET(xi)(x*,y - x) ~ 0) =}SUP:r;*ET(;r)(X*,y- x) ~0, where Xt = (1- t)x+ty. For example,if T is upper hemicontinuous( i.e., the restrictionof T to every linesegmentof K is theuscwith respecto thew*-topologyin X*) thenT is uppersign-continuous.Any strictly positivefunctionis uppersign-continuous. Proposition 28. LetT : X + 2x* bea properlyquasimonotoneoperatorwhose domaintheclosedandconvexsetK. If K weaklycompactthentheMintyvaria- tionalinequalityhasa solution. Proof.DefinethemultivaluedmappingG : K +2x*\0by G(x):= {yE K: (x*,y- x) ~ O,Vx*E T(x)}. For every Xl, x2, ..., xn E K and y E CO{XI,X2,...,xn}, properly quasimonotone impliesthat y E U7~1G(Xi)' In addition, for eachx E K, G(x) is alsoweakly compact.The well-knownKy Fan lemmaimpliesthat nxEK G(x) is a solutionof the Minty variationalinequlity. - Proposition 29. See[lj. LetK beanonempty,convexsubsetoftheBanachspace X andletT : K + 2x* bequasimonotone.Then,oneof thefolowingassertions holds (i) T is properlyquasimonotone, 24 (ii) LM(T, K) =I- 0. In addition,if K is weaklycompact,henLM(T, K) =I- 0 in bothcases. Proof. Supposethat T is notproperlyquasimonotone.Then, thereexistXl, ...,XnE K, X*; E T(X.i),i = 1,...,n, and X E CO{.TI'...,xn}suchthat (X\, X - X'i) > 0,i = 1,...,n By continuityof thefunctionalsx;, thereexistsaneighborhoodU ofx suchthat, foranyy E K n U, onehas, (X\, Y - Xi) >o. By quasimonotonicity, Vy*ET(y),(y*,y-x) ~O. Thus,x E LM(T, K) sincethepreviousinequalityholdsfor everyy E K n U. It remainsto showthat LM(T, K) =I- 0wheneverK is weaklycompactand T is properlyquasimonotone.But undersuchassumtion,it is known Proposition 28 thatM(T, K) i=0;sinceM(T, K) c LM(T, K), it followsthatLM(T, K) i=0. . Proposition 30. See[1].LetK beanonemptyconvexsubsetoftheBanachspace X' X and letT : K -* 2 beanoperator. (i) If T is pseudomonotone,thenLM(T, K) =M(T,K). (ii) If for everyx E K thereexistsa convexneighborhoodVx ofx andanupper sign-continuousoperatorSx : Vxn K -* 2x' with nonempty,w*-compactvalues satisfingSx(Y)C T(y),Vy E Vxn K, thenLM(T, K) c S(T, K). (iii) Additionally to theassumptionsof (ii), if theoperatorsSx areconvexvalued, thenLM(T,K) c S(T,K) =M(T,K). 25 Proof.(i) Letx beanelementofLM(T, K). Then,thereexistsaneighborhoodU ofx suchthatx E M(T, K nU). Foranyy E K, thereexistsz = x +t(y - x),t E ]0,1[,suchthatz E K n U. Then,foranyz* E T(z), (z*,y - z) = [(1- t)jt](z*,z - x) 2:o. By pseudomonotonicity, (y*,y - x) 2:0,Vy*E T(y). Thereforex is an elementof M(T, K). (ii) Let x beanelementof LM(T, K). Thus,thereexistsa neighborhoodU ofx v suchthat .TE M(Sx,K n V n U). Lety E K n Vx. SinceK n Vxis convex,there existsy' E]x,y] forwhich[x,y']c (K n Vxn U) andthus infuE[x,Y'jinfu*ESx(U)(u*, u - x) 2:o. By theuppersign-continuityofSx, SUpX*ESx(x)(x*, Y - x) 2:o. But Sx(.T)is w*-compactandwededucethat infYEvxnKmaxx*ESx(x)*,y - x) 2:o. (8) whichmeanthat, for all y E Vxn K, thereexistsx* E SAx) c T(x) suchthat (x*,y - x) 2:O.Therefor,x is anelementofS(T, K) since,usingtheconvexityof K, onecanproveeasilythattheaboverelationholdsforanyy E K (iii) This is a consequenceof the Sion minimaxtheoremappliedto the relation (8). . In particular, if T itself