On the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity*

Transcription

1 Ž. Joural of Mathematcal Aalyss ad Applcatos 237, Artcle ID jmaa , avalable ole at o O the Subdfferetals of Quascovex ad Pseudocovex Fuctos ad Cyclc Mootocty* Ars Dalds ad Ncolas Hadjsavvas Departmet of Mathematcs, Uersty of the Aegea, Karloass, Samos, Greece Submtted by Arrgo Cella Receved Jauary 7, 1998 The otos of cyclc quasmootocty ad cyclc pseudomootocty are troduced. A classcal result of covex aalyss cocerg the cyclc mootocty of the Ž FechelMoreau. subdfferetal of a covex fucto s exteded to correspodg results for the ClarkeRockafellar subdfferetal of quascovex ad pseudocovex fuctos. The oto of proper quasmootocty s also troduced. It s show that ths ew oto retas the characterstc property of quasmootocty Ž.e., a lower semcotuous fucto s quascovex f ad oly f ts ClarkeRockafellar subdfferetal s properly quasmootoe., whle t s also related to the KKM property of multvalued maps; ths makes t more useful applcatos to varatoal equaltes Academc Press 1. INTRODUCTION Let X be a Baach space ad f: X R 4 a lower semcotuous Ž lsc. fucto. Accordg to a relatvely recet result of Correa, Joffre, ad Thbault Žsee 7 for reflexve ad 8 for arbtrary Baach spaces., the fucto f s covex f ad oly f ts ClarkeRockafellar subdfferetal f s mootoe. I the same le of research, much work has bee doe to characterze the geeralzed covexty of lsc fuctos by a correspodg geeralzed mootocty of the subdfferetal. Thus Luc 15 ad, de- * Work supported by a grat of the Greek Mstry of Idustry ad Techology. E-mal: arsd@aegea.gr E-mal: had@aegea.gr X99 $30.00 Copyrght 1999 by Academc Press All rghts of reproducto ay form reserved. 30

2 SUBDIFFERENTIALS AND QUASICONVEXITY 31 pedetly, Aussel, Corvellec ad Lassode 2, showed that f s quascovex f ad oly f f s quasmootoe. Smlarly, Peot ad Quag 16 showed that f the fucto f s also radally cotuous, the f s pseudocovex f ad oly f f s pseudomootoe Ž the sese of Karamarda ad Schable 14, as geeralzed for multvalued operators by Yao 20.. I Secto 2, we revew these results, together wth some otato ad deftos, ad show that most cases the radal cotuty assumpto s ot ecessary. However, sce the ClarkeRockafellar subdfferetal of a covex fucto cocdes wth the classcal FechelMoreau subdfferetal 19, t s ot oly mootoe, but also cyclcally mootoe 17. I Secto 3 of ths work, we defe aalogous otos of cyclc quasmootocty ad cyclc pseudomootocty ad show that the subdfferetal of quasmootoe ad pseudomootoe fuctos have these propertes, respectvely. Cyclc geeralzed mootocty s ot just a stroger property tha the correspodg geeralzed mootocty, but t expresses a behavor of a specfc kd. I partcular, a operator ca eve be strogly mootoe wthout beg cyclcally quasmootoe. Cyclc Ž geeralzed. mootocty descrbes the behavor of a operator aroud a cycle cosstg of a fte umber of pots. I Secto 4 we cosder stead the covex hull of such a cycle. We show that the deftos of mootoe ad pseudomootoe operators ca be equvaletly stated terms of ths covex hull. Ths s ot so for quasmootoe operators; ths leads to the troducto of the ew oto of a properly quasmootoe operator. We show that ths ew oto, whle retag the mportat characterstcs of quasmootocty Ž partcular, f s quascovex f ad oly f f s properly quasmootoe. s ofte easer to hadle; partcular, t s closely related to the KKM property of multvalued maps. We show ths by a applcato to Varatoal Iequaltes. I addto, quasmootocty ad proper quasmootocty are detcal o oe-dmesoal spaces, whch s probably the reaso why the latter escaped atteto. 2. RELATIONS BETWEEN GENERALIZED CONVEXITY AND GENERALIZED MONOTONICITY We deote by X* the dual of X ad by Ž x*, x. the value of x* X* at x X. For x, y X we set x, y tx Ž 1 t. y:0t14 ad defe Ž x, y, x, y., ad Ž x, y. aalogously. Gve a lsc fucto f: X R 4 wth doma domž f. x X : fž x. 4, the

