CONSTRUCTING GENERALIZED MEAN FUNCTIONS USING CONVEX FUNCTIONS WITH REGULARITY CONDITIONS

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1 CONSTRUCTING GENERALIZED MEAN FUNCTIONS USING CONVEX FUNCTIONS WITH REGULARITY CONDITIONS YUN-BIN ZHAO, SHU-CHERNG FANG, AND DUAN LI Abstract. The geeralzed mea fucto has bee wdely used covex aalyss ad mathematcal programmg. Ths paper studes a further geeralzato of such a fucto. A ecessary ad suffcet codto s obtaed for the covexty of a geeralzed fucto. Addtoal suffcet codtos that ca be easly checked are derved for the purpose of detfyg some classes of fuctos whch guaratee the covexty of the geeralzed fuctos. We show that some ew classes of covex fuctos wth certa regularty such as S -regularty ca be used as buldg blocks to costruct such geeralzed fuctos. Key words. Covexty, mathematcal programmg, geeralzed mea fucto, self-cocordat fuctos, S -regular fuctos. AMS subject classfcatos. 90C30, 90C25, 52A4, 49J52. Itroducto. I ths paper, we deote the -dmesoal Eucldea space by R, ts oegatve orthat by R+, ad postve orthat by R++. I 934, Hardy, Lttlewood ad Pólya [3] cosdered the followg fucto uder the ame of geeralzed mea:. Υ w x = φ w φx where φ s a real, strctly creasg, covex fucto defed o a subset of R ad w = w, w 2,..., w T s a gve vector R+. Assumg that φ > 0, φ > 0 ad > 0, they showed a equvalet codto for the covexty of Υ w. Whe φ s three tmes dfferetable, Be-Tal ad Teboulle [2] establshed aother equvalet codto for Υ w beg covex see ext secto for detals. The geeralzed mea fucto. has may applcatos optmzato. Be- Tal ad Teboule [2] demostrated a terestg applcato of. a cotuous form o pealty fuctos ad dualty formulato of stochastc olear programmg problems. However, the most wdely used geeralzed meas are the logorthmc-expoetoal ad p-orm fuctos: /p f w x = log w e x, p w x = w x p for x = x,..., x T R. They correspod to the specal cases of Υ w wth φt = e t ad φt = t p, respectvely. Needless to say that the log-exp fucto has bee wdely used covex aalyss ad mathematcal programmg. For example, a geometrc program see Duff et Isttute of Appled Mathematcs, AMSS, Chese Academy of Sceces, Bejg 00080, Cha Emal: ybzhao@amss.ac.c. Ths author s work was partally supported by the Natoal Natural Scece Foudato of Cha uder Grat No ad Grat No Correspodg author. Idustral Egeerg ad Operatos Research, North Carola State Uversty, Ralegh, NC , USA Emal: fag@eos.csu.edu. Ths author s work was partally supported by the US Army Research Offce Grat No. W9NF-04-D Departmet of Systems Egeerg ad Egeerg Maagemet, Chese Uversty of Hog Kog, Shat, NT, Hog Kog Emal: dl@se.cuhk.edu.hk. Ths author s work was partally supported by Grat CUHK480/03E, Research Grat Coucl, Hog Kog.