is uppersign-continuousand has nonemptyconvex,and 26 w*-compactvalues,then wecantakein the lemmaVx=K, Sx=T. Remark From Proposition 29 and Proposition 30, with the assumptionK is weaklycompactset, T is pseudomonotoneoperatorand upper -sign continuous thenS(T, K) =I=- 0. The assumptionpseudomonotoneis strongerthan quasimono- tone, in the later theorem, the authors provethe existenceos solution of the Stampachiavariationalinequalityfor a quasimonotonemultivaluedoperator. Theorem 31. See(l). LetK bea nonemptyconvexsubsetofX. Further,letT : K -7 2x* bea quasimonotoneoperatorsuchthatthefollowingcoercivitycondition holds 3p> 0,Vx E K n B(O,p),3y E K withIlyll< Ilxll suchthatVx*E T(x), (x*,x - y) 2:O. Supposethatthereexistspi >p suchthatKn13(O,pi) is nonemptyweaklycompact. Moreover,supposethat,for everyx E K, thereexistsaconvexneighborhoodVxofx andun uppersign-continuousoperatorS;J; : VTnK -7 2x*withnonempty,convex, w*-compactvaluessatisfyingSx(y)c T(y),Vy E Vxn K. ThenSstr(T,K) =I=- 0. Proof.The setK~ := K n 13(0,pi) is nonempty,convexandweaklycompact. Accordingto PRO, LM(T, K~)=I=- 0. By Proposition30,thesetSstr(T,K~)=I=- 0. ChooseXoE Sstr(T,K~). Then 3x~E T(x) :Vy E K~,(x~,y- xo) 2:O. Accordingto coercivitycondition,thereexistsYoE K n B(O,pi)suchthat Vx~E T(xo), (x*,Xo- Yo)2:0, (if 11.1',011< pi we can takeYo= xo). It fowllowsthat (x~,Yo- xo) = O. 27 Now,foreveryy E K, thereexistst E [0,1[suchthat (1- t)y+tyoE K~;hence, (x~,(1- t)y+tyo- xo)~ O. It followsthat (x~,y - xo)~ O. I.e. XoE Sstr(T,K). . Notethat,in theTheorem31, theconditiononthecompactnessof K n 13(0,p') is satisfiedautomaticallyif K is weaklycompactor X is reflexiveand K is closed; the coercivity condition is also satisfiedautomaticallyif K bounded. Finally, the conditionon the existenceof Sx is sasfiedif T itself is upper sign-continous with nonempty,convex,w*-compactvalues.Thus, Theorem31generalizedcorre- spondingresultsfor pseudomonotoneoperator,quasimonotoneoperatorwhereK is assumedto containinnerpoint, denselypseudomonotoneoperator. 2.2. APPLICATIONS IN OPTIMIZATION (See[13].)In the sequelX is a normedvectorspace. We denoteby N(C,.T) the normal coneat x E C to a convexsubsetC of X givenby N(C,x):= {x*E X*: Vu E C, (.T*,U- x) ::; O}. It is thepolarconeofthe tangentconeT (C,x) whichis theclosureof ffi.+(C - x). Definition 32. Thelowersubdifferential,or Plastriasubdifferentialof afunction f : X -+ ffi. on a BanachspaceX at somepointx of its domaindomf := {x E 28 X : f(.r) E JR.}is theset eJ<f(x) := {x*E X* : \:IxE Sf(x), f(x) - f(x) ~ (x*,x - x)}, whereSf (x) :=Sf (f, f (x)) is thestrictsublevelsetoff atX. Definition 33. Thefollowingvariant,calledtheinfradifferentialor Gutierrezsub- differential: o~f(x) :={x*E X* : \:IxE Sf(x), f(x) - f(x) ~ (x*,x - x)}, If no pointof the levelsetLf(x) := f-l(f(X)) is a localminimizerof f, we have(}<f(x)= o~f(x). This equalityalsoholdswhenf is radiallycontinuous (i.e. continuousalong lines) and semistrictlyquasiconvexin the sensethat when f(.r) < f(y) onehasf((l - t)x + ty) < f(y) foranyt E (0,1);in particular,this equalityholdsfor convexcontinuousfunctions. Definition 34. f is a Plastriafunctionat x if its strictsublevelsetSf (x) is convexandsuchthat N(Sf(x),x) =JR.