3 32 DANIILIDIS AND HADJISAVVAS ClarkeRockafellar geeralzed dervatve of f at x domž f. 0 the drecto d X s gve by Žsee 19. fž x td. fž x. f Ž x 0, d. sup lm sup f, Ž 2.1. Ž. t 0 xf x 0 t0 db d where B Ž d. d X : d d 4, t 0 dcates the fact that t 0 ad t 0, ad x x meas that both x x ad fž x. fž x. f The ClarkeRockafellar subdfferetal of f at x s defed by 4 fž x. x* X : Ž x*, d. f Ž x, d., d X. Ž We recall that a fucto f s called quascovex, f for ay x, y X ad z x, y we have fž z. max fž x., fž y. 4. Ž 2.3. A lsc fucto f s called pseudocovex 16, f for every x, y X, the followg mplcato holds: x* fž x. : Ž x*, y x. 0 fž x. fž y.. Ž 2.4. It s kow 16 that a lsc pseudocovex fucto whch s also radally cotuous Ž.e., ts restrcto to le segmets s cotuous., s quascovex. Both quascovexty ad pseudocovexty of fuctos are ofte used Optmzato ad other areas of appled mathematcs whe a covexty assumpto would be too restrctve 5. X * Let T: X 2 be a multvalued operator wth doma DT Ž. x X : TŽ x. 4. The operator T s called Ž. cyclcally mootoe, f for every x 1, x 2,..., x X ad every x TŽ x., x TŽ x.,..., x TŽ x. we have Ý 1 1 Ž x, x x. 0 Ž 2.5. Ž where x x Ž. mootoe, f for ay x, y X, x* TŽ x., ad y* TŽ y. we have Ž y* x*, y x. 0. Ž

4 SUBDIFFERENTIALS AND QUASICONVEXITY 33 Ž. pseudomootoe, f for ay x, y X, x* TŽ x., ad y* TŽ y. the followg mplcato holds: or equvaletly, Ž x*, y x. 0 Ž y*, y x. 0 Ž 2.7. Ž x*, y x. 0 Ž y*, y x. 0. Ž 2.8. Ž v. quasmootoe, f for ay x, y X, x* TŽ x., ad y* TŽ y. the followg mplcato holds: Ž x*, y x. 0 Ž y*, y x. 0. Ž 2.9. The above propertes were lsted from the strogest to the weakest. We recall the htherto kow results coectg geeralzed covexty wth geeralzed mootocty: THEOREM 2.1. Let f: X R 4 be a lower semcotuous fucto. The Ž. f s coex f ad oly f f s mootoe 8. I ths case f s also cyclcally mootoe Žsee for stace 17.. Ž. f s quascoex f ad oly f f s quasmootoe Žsee 2 or 15.. Ž. Let f be also radally cotuous. The f s pseudocoex f ad oly f f s pseudomootoe Žsee 4 or 16.. We ow show that pseudocovexty of a fucto f mples quascovexty of f ad pseudomootocty of f, eve wthout the radal cotuty assumpto: 4 PROPOSITION 2.2. Let f: X R be a lsc, pseudocoex fucto wth coex doma. The Ž. Ž. f s quascoex f s pseudomootoe. Proof. Ž. Suppose that for some x, x domž f. 1 2 ad some y Ž x, x. we have fž y. max fž x., fž x.4. Set m maxfž x., fž x Sce f s lower semcotuous, there exsts some 0 such that fž y. m, for all y B Ž y.. From Ž 2.4. t follows Žsee also. 4 that the sets of local ad global mmzers of the fucto f cocde; hece the pot y caot be a local mmzer, so there exsts w B Ž y. such that fž w. fž y.. Applyg Zagrody s Mea Value Theorem 21, Theorem 4.3 to the segmet w, y, we obta u w, y., a sequece u u, ad