2 al. [8] ad Boyd ad Vadeberghe [6] ca be coverted to a covex programmg problem by usg the log-exp fucto so that the teror-pot algorthms ca be developed to solve geometrc programs wth great effcecy Kortaek et al. [4]. Aother example s cocered wth the odfferetable mmax problem m max g y, y D where g, =,...,, are real fuctos defed o a covex set D R m. Sce the recesso fucto of the log-exp fucto s the max-fucto see Rockafellar [20],.e., max x = lm ε 0+ εf x ε where f = f w ad w =,,...,, the above odfferetal optmzato problem ca be approxmated by solvg the followg optmzato problem m ε log e g y ε. y D Ths objectve fucto s dfferetable ad covex, f every g y s. Other applcatos of the log-exp fucto optmzato ca be foud Be-Tal [], Be-Tal ad Teboulle [3], Zag [25], Bersekas [4], Polyak [9], Fag [9, 0], L ad Fag [5], Peg ad L [7], Brbl et al. [5], Su ad L [22, 23, 24], etc. It s worth metog that the cojugate fucto of the log-exp fucto happes to be the well-kow Shao s etropy fucto [2] whch plays a vtal role so may felds ragg from the mage ehacemet to ecoomcs ad from statstcal mechacs to uclear physcs see, Buck ad Macaulay [7] ad Fag et al. []. We cosder ths paper a further geeralzato of. the followg form:.2 Γ w x = Ψ w φ x where φ : Ω R, =,...,, are covex, twce dfferetable but ot ecessarly beg strctly creasg fuctos defed o a ope covex set Ω R, Ψ : Ω R s covex, twce dfferetable ad strctly creasg, ad w R+ s a gve vector. Clearly, Υ w s a specal case of Γ w wth φ = φ 2 =... = φ = Ψ = φ. For coveece, ths paper, we stll call Γ w gve by.2 a geeralzed mea fucto, ad we call φ the er fucto ad Ψ the outer fucto of Γ w. To assure the well-defedess of Γ w, we aturally requre that Coe[φ Ω] ΨΩ, where Coe[φ Ω] deotes the coe geerated by the set φ Ω. As the case of Υ w, we would lke to derve certa suffcet ad ecessary codtos for the fucto Γ w to be covex. Moreover, we hope to fd a systematc way to explctly costruct some classes of covex Γ w. It s terestg to pot out that Γ w s by o meas a ew research subject. I fact, t was essetally studed by W. Fechel hs lecture otes of Covex Coes, Sets ad Fuctos 953 [2]. Based o the propertes of level sets ad characterstc roots of Hessa matrces of fuctos volved, Fechel derved some suffcet ad ecessary codtos for the covexty of the geeralzed mea fucto Γ w. The codtos he derved, however, are rather complcated, ad there s o smple test to decde what kd of fuctos may admt these complcated propertes. Ulke Fechel s approach, our aalyss ths paper depeds oly o the fucto value, ts frst dervatve, ad secod dervatve to provde a suffcet ad ecessary codto for Γ w beg covex. The ecessary ad suffcet codto we derve ths paper 2

3 ca be vewed as a geeralzato of that [3] cocerg the fucto.. We ca also use related suffcet codtos to explctly costruct cocrete examples of covex Γ w. Moreover, we show how the so-called S -regular fuctos to be defed ths paper ca be used to costruct covex geeralzed mea fuctos. The rest of the paper s orgazed as follows. I Secto 2, we vestgate the codtos that assure the covexty of the geeralzed mea fucto Γ w. I Secto 3, we detfy some classes of fuctos that satsfy the codtos derved Secto 2, ad llustrate how the geeralzed mea fucto Γ w ca be explctly costructed. Coclusos are gve the last secto. 2. Necessary ad Suffcet Codtos for the Covexty of Γ w. Let us start wth a smple lemma proof omtted that shows the verse of a creasg covex fucto s cocave ad creasg. Lemma 2.. Let Ω be a ope covex subset of R ad Ψ : Ω R be a real fucto defed o Ω. The Ψ s strctly covex ad strctly creasg f ad oly f ts verse Ψ : R Ω s strctly cocave ad strctly creasg. Notce that f w = 0, for some, the the term w φ x ca be removed from the expresso of Γ w x, ad t suffces to cosder Γ w defed o R. Thus, wthout loss of geeralty, we may assume that the vector w R++ throughout the rest of the paper. To study the covexty of Γ w, whe assumg that φ, =,...,, ad Ψ are twce dfferetable, we eed to check the propertes of ts Hessa matrx. Let Sce xw x Moreover, = w φ x, we have x w = w φ x. Γ w x = Ψ x w w φ x. 2 Γ w x 2 = Ψ x w w φ x 2 + Ψ x w w x, 2 Γ w x x j = Ψ x w w w j φ x φ jx j for j. Cosequetly, the Hessa matrces of Γ w becomes w x Γ w x 2 = Ψ x w 0 w 2 2x w x w φ x + Ψ x w w 2 φ 2x [w φ x, w 2 φ 2x 2,..., w φ x ]. w φ x Note that whe φ, =,...,, are all covex ad Ψ s covex ad creasg, by Lemma 2., we see that the frst term o the rght-had sde of 2. s a postve 3

