MINIMIZATION OF A CONVEX SEPARABLE EXPONENTIAL FUNCTION SUBJECT TO LINEAR EQUALITY CONSTRAINT AND BOX CONSTRAINTS

Transcription

1 ournl of Pure n Applie Mthemtics Avnces n Applictions Volume 9 Numer Pges MINIMIZATION OF A CONVEX SEPARABLE EXPONENTIAL FUNCTION SUBECT TO LINEAR EQUALITY CONSTRAINT AND BOX CONSTRAINTS Deprtment of Informtics Neofit Rilsi South-Western University 2700 Blgoevgr Bulgri e-mil stefm@swu.g Astrct In this pper we consier the prolem of minimizing conve seprle eponentil function over fesile region efine y liner equlity constrint n o constrints (ouns on the vriles. Prolems of this form re interesting from oth theoreticl n prcticl point of view ecuse they rise in some mthemticl optimiztion prolems s well s in vrious prcticl pplictions. Algorithms of polynomil computtionl compleity re propose for solving prolems of this form n their convergence is prove. Some emples n results of numericl eperiments re lso presente.. Introuction Consier the following conve seprle progrm with n eponentil oective function liner equlity constrint n oune vriles 200 Mthemtics Suect Clssifiction 90C30 90C25. Keywors n phrses eponentil function conve progrmming seprle progrmming polynomil lgorithms computtionl compleity. Receive Ferury Scientific Avnces Pulishers

2 08 (CSE m min ( ( ( c c s e ( suect to α (2 (3 where s > m > 0 > 0 ( n { n}. 0 2 m Since c ( s m e > 0 then c ( re strictly m conve functions n since c ( s m e < 0 uner the ssumptions then functions c ( re ecresing. Consier lso the conve eponentil seprle progrm with liner equlity constrint n oune vriles which is similr to prolem (CSE efine y (-(3 (CESP ( ( min c c e (4 ef suect to α (5 (6 where > 0 > 0. Since 2 c ( e > 0 then c ( re strictly conve functions n since c ( e > 0 uner the ssumptions then functions c ( re incresing.

3 MINIMIZATION OF A CONVEX SEPARABLE 09 Prolems (CSE n (CESP re conve seprle progrmming prolems ecuse the oective functions n constrint functions re conve n seprle. Prolems (CSE n (CESP efine y (-(3 n (4-(6 respectively rise in prouction plnning n scheuling in lloction of resources in the theory of serch in sugrient optimiztion in fcility loction ([ ] etc. Prolems lie (CSE n (CESP n relte to them re suect of intensive stuy. Relte prolems n methos for them re consiere in [-20]. Algorithms for resource lloction prolems re propose in [ ] n lgorithms for fcility loction prolems re suggeste in [7 9] etc. Singly constrine qurtic progrms with oune vriles re consiere in [ ] etc. n some seprle progrms re consiere n methos for solving them re suggeste in [8 9] etc. Fesile regions (2-(3 n (5-(6 of prolems (CSE n (CESP respectively re nown s the npsc polytope. Vli inequlities cutting plnes n integrlity of the npsc polytopes re consiere in [6 8] etc. Methos for solving vritionl inequlities over fesile regions efine y npsc polytopes re propose in [0 6] etc. This pper is evote to the evelopment of new efficient polynomil lgorithms for solving prolems (CSE n (CESP. The pper is orgnize s follows. In Section 2 chrcteriztion theorems (necessry n sufficient conitions for the optiml solutions to the consiere prolems re prove. In Section 3 new lgorithms of polynomil compleity re suggeste n their convergence is prove. In Section 4 we consier some theoreticl n numericl spects of implementtion of the lgorithms n give some etensions of oth chrcteriztion theorems n lgorithms. In Section 5 we present results of some numericl eperiments.