5 34 DANIILIDIS AND HADJISAVVAS u fž u., such that Žu, y u. 0. Sce y cox, x t follows that Žu, x u. 0, for some 1, 24. Usg relato Ž 2.4. we get m fž x. fž u. ad, sce f s lower semcotuous, m fž u.. Ths clearly cotradcts the fact that u B Ž y.. Ž. Let x* fž x. be such that Ž x*, y x. 0. By part Ž., f s quascovex, so applyg Theorem 2.1Ž. we coclude that f s quasmootoe. Hece Ž y*, y x. 0, for all y* fž y.. Suppose to the cotrary that for some y* fž y. we have Ž y*, y x. 0. From relato Ž 2.4. we obta fž x. fž y.. O the other had, sce f Ž x; y x. 0, there exsts 1 0, such that for some x x, t 0 ad for all y B Ž y., we have fžx t Ž y 1 x.. fž x.. Quascovexty of f mples fž y. fž x., for every y B Ž y.. I partcular fž y. fž x. Ž sce f s lsc., hece fž y. fž y. 1. The latter shows that y s a local mmzer, hece a global oe. Ths s a cotradcto, sce we have at least fž y. fž x.. It s stll a ope questo whether pseudomootocty of f mples pseudocovexty of f, wthout the radal cotuty assumpto. As a partal result, we have the followg proposto, whch wll be of use the ext secto. PROPOSITION 2.3. Let f be a lsc fucto such that f s pseudomootoe. The f has the followg propertes: Ž. If 0 fž x., the x s a global mmzer Ž. x* fž x.: Ž x*, y x. 0 fž y. fž x.. Proof. Ž. Suppose that fž y. fž x.. The usg aga Zagrody s Mea Value Theorem, we ca fd a sequece z z y, x. ad z fž z., such that Žz, x z. 0. By pseudomootocty, Ž x*, x z. 0 for all x* fž x.,.e., 0 fž x.. Ž. Let us assume that for some x* fž x. we have Ž x*, y x. 0. We may choose 0 such that Ž x*, y x. 0, for all y B Ž y.. Sce f s obvously quasmootoe, from Theorem 2.1Ž. we coclude that f s quascovex; t the follows that fž y. fž x. Žsee for stace Theorem Suppose to the cotrary that fž x. fž y.. The fž y. fž x. fž y., so f has a local mmum at y. It follows that 0 fž y. Žsee for stace 21, Theorem 2.2Ž.. c. However, f s pseudomootoe, hece we should have Žsee relato Ž Ž y*, y x. 0, for all y* fž y., a cotradcto.

6 SUBDIFFERENTIALS AND QUASICONVEXITY GENERALIZED CYCLIC MONOTONICITY We frst troduce cyclc quasmootocty. DEFINITION 3.1. A operator T: X 2 X * s called cyclcally quasmootoe, f for every x, x,..., x X, there exsts a 1, 2,..., such that Ž x, x x 0, x 1. TŽ x. Ž 3.1. Ž where x x It s easy to see that a cyclcally mootoe operator s cyclcally quasmootoe, whle a cyclcally quasmootoe operator s quasmootoe. Cyclc quasmootocty s cosderably more restrctve tha quasmootocty Ž see Example 3.5 below.. However, ths property characterzes subdfferetals of quascovex fuctos, as show by the ext theorem. 4 THEOREM 3.2. Let f: X R be a lower semcotuous fucto. The f s quascoex f ad oly f f s cyclcally quasmootoe. Proof. I vew of Theorem 2.1Ž., we have oly to prove that f f s quascovex the f s cyclcally quasmootoe. Assume to the cotrary that there exst x, x,..., x DŽ f. 1 2 ad x fž x. such that Žx, x x. 0, for 1, 2,..., Ž 1 where as usual x x.. It follows that f Ž x, x x I partcular, for every there exsts 0, 0 such that fž x td. fž x. lm sup f 0. Ž 3.2. Ž. t x x db x1x f t0 We set m 1, 2 ad m 1, 2. For ay y B Ž x. ad x B Ž x. we have y x B Ž x x ; hece we ca choose x B Ž x. ad t Ž 0, 1. such that 2 Ž. Ž 1. Ž. f x t x x f x f 0 Ž 3.3. Ž. t x 1B 2 x1 or equvaletly Ž. Ž 1. Ž f x t x x f x t, x B Ž x. Ž 3.4. for 1, 2,...,.