4 semdefte matrx multpled by a postve coeffcet Ψ x w, whle the secod s a rak oe matrx multpled by a egatve coeffcet Ψ x w. Some codtos for covexty of the fucto Υ w x has already bee studed [3] ad [2]. We summarze ther results here. Theorem 2.2. [3] Uder the codtos of φ > 0, φ > 0 ad > 0, the fucto Υ w x defed by. s covex f ad oly f the followg codto holds: w [φ x ] 2 x [φ y] 2 y for y = Υ w x. Be-Tal ad Teboulle [2] also provded a dfferet suffcet ad ecessary codto, uder certa assumptos, for the covexty of the fucto Υ w x. Theorem 2.3. [2] Let φt C 3. Υ w x s covex f ad oly f /ρt s covex, where ρt = /φ. It s possble to exted the aalyss [2] to derve suffcet codtos for more geeral stuatos where the covexty of Γ w x defed by.2 s cosdered. Actually, the followg result ca be proved alog the le of the proof of Lemma ad Theorem [2]. Theorem 2.4. [2] Let Ψt C 3 ad φ t C 3 be strctly creasg ad ρt = Ψ t/ψ t. If /ρt s covex ad Ψ φ t s covex, for =,...,, the Γ w x gve by.2 s covex. Note that f φ s suffcetly smooth, /ρt s covex, where ρt = t/φ t, f ad oly f ts secod dervatve s oegatve,.e., = φ 3 2φ 2 + φ 2 ρ 2 0. Thus, to check the covexty of /ρt, t s usually eeded to check the above equalty volvg the thrd ad forth dervatve of the fucto φ. Note that Theorem 2.2, however, does ot requre the thrd or fourth dfferetablty of the fucto φ. I what follows, we geeralze the results Theorem 2.2 for the fucto Γ w x. Although the basc dea of our proof s closely related to that of [3], the proof s ot straghtforward. For completeess, we gve a detaled proof for the result. Theorem 2.5. Let Ω R be ope ad covex, Ψ : Ω R be covex, twce dfferetable ad strctly creasg, φ : Ω R, =,...,, be strctly covex ad twce dfferetable, ad w R++ be a gve vector. The the geeralzed mea fucto Γ w x = Ψ w φ x {}}{ s covex o Ω := Ω Ω f ad oly f Ψ [φ y w x ] x [Ψ y] 2 for x Ω ad y = Γ w x. Moreover, Γ w x s strctly covex f ad oly f the equalty 2.2 holds strctly. Proof. Let y = Γ w x = Ψ x w. The x w = Ψy ad 2.3 Ψ x w Ψ y =. 4

5 Dfferetatg both sdes wth respect to y ad use the above relatos, we have Therefore, 0 = Ψ x w [Ψ y] 2 + Ψ x w Ψ y = Ψ x w [Ψ y] 2 + Ψ y Ψ y. 2.4 Ψ x w = Ψ y [Ψ y] 3. Combg 2.3 ad 2.4 yelds 2.5Ψ x w + Ψ x w [φ w x ] 2 [Ψ y] 2 x = w [φ x] 2 Ψ y x. [Ψ y] 3 Frst we prove that Γ w x s covex, f 2.2 holds. It suffces to show that the Hessa matrx of Γ w x s postve sem-defte. For ay d R ad x Ω, the Cauchy-Schwartz equalty mples that 2 w φ x d = [ ] w x d w x d 2 w x φ x w [φ x ] 2 x By Lemma 2., we kow Ψ s cocave ad hece Ψ x w 0 for all x w. Combg ths fact wth the above equalty, we see that, for ay d R, 2. d T 2 Γ w x 2 d 2 = Ψ x w w x d 2 + Ψ x w w φ x d Ψ x w = = 0. w x d 2 w x d 2 w x d 2 [ + Ψ x w Ψ x w + Ψ x w [Ψ y] 2 w [φ x]2 x [Ψ y] 3 w x d 2 Ψ y w [φ x ] 2 x ] [φ w x ] 2 x The last equalty follows from 2.5 ad the last equalty follows from the fact that the frst quatty o the rght-had sde,.e., w x d 2, s oegatve, ad the secod quatty s also oegatve due to our assumpto. Cosequetly, we have prove that the Hessa matrx 2 Γ w x s postve sem-defte, as desred. 2 5