4 0 2. Chrcteriztion Theorems 2.. Prolem (CSE First consier prolem (CSE efine y (-(3. Suppose tht the following ssumptions re stisfie. (. for ll. If for some then the vlue is etermine in vnce. (. α. Otherwise the constrints (2-(3 re inconsistent n X 0/ where X is efine y (2-(3. The Lgrngin for prolem (CSE is m L( u v ( s e α u ( v ( where R ; u v R n n R n consists of ll vectors with n rel components. The Krush-Kuhn-Tucer (KKT necessry n sufficient optimlity conitions for the minimum solution ( of prolem (CSE re m s me u v 0 (7 u ( 0 (8 v ( 0 (9 α (0

5 MINIMIZATION OF A CONVEX SEPARABLE ( u R R v (2 where u v re the Lgrnge multipliers ssocite with the constrints (2 respectively. If or for some we o not consier the corresponing conition (8 [conition (9] n Lgrnge multiplier respectively]. u [Lgrnge multiplier v Since u 0 v 0 n since the complementry conitions (8-(9 must e stisfie in orer to fin from systems (7-(2 we hve to consier ll possile cses for u v ll u v equl to 0; ll u v ifferent from 0; some of them equl to 0; n some of them ifferent from 0. The numer of these cses is 2 2n where 2n is the numer of ll u v where n. This is n enormous numer of cses especilly for lrge-scle prolems. For emple when n 500 we hve to consier 2 0 cses. Moreover in ech cse we hve to solve lrge-scle system of nonliner equtions in u v. Therefore the irect ppliction of the KKT theorem y using eplicit enumertion of ll possile cses for solving lrge-scle prolems of the consiere form woul not give result n we nee efficient methos for solving prolems uner consiertion. The following Theorem gives chrcteriztion of the optiml solution to prolem (CSE. Its proof is se on the KKT theorem. As we will see in Section 5 y using Theorem we cn solve prolem (CSE with n 500 vriles for ten-thousnth of secon on personl computer. Theorem (Chrcteriztion of the optiml solution to prolem (CSE. A fesile solution ( X where X is efine y

6 2 (2-(3 is the optiml solution to prolem (CSE if n only if there eists some R such tht ef m s me (3 ef m s me (4 ln( sm ln( m ef m sme < < m sme. (5 Remr. We will show elow tht > 0 so tht the epressions of in (5 (especilly epressions uner the logrithm sign re correct. Proof. Necessity. Let ( e the optiml solution to prolem (CSE. Then there eist constnts u v such tht KKT conitions (7-(2 re stisfie. Consier ll possile cses. ( If then u 0 ccoring to (2 n v 0 ccoring to (9. m Therefore (7 implies s m e u. then Since > 0 m s me m s me. ( If then u 0 ccoring to (8 n v 0 ccoring to m (2. Therefore (7 implies s m e v. Hence

7 MINIMIZATION OF A CONVEX SEPARABLE 3 m s me m s me. (c If < < then u v 0 ccoring to (8 n (9. m Therefore (7 implies s m e. ln( s m ln(. m m s me Hence n Since s > 0 m > 0 > 0 y the ssumption n > > it follows tht s m e m < s m e m s m e m > s m e m tht is s m e m < < s m e m. In prticulr if we ssume tht 0 since s 0 m > 0 > > then oviously 0/ n. Similrly if we ssume tht < 0 since s 0 m > 0 > 0 then 0/ n 0. > 0 0 In orer to escrie the cses ( ( (c it is convenient to introuce the ine sets efine y (3 (4 n (5 respectively. It is ovious tht. The necessity prt is prove. Sufficiency. Conversely let (3 (4 n (5 where R. X n components of stisfy

8 4 Set m sm e otine from ln( s m ln( m α; u v 0 for ; u m s me ( 0 ccoring to the efinition of v 0 for ; m u 0 v s me ( 0. ccoring to the efinition of for By using these epressions it is esy to chec tht conitions (7 (8 (9 n (2 re stisfie; n conitions (0 n ( re lso stisfie ccoring to the ssumption X. We hve prove tht u v stisfy KKT conitions (7-(2 which re necessry n sufficient conitions for fesile solution to e n optiml solution to conve minimiztion prolem. Therefore is n optiml solution to prolem (CSE n since ( c is strictly conve function then this optiml solution is unique. In view of the iscussion ove the importnce of Theorem consists in the fct tht it escries components of the optiml solution to prolem (CSE only through the Lgrnge multiplier ssocite with the equlity constrint (2. Since we o not now the optiml vlue of from Theorem we efine n itertive process with respect to the Lgrnge multiplier n we prove convergence of this process in Section 3 The Algorithms.