7 36 DANIILIDIS AND HADJISAVVAS Now for every we choose x x, hece Ž 3.4. becomes 1 1 Ž Ž.. f x t x x fž x. t Ž for 1, 2,...,. Sce f s quascovex, Ž 3.5. mples that Ž. Ž Ž.. f x f x t x x Ž for 1, 2,...,. Combg wth Ž 3.5. ad addg for 1, 2,...,, we Ž get 0 Ý t., a cotradcto. 1 I 18 t was proved that the subdfferetal of a covex fucto s a maxmal mootoe ad maxmal cyclcally mootoe operator. A aalogous property does ot hold for quascovex fuctos, sce, for the quascovex fucto fž x. sgž x. ' x, x R, t s kow Žsee 15. that f s ot maxmal quasmootoe. The followg proposto shows that t s ether maxmal cyclcally quasmootoe: PROPOSITION 3.3. quasmootoe. Eery quasmootoe operator T: R 2 R s cyclcally Proof. We assume to the cotrary that the operator T s quasmoo- toe ad there exst x, x,..., x R, x TŽ x., such that 1 2 Ž x, x1 x. 0 Ž 3.7. for 1, 2,..., Ž where x x Set xm max 1,2,..., x. The relato Ž 3.7. mples that xm 0. O the other had, sce xm1 x M, we coclude from Ž 3.7. that x 0. Thus Ž x, x x. M 1 M1 M M1 0, whle Ž x, x x. M M M1 0, whch cotradcts the defto of quasmootocty. We ow troduce cyclc pseudomootocty: DEFINITION 3.4. A operator T: X 2 X * s called cyclcally pseudomootoe, f for every x 1, x 2,..., x X, the followg mplcato holds: 1,2,..., 4, x TŽ x. : Ž x, x1 x. 0 j 1,2,..., 4, x TŽ x. : Ž x, x x. 0 Ž 3.8. j j j j1 j Ž where x x Oe ca easly check that every cyclcally mootoe operator s cyclcally pseudomootoe, whle every cyclcally pseudomootoe operator s pseudomootoe ad cyclcally quasmootoe. O the other had, the

8 SUBDIFFERENTIALS AND QUASICONVEXITY 37 followg example shows that cyclc geeralzed mootocty dffers essetally from geeralzed mootocty: 2 2 a b Ž. Ž. 2 2 EXAMPLE 3.5. Let T: R R be defed by T a, b b, a. The the operator T s mootoe Žad eve strogly mootoe,.e., Ž Ž. Ž satsfes T x T y, x y k x y for all x, y R where k s a postve costat.. I partcular, T s pseudomootoe ad quasmootoe. However, t s ot cyclcally quasmootoe, as oe sees by cosderg the pots x Ž 1, 0., x Ž 0, 1., x Ž 1, 0., ad x Ž 0, Ž. We ow show the followg stregtheg of Theorem THEOREM 3.6. Let f: X R be a lsc fucto. If f s pseudocoex, the f s cyclcally pseudomootoe. Coersely, f f s pseudomootoe ad f s radally cotuous, the f s pseudocoex. Proof. Aga we have oly to show that f f s pseudocovex the f s cyclcally pseudomootoe. Assume to the cotrary that there exst x, x,..., x DŽ f. ad x fž x. such that Žx, x x , for 1, 2,..., Ž where x x., whle for some ad some x fž x. 1 1 o o o we have Ž x, x x. 0. Ž 3.9. o o1 o By the defto of pseudocovexty Žrelato Ž we have fž x. 1 fž x., for 1, 2,...,, hece all fž x. are equal. I partcular, fž x. o1 fž x., whch cotradcts Ž 3.9. vew of Proposto 2.3. o 4. PROPER QUASIMONOTONICITY The deftos of mootocty ad pseudomootocty have a equvalet formulato, whch volves a fte cycle of pots ad ts covex hull: PROPOSITION 4.1. Ž. A operator T s mootoe, f ad oly f for ay x 1, x 2,..., x X ad eery y Ý 1 x, wth Ý 1 1 ad 0, oe has Ý 1 x TŽ x. sup Ž x, y x. 0. Ž 4.1. Ž. A operator T wth coex doma DŽ T. s pseudomootoe, f ad oly f for ay x, x,..., x X ad eery y Ý x, wth 1 2 1