6 Coversely, we would lke to show that equalty 2.2 holds, f Γ w x s covex. For ay vector 0 d R, kowg 2.3, 2.4 ad the covexty of Γ w x, we have 0 d T 2 Γ w x 2 d = 2 Ψ x w w x d 2 + Ψ x w w φ x d = 2 Ψ w x d 2 Ψ y y Ψ y 3 w φ x d [ ] = w x d 2 Ψ y Ψ y [ w φ x d ] Ψ y 3 w x d 2. Notce that the above equalty holds for ay vector d R. I partcular, let The, we have d = w φ x d =, φ x x k= w [φ k x k] 2 k k x k w x d 2 = As a result, the equalty 2.6 reduces to [ 0 w [φ x]2 Ψ y Ψ y Ψ y 3 x, =,...,. w. [φ x]2 x ] [φ w x ] 2 x We see that equalty 2.2 deed holds. The result about strct covexty ca be easly checked out. Theorem 2.5 geeralzes the result of Theorem 2.2 cocerg Υ w x to the more geeral fucto Γ w x, whle Theorem 2.4 geeralzes the suffcet codto of Theorem 2.3 cocerg Υ w x to the fucto Γ w x, uder dfferet assumptos. Except for some very smple cases, such as e t or x p, these results, however, do ot gve us a cocrete class of fuctos whch ca be used to costruct specfc geeralzed mea fuctos. The purpose of the remader of ths paper s to provde a systematc way to detfy the desred class of fuctos. Our aalyss here the paper s based oly o the result of Theorem 2.5. We beleve that there should also exst some parallel results based o Theorem 2.4. To ths ed, two related suffcecy results of Theorem 2.5 are derved below for ther coveet usage costructg covex Γ w see ext secto. Theorem 2.6. Let Ω be a ope covex subset of R, Ψ : Ω R be strctly creasg, twce dfferetable ad covex, φ : Ω R, =,...,, be strctly covex ad twce dfferetable, ad w R++ be a gve vector. Assume that there exsts a scalar α R such that 2.7 αψtψ t [Ψ t] 2 for t Ω. The the fucto Γ w s covex o Ω f 2.8 w [φ x ] 2 x αψy for x Ω, 6

7 where y = Γ w x. Proof. Multplyg both sdes of 2.8 by Ψ y ad applyg 2.7, we see that codto 2.2 holds. The result follows from Theorem 2.5 mmedately. Theorem 2.7. Let Ω be a ope covex subset of R, Ψ : Ω R be strctly creasg, twce dfferetable ad covex, φ : Ω R, =,...,, be strctly covex ad twce dfferetable, ad w R++ be a gve vector. Assume that there exst 0 α R, =,...,, holdg the same sg such that 2.9 α φ t t [φ t] 2 for t Ω, ad there exsts a α R such that the equalty 2.7 holds. The the fucto Γ w s covex f 2.0 or 2. α max α α m α whe α > 0 for all, whe α < 0 for all. Proof. Takg y = Γ w x, we see two cases. Case : α > 0 for =,...,. I ths case, 2.9 mples that φ t 0 for t Ω ad 2.0 mples that [φ w t]2 x w α φ x max α w φ x αψy. Case 2: α < 0 for =,...,. I ths case, 2.9 mples that φ t 0 for t Ω ad 2. mples that [φ w t]2 x w α φ x m α w φ x αψy. Both cases yeld 2.8 ad the desred result follows from Theorem 2.2. A specal case of φ t = φ 2 t =... = Ψt mmedately leads to the ext result. Corollary 2.8. Let Ω be a ope covex set R, φ : Ω R be a covex, twce dfferetable ad strctly creasg fucto, ad w R++ be a gve vector. If there exsts a α 0 such that 2.2 [φ t] 2 = αφt t for t Ω, the the fucto Υ w x = φ w φx s covex o Ω. Ths result ca also follow drectly from the aforemetoed Theorem 2.3 due to Be-Tal ad Teboulle [2]. I fact, t s easy to verfy that the relato 2.2 mples that the secod dervatve of φ / s equal to zero, ad thus by Theorem 2.3 the fucto Υ w x s covex. Remark 2.. The fuctos satsfyg a dfferetal equalty such as 2.7 are related to the so-called self-cocordat barrer fucto troduced by Nesterov ad Nemrovsky [6]. Recall that a C 3 fucto ξ : 0, R s sad to be self-cocordat f ξ s covex ad there exsts a costat µ > 0 such that 2.3 ξ t µ ξ t 3 2 for t 0,. 7