+o<f(x). (9) Definition 35. f is a Gutierrezfunctionatx if its sublevelsetSf(x) zsconvex andsuchthat N(Sf(x),x) =JR.+o~f(x). (10) Sinceo<f(x) and o~f(x) areshadyin the sensethat they are stableun- der multiplicationby any t E [1,00),relations(9) and (10)areequivalentto N(Sf(x),x) = [O,l](}<f(x)andN(Sf(x),x) = [0,1](}~f(x) respectively.These conditionsbeing rather stringent,it may be usefulto replacef by its extension 29 by +00outsidesomeball. Howeverwe providethreecriteria. The first onedeals with convextransformablefunctions,an importantclassof quasiconvexfunctions. Proposition 36. See[i3}. Supposef is aproperconvexfunctionandx E domf := f-1(JR) issuchthatf(x) > inff(X) andJR+(domf -x) =X. Thenf is a Gutierrez function and a Plastria function: N(Sr(x),x) = JR+o~f(x) =JR+of(x)=JR+o<f(x)=N(Sf(x),x). Moregenerally,if f := hog, where9 : JR-. JRoois a convexfunction andh : T-. JRoois an increasingfunction onsomeintervalT of JRoocontainingg(X), with h(+00)=+00,thenf is a GutierrezfunctionandaPlastriafunctionatxprovided f(x) >inff(X). Moreover,JR+o<f(x)=JR+o<g(x)= JR+o~g(x)= JR+o~f(x). The secondoneis a slight improvementof previousresultsin [22],[16].It uses the now classicalnotion of calmness:f : X -.JR is said to be calmwith ratec at w E X if f(w) is finite and if 'ixEX f(x) - f(w) ~ -c Ilx - wll. Such a condition is obviouslysatisfiedif f is Lipschitzian with rate c or if f is Stepanovian(orstable)with ratecat w in thesensethat If(x) - f(w)1 :::;c Ilx - wll for anyx EX. Proposition 37. See[i3}. Assumethatf is radially continuouson X and calm withratec E JR+at eachpointof thelevelsetL.r(x) :=f-l(f(x)) andthatS.r(x) andSf (x) areconvex.Then N(Sf(x),x)\cBx=o<f(x)\cBx, N(S.r(x),x)\cBx=o~f(x)\cBx 30 If moreoverN (S1(x),x) -1={O},thenf is a Plastriafunctionwhileif N (Sj (x),x) -1= {a}and a Gutierrezfunction at x. The condition N(Sj(x),x) -1={a} (or N(S1(x),x) =I-{a}) is a rather mild conditionwhenX is finitedimensional.However,whenX is infinitedimensional, it mayoccurthata closedconvexsetC =I- X is suchthatN (C,x) = X* forsome xE C. The third criterionusesa differentiabilityassumptionand bringssomesupple- mentto [19,Prop. 15]. Proposition 38. See{13}.Supposef is quasiconvex,differentiableat x with a nonnul derivative.If jJ<f(x) is nonempty,thenf is a Plastriafunctionatx and thereexistssomer 2: 1 suchthat8<f(x) = [r,oo)1'(x).If 8~f(x) is nonempty, thenf is a Gutierrezfunctionatx andthereexistssome"82:1 suchthat8~f (x) = ["8,00)1'(x). Thefollowingexampleshowsthatonemayhaver > 1. Example. Let X =IR and for c <0 let f begivenby f(x) =C3for x E (-00, c), f(x) =X3 for x 2:c. Then, for x =1,wehave8<f(x) =8~f(x) = [r,oo)with r = max(3,1+c+C2). Lemma 39. See{13}.Let (9i)iEI beafinite family of quasiconvexGutierrezfunc- tions at somex E X. For i E I, letCi := 9;1((-00,0]). Assume9i is u.s.c. atx, 9i(x) = 0 for eachi E I andeither (a) there existsomek E I and somez E Ck such that 9i(Z) < 0 for each i E 1\{k}(Slatercondition),or (b) Ci is closedfor eachi E I andIR+ (IJ.