9 38 DANIILIDIS AND HADJISAVVAS Ý 1 1 ad 0, the followg mplcato holds: 1,2,..., 4, x TŽ x. : Ž x, y x. 0 j 1,2,..., 4, x TŽ x. : Ž x, y x. 0. Ž 4.2. j j j j Proof. If the operator T satsfes codto Ž 4.1. Žrespectvely Ž 4.2.., the by choosg y Ž x x , we coclude that t s mootoe Ž respectvely, pseudomootoe.. Hece t remas to show the two opposte drectos. Let us frst suppose that T s mootoe. The for ay x 1, x 2,..., x X, Ž. Ž. ay x T x for 1, 2,..., ad ay y Ý j1 jx j, wth Ý j1 j 1 ad j 0, we have Ý Ž x, y x. Ý Ý jž x, x j x. 1 1 j1 Ý j x, x j x x j, x x j j Ý x x j Ž j, x j x. 0, j Ž. Ž. where the last equalty s a cosequece of the mootocty of T. Hece T satsfes relato Ž We ow suppose that the operator T s pseudomootoe. If relato Ž 4.2. does ot hold, the there exst x, x,..., x X, x TŽ x. 1 2 for 1, 2,...,, ad some y Ý j1 jx j wth Ý j1 j 1 ad j 0, such that Ž x, y x. 0 Ž 4.3. whle for at least oe Ž say 1,. Ž x 1, y x1. 0. Ž 4.4. I partcular we have x, x,..., x DT, Ž. hece TŽ y Choose ay y* TŽ y.. Relatos Ž 2.7. ad Ž 4.3. show that Ž y*, y x. 0 Ž 4.5. for all y* TŽ y. ad all s. Sce Ý Ž y*, y x. 0, relatos Ž 4.5. show that Ž y*, y x. 0 for all s. O the other had, relato Ž 4.4. together wth relato Ž 2.8. mply that Ž y*, y x1. 0, a cotradcto. I vew of Proposto 4.1, oe could seek a equvalet formulato for the defto of quasmootocty, whch would volve aga the covex

10 SUBDIFFERENTIALS AND QUASICONVEXITY 39 hull of a fte cycle. However, cotrast to mootoe ad pseudomootoe operators, ths leads to a dfferet, more restrctve defto: DEFINITION 4.2. A operator T: X 2 X * s called properly quasmootoe, f for every x 1, x 2,..., x X ad every y Ý 1 x, wth Ý 1 ad 0, there exsts such that 1 x T x : x Ž. Ž, y x. 0. Ž 4.6. Choosg y Ž x x , we see that a properly quasmootoe operator s quasmootoe. As Proposto 3.3, t s easy to show that the coverse s true wheever X R; however, t s ot true geeral, as the followg example shows. 2 EXAMPLE 4.3. Let X R, x Ž 0, 1., x Ž 0, 0., x Ž 1, We 2 2 defe T: R R by TŽ x. Ž 1, 1., TŽ x. Ž 1, 0., TŽ x. Ž 0, , ad TŽ x. 0 otherwse. It s easy to check that T s quasmootoe but ot properly quasmootoe Žt suffces to cosder y Ž x x x The class of properly quasmootoe operators, though strctly smaller tha the class of quasmootoe operators, s a sese ot much smaller. Ths s show the ext proposto. PROPOSITION 4.4. Ž. Eery pseudomootoe operator wth coex doma s properly quasmootoe. Ž. Eery cyclcally quasmootoe operator s properly quasmootoe. Proof. Ž. Ths s a obvous cosequece of Proposto 4.1Ž.. Ž. Suppose that the operator T s ot properly quasmootoe. The there would exst x, x,..., x DT, Ž. x TŽ x. 1 2, ad y Ý x wth 0, such that 1 Ž x, y x. 0 Ž 4.7. for 1, 2,...,. Set x x. Relato Ž 4.7. mples that Ý Žx Ž1. 1 j j Ž1., x j x. 0. It follows that for some x x we have Žx, x x. Ž1. j 1 Ž1. j Ž1. 0. We set x x ad apply relato Ž 4.7. Ž2. j aga. Cotug ths way, we defe a sequece x Ž1., x Ž2.,... such that Ž x Ž k., xž k1. xž k.. 0 Ž 4.8. for all k N. Sce the set x, x,..., x s fte, there exst m, k N, m k such that xž k1. x Ž m.. Thus, for the fte sequece of pots x Ž m., x,..., x, relato Ž 4.8. Ž m1. Ž k. holds. Ths meas that T s ot cyclcally quasmootoe.