8 Moreover, the self-cocordat fucto ξ s called a self-cocordat barrer fucto f there exsts a costat µ 2 > 0 such that 2.4 ξ t µ 2 [f t] 2 for t 0,. Combg 2.3 ad 2.4 yelds ξ tξ t µ[ξ t] 2. Ths dcates that the frst-order dervatve fucto of a self-cocordat barrer fucto,.e., gt := ξ t, satsfes the equalty 2.7. A self-cocordat fucto ξ tself may also satsfy a equalty lke 2.7 or 2.9. Remark 2.2. The fuctos satsfyg a dfferetal equalty such as 2.7 also appear covexty theory. doma Ω, we cosder the covexty of the fucto ht := φt Gve a twce dfferetable fucto φt > 0 o ts o Ω. Notce that h t = 2[φ t] 2 φt t [φt] 3 for t Ω. Hece the fucto ht = φt s covex f ad oly f the equalty φtφ t 2[φ t] 2 holds o Ω. Moreover, f φt t [φ t] 2, the covex fucto ht satsfes a reverse equalty,.e., hth t [h t] 2 o Ω. From ths observato, a related questo arses. Gve a fucto φt > 0 o Ω ad a costat r > 0, whe wll the fucto ht := φt become covex ad satsfy r a equalty such as 2.9? A straghtforward aalyss leads to the ext result. Theorem 2.9. Let Ω be a covex subset of R ad φ : Ω 0, be a fucto. If φt t [φ t] 2 for t Ω, the, for ay r > 0, the fucto ht := φt s r covex ad hth t [h t] 2 for t Ω. Coversely, f there exsts a r > 0 such that ht := φt s covex ad hth t [h t] 2 for t Ω, the φt t [φ t] 2 r for t Ω. Let Ω be a covex subset of R, τ > 0, ad φ : Ω τ, be a fucto. If φt t [φ t] 2 for t Ω, the, for ay scalar r > 0 ad T > 0, the fucto h T t := T + φt s covex ad αh r T th T t [h T t]2 for t Ω, where α = T τ r +. Proof. For case, t s suffcet to see that ad h t = r2 φ t 2 + r[φ t 2 φt t] φt r+2, hth t [h t] 2 = r[φ t 2 φt t] φt 2r+. For case, t s easy to verfy that h T t = h t ad T φt r h T th T t [h T t] 2 = r[φ t 2 φt t]. + φt 2r+ The the desred result follows. The above results dcate that f we have a fucto φ satsfyg the equalty 2.7 wth α =, the we may costruct a fucto h from φ such that h satsfes the 8

9 coverse dfferetable equalty αhth t [h t] 2 for some costat α. Moreover, f we take a T-traslato of the value of the fucto h, the the resultg fucto satsfes the coverse dfferetable equalty wth a α that ca be reduced to be smaller tha ay threshold gve 0, provded a sutable choce of T > 0. Ths fact wll be used ear the ed of Secto Costructg covex geeralzed mea fuctos Γ w. I ths secto, we develop procedures to detfy some classes of fuctos that satsfy equalty 2.7 ad/or equalty 2.9 so that we have buldg blocks for costructg cocrete covex fucto Γ w x. Frst, we gve a result that detfes fuctos satsfyg the equato 2.2. Obvously, ths class of fuctos satsfes both equaltes 2.7 ad 2.9. Theorem 3.. Let Ω be a ope set R ad φ : Ω R be a covex, twce dfferetable ad strctly creasg fucto satsfyg equato 2.2 wth a costat α 0. The, whe α =, φ s the form of φt = γe t β for some γ > 0 ad β > 0. whe 0 < α wth v α := sup t Ω α t beg fte, φ s the form of α α φt = γ α t + β α for some γ > 0 ad β v. whe α < 0 wth u α := sup t Ω α t beg fte, φ s the form of φt = γ β α α α t α for some γ > 0 ad β u. Note that results ad were poted out [2] ad [3] ad result ca be easly derved. The above result leads to the followg cosequece related to Υ w. Corollary 3.2. The followg fuctos ca be used to explctly costruct a covex geeralzed mea fucto Υ w x = φ w φx over Ω : φt = γe t β over Ω = R wth γ > 0 ad β > 0. φt = γ φt = p p t + β over Ω = η, wth p >, γ > 0 ad β η γ over Ω =, η wth p > 0, γ > 0 ad β η β p tp p. v φt = γβ p tp over Ω =, η wth 0 < p <, γ > 0 ad β η p. Aga, results ad were gve [2] ad [3] ad results ad v ca be easly derved. The fuctos lsted Corollary 3.2 actually form a complete bass the sese that the fucto φ case satsfes codto 2.2 wth α = ; the fucto φ case satsfes codto 2.2 wth α = > ; the fucto φ case satsfes codto 2.2 wth α = p p+ 0, ; ad the fucto φ v satsfes codto 2.2 wth α = p p < 0. We ow try to detfy some class of fuctos that satsfy equaltes 2.7 ad/or 2.9. For smplcty, we oly cosder covex, twce dfferetable, strctly creasg fuctos ϑ o Ω = 0,. Let us frst defe the followg four categores of such fuctos: p p U = {ϑ : There exsts α R such that αϑtϑ t [ϑ t] 2 for t Ω}; U 2 = {ϑ : There exsts α R such that αϑtϑ t [ϑ t] 2 for t Ω}; 9 p.