- IICi) = X I, whereIJ. is theiEI 31 diagonalof X I. Then,g : =maXiEI gi is a Gutierrezfunctionatx andonehas ~+a::;g(x)=L ~+a::;gi(X). iEI (11) Optimality conditions for constrained problems In the presentsectionweconsiderthe minimizationproblem (C) minimizef(x) subjectto x E C wheref :X -+ ~ is a functiononthen.v.s.X andC is a convexsubsetofX. Proposition 40. See [18]. Suppose f is an U.S.c. Plastria function at x and x is a solution to (C) but is not a local minimizer of j. Then one has 0 E a<j(x) +N(C, x). (12) Proof.Sincef is quasiconvexandU.s.c.,thestrictsublevelsetSj(x) is openand convex;it is nonemptysincex is notaminimizerof j. Sincex is asolutionto (C), this sublevelset is disjoint from C. Thus, the Hahn-Banachseparationtheorem yieldssomec E ~ and u* in the unit sphereof X* suchthat (u*,x-x) 2::c2::(u*,w-x) 'l/wE Sf (x),'l/xE C. (13) Takingx = x, weseethat c :::;O.Moreover,sincex is not a localminimizerof f, thereexistsa sequence(wn)-+ x such that Wn E Sf (x) foreachn. Therefore c = O.Thenwehaveu* E N(Sf(x), x) =~+a<j(x) and sinceu* =J0, we can find x* E a<f (x) and r E ~+ suchthat x* =ru*. On theotherhand,thefirst inequalityof (13) meansthat -u* E N( C, x). Thus, x* +ru* = 0 and (12) is satisfied. . 32 Now let us givea sufficientcondition. Observethat no assumptionis required on f besidesfinitenessatx. Proposition 41. See[18).Supposethatf :X ---t IRU{oo} is an arbitraryfunction andf is finiteatx andsatisfiesrelation(12). Thenx is a solutionto (C). Proof.Let x* E fJ<f(x) be suchthat -x* E N(C,x). Assumethat x is not a solutionto (C): thereexistssomex E C suchthatf(x) < f(x). Thenonehas,by thedefinitionsof [)<f (x) andN (C,x), 0> f(x) - f(x) 2::(x*,x - x), (x*,.r- x) 2::0, a contradiction. . A slightsupplementto theprecedingresultscanbegiven.It dealswith strict solutionsto (C), i.e. pointsx E C suchthat f(x) < f(x) for each.r E C\{.r}. For thesufficientconditionweassumethatC is strictlyconvexat x in thesense that (x*,x - x) <a foreveryx E C\ {x}andx* E N(C,x)\ {a}.Observethatif N(C,x)\{O} is nonempty(in particularif C is a convexsubsetof a finitedimen- sionalspace)andif C is strictlyconvexatx, thenx is anextremalpointofC (i.e. C\ {x}is convex). Proposition 42. See[18).Givenafunctionf :X ---t IRU {oo}finiteatx anda subsetC ofX whichis strictlyconvexatx, thefollowingrelationimpliesthatx is a strictsolutionto (C) or a globalminimizeroff onX : aE [):50f(x)+N(C,x). (14) Conversely,whenX isfinitedimensional,C is aconvexsubsetofX notreduced to {x},x is an extremalpointof C andf is a Gutierrezfunctionatx, relation 33 (14)is necessaryin orderthatx bea strictsolutionto (C) or a globalminimizer off onx. Proof.Supposerelation(14)holdsandC isstrictlyconvexatx. If x isnotaglobal minimizerof f onX thereexistssomex* E [)~f(x) suchthat -x* E N(C,x) and x* i- O. Then, if x E C\{x} is such that f(x) :::;f(x) we have (x*,x - x) :::; f(x) - f(x) :::;0 sincex* E [)~f(x) and (-x*,.'E-x) < 0 since-x* E N(C,x)\{O}, a contradiction.Thusx is a strictsolutionto (C). Whenx is a strictsolutionto (C), thesetsC\ {x}andSj(x) aredisjoint.If moreoverf is a Gutierrezfunctionatx andx is anextremalpointofC butis not a globalminimizerof f onX, andC i- {x},thesesetsareconvexandnonempty. Thus, whenX