9 MINIMIZATION OF A CONVEX SEPARABLE 5 From 0 s > 0 m > 0 n it follows tht > u ef m sme m s me ef l for the epressions y which ine sets re efine. The prolem how to ensure fesile solution to prolem (CSE which is n ssumption of Theorem is iscusse in Susection Prolem (CESP Consier the conve eponentil seprle progrm with liner equlity constrint n o constrints (CESP (4-(6. Assumptions (2. for ll. (2. α. Otherwise the constrints (5-(6 re inconsistent n the fesile region X efine y (5-(6 is empty. The KKT conitions for prolem (CESP re e u v 0 u ( 0 v ( 0 α u R v R. In this cse the following Theorem 2 which is similr to Theorem hols true.

10 6 Theorem 2 (Chrcteriztion of the optiml solution to prolem (CESP. A fesile solution ( X where X is efine y (5-(6 is the optiml solution to prolem (CESP if n only if there eists some R such tht ef e (6 ef e (7 ln ef e < < e. (8 Remr. As we will show elow < 0 so tht the epressions of in (8 (especilly epressions uner the logrithm sign re correct. The proof of Theorem 2 is omitte ecuse it is similr to the proof of Theorem. 3. The Algorithms 3.. Anlysis of the optiml solution to prolem (CSE Before the forml sttement of the lgorithm for prolem (CSE we iscuss some properties of the optiml solution to this prolem which turn out to e useful. Using (3 (4 n (5 the KKT conition (0 cn e written s follows

11 MINIMIZATION OF A CONVEX SEPARABLE 7 ln( s m ln( m α. ( 0 Since the optiml solution components of R to prolem (CSE epens on cn e consiere s functions of for ifferent ( (9 ln( s m ln(. m Functions ( re piecewise monotone nonincresing piecewise ifferentile functions of m sm e sme n. m with two repoints t Let ef δ( ln( s m ln( α. m (20 If we ifferentite δ ( with respect to we get δ ( < 0 m (2 ccoring to the remr (fter sttement of Theorem tht > 0 when 0/ n δ ( 0 when 0/. Hence δ ( is monotone nonincresing function of R. From the eqution δ ( 0 where δ ( is efine y (20 we re le to otin close form epression for

12 8 ep m m sm ln α (22 ecuse δ ( < 0 ccoring to (2 when 0/ (it is importnt tht δ ( 0. This epression of shows tht > 0 n it is use in the lgorithm suggeste for solving prolem (CSE. It turns out tht without loss of generlity we cn ssume tht δ ( 0 tht is δ ( epens on which mens tht 0/. At itertion of the implementtion of the lgorithms enote y ( the vlue of the Lgrnge multiplier ssocite with constrint (2 [constrint (5 respectively] y α ( the right-hn sie of (2 [of (5 respectively]; y ( ( ( ( the current sets respectively Algorithm (for prolem (CSE The following lgorithm for solving prolem (CSE is se on Theorem Algorithm (for prolem (CSE (Initiliztion { } ( 0 ( 0 ( 0 n 0 α α n n 0 / 0. / If α go to 2 else go to 9. 2 ( (.. Go to 3. Clculte ( y using the eplicit epression (22 of 3 Construct the sets ( ( ( through (3 (4 (5 (with ( inste of n fin their crinl numers ( ( ( respectively. Go to 4.

13 MINIMIZATION OF A CONVEX SEPARABLE 9 4 Clculte ( ( ( ( ( ( ( (. ln ln m m s α δ Go to 5. 5 If ( ( 0 δ or ( 0/ then ( ( ( ( go to 8 else if ( ( 0 > δ go to 6 else if ( ( δ 0 < go to 7. 6 ( ( ( ( for α α ( ( ( \ ( ( ( (. n n Go to 2. 7 ( ( ( ( for α α ( ( ( \ ( ( ( (. n n Go to 2. 8 for ; for ( ( m m s ln ln ; for. Go to 0. 9 Prolem (CSE hs no optiml solution ecuse the fesile set X efine y (2-(3 is empty. 0 En.