10 U 3 = {ϑ : There exst α α 2 such that α ϑtϑ t [ϑ t] 2 α 2 ϑtϑ t for t Ω}; U 4 = {ϑ : There exsts α R such that αϑtϑ t = [ϑ t] 2 for all t Ω}. It s evdet that U 4 U 3 U 2 U. As poted out Theorem 3., the class U 4 ca be gve explctly. By allowg α α 2, we show that U 3 s much broader tha U 4. I fact, may covex fuctos wth certa regulartes fall to the category U 3. To start, we troduce a ew class of fuctos wth certa regularty propertes. Defto 3.3. A covex, twce dfferetable, strctly creasg fucto δt : 0, R s called a S -regular fucto f δt vashes at t = 0 the sese of lm δ0 = lm δ 0 = lm δ 0 = 0; t 0 + t 0 + t 0 + ad there exst postve costats 0 < β β 2, p ad q such that 3. β [t + p t + q ] δ t β 2 [t + p t + q ], t > 0. Note that codto 3. actually mples the strct covexty of a S -regular fucto o 0,. I partcular, settg β = β 2, codto 3. reduces to a equato 3.2 δ t = t + p t + q. Takg tegrato twce ad otg that lm t 0+ δ0 = lm t 0+ δ 0 = 0, the uque soluto to equato 3.2 s 3.3 p,q t = t + p+ pp + t + q qq p + q t for p ad q >. pq I addto, sce lm q + [ t + q ]/q = lt +, we have 3.4 p, t = t + p+ pp + + lt + p + t for p. p Takg p = 3.4, we have 3.5, t = t Moreover, takg p = ad q = yelds + lt + 2t = 2 t2 t + lt ,2 t = 2 [ t + 2 t + 3t ]. I terms of ths partcular soluto p,q t, codto 3. ca be wrtte as 3.7 β p,qt δ t β 2 p,qt. By tegratg ad otg that lm t 0+ δ 0 = lm t 0+ δ0 = 0, we further have 3.8 β p,qt δ t β 2 p,qt 0