is finite dimensional,a separationtheoremyieldssomec E IR and u* in the unit sphereof X* suchthat (u*,x-x) :::::c:::::(u*,w-x) 'i/wE Sj(x), 'i/xE C\ {x}. (15) Sincex can be arbitrarily closeto x, we havec :::;O. On the other hand, since we cantakew = x, we havec ::::: 0, hencec = O.Thus -u* E N(C,x) and u* E N(Sr(x), x) =IR+[)~f(x) sincef is a Gutierrezfunctionat x. Sinceu* i- 0, onecanfindr > 0 andx* E [)~f(x) suchthatu* = rx* and-x* E N(C,x), so that relation(14)holds. Whenx is a globalminimizerof f on X, wehave 0 E [)~f(x)n(-N(C,x)). . Necessary condition for the mathematical programming problem Let us considernow the casethe constraintset C is definedby a finite family of inequalities,so that problem (C) turns into the mathematicalprogramming 34 problem (M) minimizef (x) subjectto x E C := {xEX: gl (x) :::;0, ..., gn(x) :::;O}. Let us first considerthe caseof a singleconstraint. Lemma 43. See[13J.Letx bea solutionto (M) in whichgl =... =gn =9 and x is nota localminimizeroff. Assumethatf is a Plastriafunctionatx,that9 is U.S.c.atx anda GutierrezfunctionatX. Theng(x)= 0 andthereexistssome y E ~+suchthat 0 E (;<f(x)+y8s'g(x). Proof.By Proposition40,thereexistsx* E 8<f(x) suchthat -x* E N(C,x). If g(x) < 0, since9 is U.S.c.atx, x belongsto theinteriorof C, hencex is a local minimizerof f, andourassumptiondiscardsthatcase.Thusg(x)= 0,andsince 9 is a Gutierrezfunctionat x, wehaveN(C,x) = ~+8s'g(x).Thus thereexists Y E ~+suchthat -x* E y8s'g(x). . Now let us turn to the generalcase.We will usethe Lemma39. Proposition 44. See[13J.Letx bea solutionto (M) andx is nota localmin- imizerof f. Assumethatf is a Plastriafunctionatx, gl,...,gn are Gutierrez functionsatx andU.S.c.atX. Let I := {i EN: gi(X)= O},Ci :=g,;,-l((-oo,0]) andassumethatoneof theassumptions(a) or (b)ofLemma39is satisfied.Then, thereexistsomeY1,...,YnE ~+ suchthat 0 E 8<f(x)+Y18s'gl(x)+ ...+Yn8s'gn(x), for i = 1,...,n, Yig'i(x) = O. 35 Proof. Let h := maxlsisngi,let D := h-1((-00,0]). Let I := l(x) := {i E {1, ...,17,}: g'i(X) = h(x)}. Then, for i E {1,...,17,}\1 the point x belongsto the in- terior of Ci := gjl(( -00,0]), sothat for anyx E C := g-I(( -00,0]) andany t >0 smallenoughwehavex+t(x- x) E D. It followsthatN(D,x) = N(C,x).By Proposition40thereexistssomex* E a<f(x)suchthat-x* E N(D, x) = N(C,x).. Let ussetg := maXiEIgi. Theng is u.s.c.at x andis a Gutierrezfunctionat x by Lemma11. Then, by relation(11),thereexistYi E JR;.+,y7E aSgi(x) suchthat -x* =YIY~+ ...+YnY~andtheresultis proved. - Proposition 36 showsthat the precedingstatementencompassesthe classical result for convexmathematicalprogramming. A link with the classicalKarush, Kuhn and Tucker Theoremis delineatedin the next statement. Corollary 45. Seeli8}. Supposetheassumptionsof theprecedingpropositionare satisfiedand thatf, gl, ...,gn aredifferentiableatx withnon null derivatives.Then thereexistsomeAI, ...,An E JR;.+suchthat f'(x) +Alg~(X)+...+Ang~(x)=0, Aigi(x) = 0 for i = 1,...,17,. Proof. By Proposition 38andtheprecedingresult,thereexistsomer 2:1,Yi E JR;.