14 Convergence n compleity of Algorithm The following Theorem 3 sttes convergence of Algorithm. Theorem 3. Let ( e the sequence generte y Algorithm. Then (i if δ( ( > 0 then ( ( ; (ii if δ( ( < 0 then ( (. ( Proof. Denote y the components of itertion of implementtion of Algorithm. ( ( ( t (i Let δ( ( > 0. Using Step 6 of Algorithm (which is performe when δ( ( > 0 we get ( ( ( ( ( ( ( α. ( \ ( ( (23 Let (. Accoring to efinition (3 of ( we hve m sme ( ( m sme. Multiplying this inequlity y > 0 we otin m s m e ( m e m e ( ( m. e Therefore ( ecuse m > 0 n ccoring to properties of the eponentil function. get From (23 y using tht 0 ( ( > n Step 6 we ( ( ( ( ( ( α α α. ( ( ( (

15 MINIMIZATION OF A CONVEX SEPARABLE 2 Since > 0 then there eists t lest one ( 0 such tht ( (. Then 0 0 ( ( ( m m s 0 m 0 e s 0 m 0 e 0 (. 0 0 We hve use tht the reltionship etween ( ( n is given y (5 for ( ccoring to Step 2 of Algorithm n > 0 s > 0 m > 0. The proof of prt (ii of Theorem 3 is omitte ecuse it is similr to the proof of prt (i. Consier the fesiility of ( generte y Algorithm. Components n oviously stisfy (3. From m sme < m s me < m s me n > 0 s > 0 m > 0 it follows tht < < for. Hence ll stisfy (3. Since t ech itertion ( is etermine from the current equlity constrint (2 (Step 2 of Algorithm n since re etermine in ccornce with ( t ech itertion (Steps n 8 of Algorithm then stisfies (2 s well. Therefore Algorithm genertes which is fesile for prolem (CSE which is n ssumption of Theorem.

16 22 Remr. Theorem 3 efinitions of ( 3 ( 4 n ( 5 n Steps 6 7 n 8 of Algorithm llow us to stte tht ( ( ( ( n ( (. This mens tht if elongs to current ine set ( then elongs to the net ine set ( n continuing in the sme mnner elongs to the optiml ine set ; ( the sme hols true out the sets n. Therefore ( converges to the optiml vlue of of Theorem n ( ( ( converge to the optiml ine sets respectively. This mens tht clcultion of opertions ( (Step 6 ( (Step 7 n the construction of ccornce with Theorem. re in At ech itertion of Algorithm we etermine the vlue of t lest one vrile (Steps n t ech itertion we solve prolem of the form (CSE ut of less imension (Steps 2-7. Therefore Algorithm is finite n it converges with t most n itertions tht is the itertion compleity of Algorithm is O ( n. Step (initiliztion n checing whether the fesile set X is empty tes time O ( n. The clcultion of ( requires constnt time (Step 2. Step 3 tes O ( n time ecuse of the construction of ( (. ( Step 4 lso requires O ( n time n Step 5 requires constnt time. Ech of Steps 6 7 n 8 tes time which is oune y O ( n ecuse t these steps we ssign some of s the finl vlue n since the numer of ll s is n then Steps 6 7 n 8 te time O ( n. Hence 2 Algorithm hs O ( n running time n it elongs to the clss of strongly polynomilly oune lgorithms.

17 MINIMIZATION OF A CONVEX SEPARABLE 23 As the computtionl eperiments show the numer of itertions of the lgorithm performnce is not only t most n ut it is much much less thn n for lrge vlues of n. In fct the numer of itertions of lgorithm performnce oes not epen on n ut only on the three ine sets efine y (3 (4 n (5. In prctice Algorithm hs O ( n running time Algorithm 2 (for prolem (CESP n its convergence After nlysis of the optiml solution to prolem (CESP similr to tht prolem (CSE we suggest the following lgorithm for solving prolem (CESP Algorithm 2 (for prolem (CESP (Initiliztion { } ( 0 ( 0 n 0 α α n n ( 0 0 / 0. / If α go to 2 else go to 9. 2 ( (. Clculte ( y using the eplicit epression ( ep ( α ( ( ln ( < 0. Go to 3. 3 Construct the sets ( ( ( through (6 (7 (8 (with ( inste of n fin their crinl numers ( ( (. Go to 4. 4 Clculte

18 24 ( ( ( ( ( ( δ ln ( ( (. ln ln α Go to 5. 5 If ( ( 0 δ or ( 0/ then ( ( ( ( go to 8 else if ( ( 0 > δ go to 6 else if ( ( 0 < δ go to 7. 6 ( ( ( ( for α α ( ( ( \ ( ( ( (. n n Go to 2. 7 ( ( ( ( for α α ( ( ( \ ( ( ( (. n n Go to 2. 8 for ; for ln ; for. Go to 0 9 Prolem (CESP hs no optiml solution ecuse the fesile set X efine y (5-(6 is empty. 0 En.