11 ad 3.9 β p,q t δt β 2 p,q t. Therefore, we ca see that the class of S -regular fuctos s qute broad. Later, by usg , we show that S -regular fuctos fall to the category U 3. It s worth metog that for ay p, q > cludg the case of q + the S -regular fucto p,q t s ot self-cocordat. I fact, the fucto p,q t does ot satsfy the equalty 2.3 sce δ t 0 ad δ t p + q as t 0 +. S -regular fuctos are somewhat aalogous to but dfferet from the selfregular fuctos defed [8]. As we have metoed above, S -regular fuctos are ot self-cocordat. The class of self-regular fuctos, however, has a large overlap wth self-cocordat fuctos. I what follows, we dsplay the relato amog the frst ad secod dervatves of S -regular fuctos, whch shows that ay S - regular fucto belogs to the category U 3. It should be metoed that the relatos amog the frst ad secod dervatves for self-regular fucto have bee studed [8]. Theorem 3.4. Let δt : 0, R be S -regular o 0,. The there exst c 2 c > 0 such that 3.0 c δtδ t [δ t] 2 c 2 for all t 0,,.e., the fucto δt U 3. Proof. We show that a S -regular fucto p,q t satsfes the property 3.0. Actually, we have p,q t p,qt [ p,qt] 2 = t+ p+ pp+ t+ q qq t+ p p p+q pq t + t+ q q [t + p t + q ] 2. p+q pq Dvdg the umerator ad deomator of the rght-had sde of the above equato by t + 2p = t + p+ t + p, we have p,qt p,qt [ p,qt] 2 = Therefore, 3. t+ p+ pp+ + t+ p+ t+ p+q qq p+qt t+ p+q pqt+ p p+q p qt+ p+q pqt+ p p,q t p,qt lm t [ p,qt] 2 = p p +. Sce p,qt = t + p t + q, we have lm t 0+ p,qt = p + q. Sce p,qt 0, p,qt 0 ad p,q t 0 as t 0 +, we have p,qt 2 [ p,qt 2 ] lm t 0 + = lm p,qt t 0 + [ p,qt] = lm Hece, we have lm t 0 + p,q t 2 p,qt p,qt = lm t 0 + t p,qt p,qt = 2p + q. p,qt p,qt 2[ p,qt] p,qt p,qt = 6p + q.

12 Usg the above relatos, we further have 3.2 p,q t p,qt p,qt p,qt + p,q t p,qt lm t 0 + [ p,qt] 2 = lm t p,qt p,qt = 2 + lm p,q t t p,qt p,qt lm t 0 + p,qt = 2 3. Notce that p,q t > 0, p,qt > 0 ad p,qt > 0 0,. From 3. ad 3.2, we ca see by cotuty that there exst two costats µ 2 µ > 0 such that µ p,qt p,qt [ p,qt] 2 µ 2 for t 0,. Together wth 3.7 through 3.9, ths mples that a S -regular fucto δt satsfes the followg equalty: 0 < µ β δtδ t [δ t] 2 β 2 µ 2, Therefore, 3.0 holds wth c := µ β ad c 2 := µ 2 β 2. A fact that should be poted out here s that ew fuctos U or U 2 ca be costructed by usg the basc operatos addto, multplcato, dvso ad composto o kow fuctos. The proof of the followg fact s omtted. Lemma 3.5. If φ : 0, 0,, φ U wth α = α ad ϕ : 0, 0,, ϕ U wth α = α 2, the φ + ϕ U wth α = 2 max{α, α 2 }. If φ : 0, 0,, φ U wth α 0, ] ad ϕ : 0, 0,, ϕ U wth α 2 0, ], the the multplcatve fucto φt ϕt U wth α =. Smlarly, f φ U 2 wth α ad ϕ U 2 wth α 2, the φt ϕt U 2 wth α =. If φ : 0, 0,, φ U 2 wth α ad ϕ : 0, 0,, ϕ U wth α 2 0, ], the the fucto φ ϕ U 2 wth α =. Smlarly, f φ U wth α 0, ] ad ϕ U 2 wth α 2, the φ ϕ U wth α =. v Let ϕ : 0, Ω R ad φ : Ω 0, be two covex fuctos. If φ U wth α > 0, the the composte fucto φ ϕt = φϕt U wth the same costat α. The ext result shows that the composte fuctos of e t belog to U 3. Lemma 3.6. Deote the expoetal fucto e t by expt ad the composto of m m expoetal fuctos by θ m t := m {}}{ exp exp expt. The 3.3 m θ mtθ mt [θ mt] 2 θ m tθ mt for t R. Proof. Let α m t := [θ mt] 2 /θ m tθ mt for t R. Sce α t, we ca prove the rght-had sde equalty of 3.3 usg v of Lemma 3.5 ad mathematcal ducto. For the left-had sde equalty, otce that θ mt = θ m tθ m t, θ mt = θ m tθ m t 2 + θ m tθ m t for t R. 2