+ and someY7E aSgi(x) for i = 1,...,17,suchthat r f'(x) +YIY~+... +YnY~= 0; alsoy.:= Sig~(X)forsomeSi2:1.SettingAi=r-l S'iYi,we get the result. - Let us give a simple sufficientconditionfor the mathematicalprogramming problem(M). 36 Proposition 46. See[13). Ifx E C is suchthatthereexistY-iE JR+for i = 1,...,n suchthatthefollow1:ngconditionsaresatisfied,thenx is a solutiontoproblem(M): 0 E a<f(x)+Yla:S;gl(x)+ ... +Yna:S;gn(x), gl(x) ::;0,...,gn(x)::;0, Ylgl(X)=0,...,Yngn(x)=O. Proof.Supposeon thecontrarythatthereexistssomex E C suchthat f(x) < f(x). Letx* E a<f(x), x7E Y-ia:S;g'i(X)fori = 1,...,n besuchthat -* + -* + -* 0x YIXI ...+ Ynxn= . Let l(x) := {i E {I, ...,n} : g.;(x)= O}.Thenfor i E l(x), by thedefinitionsof a<f(x), a:S;gi(X)wehave,sincef(x) <f(x), g'i(X)::;0=g'i(X), (x*,.1:- x) ::;f(x) - f(x), (x,~,x - x) ::;gi(X)- gi(X) i = 1,...n. Multiplying eachsideof the last inequalityby Yi and addingthe obtainedsidesof the obtainedrelationsto the first onewe get,sinceYi =0 if i E {I, ...,n}\l(x), n n 0 =(x*,x - x)+L Yi(x:,x - x) ::;f (x) - f (x)+L Yi (gi(x) - gi(x)) i=l i=l ::;f(x) - f(x), acontradiction.- 37 2.3. RELATION BETWEEN THE MINTY VARIATIONAL IN- EQUALITY AND OPTIMIZATION Given X be a Banach space,f : K c X ~ R U {oo},cp: R ~ R andthe operatorT :K c X ~ X*. Weconsidertheopimizationproblem(M): minf(x),xEK and the Minty variatinalinequalityW.r.tcp(cp-MVI) : Find x E K: (y*,y-x) ~ cp(llx-yll),Vy E K,Vy* E T(y). Whencp(t)- 0,Vt E dom(cp),(cp-MVI)Minty variatinalinequalityW.r.tcpbecomes Minty variationalinequality. Theorem 47. LetX bea Banachspace& reliable,f is l.s.candcp-quasiconvex andcpis continuousandholdVAE (0,1): cp(Allull)~ Acp(llull),VuE X. If K = N whereN is open,convexneighborhoodfx orK = X thenthefollowing statementsareequivalent: (i) x is anoptimalsolutionof (M); (ii) .r is a cp-Mintysolutionof &f(x) withcpo Proof.(i) =?(ii). Let x beanoptimalsolution(M). Assumethatx is a strictminimumof (M) i.e.Vx E K :x =I- x : f(x) > f(x). Applying the mean valuedtheoremto segment[x,x], we get :Jc E [x,xl, Cn ~ C,cn*E &f(cn) suchthat liminf(cn*,x- x) >0,n liminf(cn*,C- cn)> O.n 38 With d =c+t(x - .f), weget (cn*,d- cn)>O. Taked=x onehas (Cn*,x - cn) >o. Sincef is <p-quasimonotoneimplyingthat=?8f is <p-quasimonotone.It follows '\Ix*E 8f(x) : (x*,x - cn)2:<p(llx- cnll), Since<pis continuous,wehave (x*,x - c) - <p(llx- cll) 2:O. Thereis BE (0,l]suchthat( x - x)B=x-c. So,onehas B(x*,x - x) - <p(llx- xllB)2:0, B(x*,x- x) 2:<p(llx- xlIB)2:B<p(llx- xii). (X*,.T- x) 2:<p(llx- xii). Supposethatx is nota strictminimum,setthefunctionh : K --7 R U {oo}such that h(x) ={ ~(x) if x =x, if x =I- x where a <f(x). Sincefisc, <p-quasiconvex=?h is lsc,<p-quasiconvex '\Ix=I-x: (x*,x - x) 2:<p(llx- xll),'\Ix*E 8h(x)=8f(x), '\IxE K : (x*,x - x) 2:<p(lIx- xii),'\Ix*E 8f(x). (ii) =?(i) If x is notoptimalsolution(M), wehave 3x E K : f(x) <f(x). 39 We get 3c E [x,x], Cn ---+ C,Cn* E af( cn): (cn*,d - cn) = (cn*, C - cn) +(cn*,X - x) >0, with d = C+ t(x - x). Since K convex, open neighborhood of x, for n so large [x,x] c K, CnE K. Taked = .1:,oneget (Cn*, X - cn) >0 =? (cn*, Cn- x) <0 : a contracdition with (ii) . REMARK. If cp- 0 i.e f quasiconvex then the solutions of (M) and the solutions of (MVI) arethe same([10]). 40