11 40 DANIILIDIS AND HADJISAVVAS Combg Proposto 4.4Ž. ad Theorem 3.2, we get the followg corollary. COROLLARY 4.5. A lower semcotuous fucto f s quascoex f ad oly f f s properly quasmootoe. The coverse of Proposto 4.4 does ot hold. For stace, the operator T defed Example 3.5 s properly quasmootoe Žsce t s mootoe, hece pseudomootoe., but ot cyclcally quasmootoe. O the other had, ay subdfferetal of a cotuous quascovex fucto f s properly quasmootoe, but ot pseudomootoe uless f s also pseudocovex. Thus, betwee the varous geeralzed mootocty propertes we cosdered, the followg strct mplcatos hold, ad oe other: cyclcally mootoe mootoe cyclcally pseudomootoe pseudomootoe cyclcally quasmootoe properly quasmootoe quasmootoe Note that the mplcato Žpseudomootoe properly quasmootoe. holds uder the assumpto that the doma of the operator s covex. X * We recall that a multvalued mappg G: X 2 s called KKM 11, f for ay x, x,..., x X ad ay y cox, x,..., x oe has y Ž. X * G x. It s easy to see that a operator T: X 2 s properly quasmootoe f ad oly f the multvalued mappg G: X 2 X * defed by ½ 5 GŽ x. y K : sup Ž x*, y x. 0 Ž 4.9. x*tž x. s KKM. Ths suggests a obvous applcato to varatoal equaltes. All kow theorems of exstece of solutos for quasmootoe varatoal equalty problems requre extra assumptos o the doma of the operator Žsee 12. ad, the case of a multvalued operator, o ts values Žsee. 9. As the followg theorem shows, exstece of solutos for properly quasmootoe operators requres very weak assumptos. We frst recall from 1 the followg defto. DEFINITION 4.6. The operator T: X 2 X * s called upper hemcotuous, f ts restrcto to le segmets of ts doma s upper semcotuous, whe X* s equpped to the weak- topology.

12 SUBDIFFERENTIALS AND QUASICONVEXITY 41 We ow have: THEOREM 4.7. Let K be a oempty, coex ad w-compact subset of X. If T s a properly quasmootoe, upper hemcotuous operator wth K DT, Ž. the there exsts a x0 K, such that for eery x K, there exsts x* TŽ x. such that 0 Ž x*, x x0. 0. Ž Proof. Sce the multvalued map G defed by Ž 4.9. s KKM, ad the sets GŽ x. are obvously weakly closed, by Ky Fa s Lemma 10 oe has GŽ x.. Take ay x GŽ x. x K 0 x K. We shall show that x0 s actually a soluto of Ž We assume, to the cotrary, that for some x K ad all x* TŽ x. 0 we have Ž x*, x x. 0. The set V x* X*: Ž x*, x x s a w*- eghborhood of TŽ x.; hece, f we set x tx Ž 1 t. 0 t x 0, by the upper hemcotuty assumpto, we have TŽ x. t V for all t suffcetly small. Sce x x tž x x., ths meas that Ž x*, x x. t 0 0 t 0 0 for all x* TŽ x.,.e., x GŽ x. t 0 t. Ths cotradcts the defto of x 0. We coclude wth a fal remark. The oto of a quasmootoe operator was troduced to descrbe a property that characterzes the subdfferetal of a lsc quascovex fucto. Sce proper quasmootocty does exactly the same thg ad s drectly related to the KKM property, t s possbly a good caddate to replace quasmootocty most theoretcal ad practcal applcatos. REFERENCES 1. J. P. Aub ad A. Cella, Dfferetal Iclusos, Sprger-Verlag, Berl, D. Aussel, J.-N. Corvellec, ad M. Lassode, Subdfferetal characterzato of quascovexty ad covexty, J. Coex Aal. 1 Ž 1994., D. Aussel, J.-N. Corvellec, ad M. Lassode, Mea value property ad subdfferetal crtera for lower semcotuous fuctos, Tras. Amer. Math. Soc. 347 Ž 1995., D. Aussel, Subdfferetal propertes of quascovex ad pseudocovex fuctos: A ufed approach, J. Optm. Theory Appl. 97 Ž 1998., M. Avrel, W. E. Dewert, S. Schable, ad I. Zag, Geeralzed Cocavty, Pleum Publshg Corporato, New York, F. H. Clarke, Optmzato ad Nosmooth Aalyss, Wley-Iterscece, New York, R. Correa, A. Joffre, ad T. Thbault, Characterzato of lower semcotuous covex fuctos, Proc. Amer. Math. Soc. 116 Ž 1992., R. Correa, A. Joffre, ad T. Thbault, Subdfferetal mootocty as characterzato of covex fuctos, Numer. Fuct. Aal. Optm. 15 Ž 1994.,

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