13 Ths dcates that α m t = + α m tθ m t > + α m t for t R. It s easy to check that α 2 t 2,. The desred result follows by ducto. To costruct examples of the covex fucto Γ w, Theorem 2.7 tells us that t suffces to fd fuctos satsfyg the equaltes 2.7 ad 2.9 ad compare ther α values. The ext result s to estmate the α values, or equvaletly, to estmate the values of c ad c For smplcty, we use the S -regular fuctos wth p = ad q =, 2 to estmate requred c ad c 2. I fact, we have the followg result. Its proof s omtted here. Lemma 3.7. The S -regular fuctos, t ad,2 t gve by 3.5 ad 3.6, respectvely, satsfy codto 3.0 wth c = 2 ad c 2 = 2 3, that s, ,t,t [,t] 2 2, t,t, ,2t,2t [,2t] 2 2,2 t,2t for t 0,. We ow gve the last result o how to costruct some covex fuctos Γ w. Theorem 3.8. Let Ω be a ope covex subset of R. Let φ : Ω 0, be a covex, twce dfferetable, strctly creasg fucto o Ω. If φt t [φ t] 2 for t Ω, the the geeralzed mea fucto Γ w x := φ w φx r s covex o Ω for ay gve w R ++ ad r > 0. Let κ > 0 be a costat ad φ : Ω κ, be a covex, twce dfferetable, strctly creasg fucto satsfyg the equalty φt t [φ t] 2 for t Ω. The, for ay gve w R ++ ad T > 0, r > 0, the fucto {}}{ Γ 2 w x := l l l l w T + φx r s covex o Ω for ay postve teger l T κ r +. I fact, result comes from part of Theorem 2.9 ad Theorem 2.7. Result follows from Lemma 3.6 ad Theorem 2.7, ad part of Theorem 2.9. I fact, t suffces to take the er fucto h T t = T + φt ad outer fucto θ r m t, as m {}}{ l l lt. defed Lemma 3.6, whose verse fucto s gve by The above result partally aswers the followg terestg questo: Gve a covex fucto, how may tmes of log-trasformatos ca be appled whle retag the covexty? Usg Theorems 2.7, 2.9, 3.8 ad Lemma 3.7, we have the followg examples of covex Γ w. 3

14 Example 3.. [ ],j,j x r, l,j x r, l m+ {}}{ l l l m + e rx, x 0,. v l {}}{ l l l [ ] m +, x r, x τ,, l m,τ r +, τ > 0. It follows from Corollary 3.2 that the fucto x p over 0, satsfes 2.2 wth α = p 29 p. Hece, whe < p 2, we have α 2, ad whe < p 7, we have α By Theorems 3.7, both,2t ad, t satsfy codto 2.9 wth α = 2. From Theorem 2.7, we see the fuctos below are examples of covex Γ w. Example 3.2. Let < p 2 ad δ t =,2 t or, t, for t 0, ad =,...,. The Γ w x = w δ x p s covex o 0,. Before closg ths secto, we brefly llustrate a possble applcato of volvg fucto Γ w the regularzato method for solvg a olear programmg problem: m{f 0 x : x C}. For smplcty, we assume that C s a covex set ad f 0 s a covex fucto. Let µ > 0 be a postve parameter. Gve a strctly covex fucto Γ w, we cosder the followg problem: m{f 0 x + µγ w x : x C}. Ths problem becomes a strctly covex programmg problem wth a uque soluto, deoted by xµ, whch comprses of a cotuato trajectory {xµ : µ > 0}. Uder sutable codtos of f 0, Ψ ad φ, ths trajectory becomes bouded. I ths case, by settg µ 0, ay accumulato pot of xµ, as µ 0, s a soluto to the orgal problem. Thus, a path-followg algorthm ca be desged to follow ths trajectory to acheve the soluto of the orgal problem. The performace of such path followg algorthm certaly depeds o the choce of the fucto Γ w wth regularty codtos. 4. Coclusos. I ths paper, we have further exteded the theoretcal foudato for the geeralzed mea fucto. We have establshed a ecessary ad suffcet codto for such a geeralzato to be covex. Moreover, a systematc way to explctly costruct covex Γ w has bee developed. To ths ed, the cocept of S -regular fuctos has bee troduced. It should be oted that ay S -regular fucto s ot self-cocordat [6]. Ackowledgmets. We would lke to thak the two aoymous referees for ther sghtful commets whch help mprove sgfcatly the results ad presetato of the paper. Especally, we are grateful to oe of the referees who brought the refereces [2] ad [3] to our atteto, ad provded Theorem 2.4 ths paper. 4

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