19 MINIMIZATION OF A CONVEX SEPARABLE 25 To voi possile enless loop when progrmming Algorithms n 2 the criterion of Step 5 to go to Step 8 t itertion usully is not ( ( δ 0 ut ( ( δ [ ε ε] where ε > 0 is some (given or chosen tolernce vlue up to which the equlity δ( 0 must e stisfie. A theorem nlogous to Theorem 3 hols for Algorithm 2 which gurntees the convergence of ( ( ( ( to the optiml respectively. Theorem 4. Let ( e the sequence generte y Algorithm 2. Then (i if δ( ( > 0 then ( ( ; (ii if δ( ( < 0 then ( (. The proof of Theorem 4 is omitte ecuse it is similr to the proof of Theorem 3. 2 It cn e prove tht Algorithm 2 hs O ( n running time n point ( generte y this lgorithm is fesile for prolem (CESP which is n ssumption of Theorem Etensions 4.. Theoreticl spects Up to now we hve require > 0 in constrints (2 n (5 of prolems (CSE n (CESP respectively. However if it is llowe 0 for some in prolems (CSE n (CESP then for such inices we cnnot construct the epressions m sme n m sme for

20 26 prolem (CSE n e n e for prolem (CESP y mens of which we efine ine sets for the corresponing prolem. In such cses s re not involve in (2 [in (5 respectively] for such inices. It turns out tht we cn cope with this ifficulty n solve prolems (CSE n (CESP with 0 for some s. Denote Z 0 { 0}. Here 0 enotes the computer zero. In prticulr when Z0 n α 0 then the set X is efine only y (3 (y (6 respectively. Theorem 5 (Chrcteriztion of the optiml solution to prolem (CSE An etene version. Prolem (CSE cn e ecompose into two suprolems (CSE for Z0 n (CSE2 for \ Z0. The optiml solution to (CSE is Z0 (24 tht is suprolem (CSE itself is ecompose into n0 Z0 inepenent prolems. The optiml solution to (CSE2 is given y (3 (4 n (5 with \ Z0. Proof. Necessity. Let ( e the optiml solution to (CSE. ( Let Z0 tht is 0 for this. The KKT conitions re m s me u v 0 Z0 from (7 ( 7 n (8 - (2.

21 MINIMIZATION OF A CONVEX SEPARABLE 27 ( If then u 0 v 0. From ( 7 it follows tht m s me u 0 which is impossile ecuse s 0 m > 0 n > e m > 0. ( If then u 0 v 0. Therefore s m e v 0 which is lwys stisfie for s > 0 m > 0. m (c If < < then u v 0. Therefore s me 0 tht is s 0 which is impossile ccoring to the ssumption m s > 0 m > 0. m As we hve oserve only cse ( is possile for Z0 n Z0. (2 Components of the optiml solution to (CSE2 re otine y using the sme pproch s tht of the proof of necessity prt of Theorem ut with the reuce ine set \ Z0. Sufficiency. Conversely let X n components of stisfy (24 for Z0 n (3 (4 (5 with \ Z0. Set u m 0 v s m e ( > 0 for Z0. If 0 set u v 0 m s me ( ( > 0 from (5; for < < \ Z0; u m sme ( 0 v 0 for \ Z0; u 0 v m s me ( 0 for \ Z0.

22 As in the proof of Theorem 0/. It cn e verifie tht u v stisfy the KKT conitions (7-(2. Then with components (24 for Z0 n (3 (4 (5 for \ Z0 is the optiml solution to prolem ( CSE ( CSE ( CSE2. Similr result hols for prolem (CESP. Theorem 6 (Chrcteriztion of the optiml solution to prolem (CESP An etene version. Prolem (CESP cn e ecompose into two suprolems (CESP for Z0 n (CESP2 for \ Z0. The optiml solution to (CESP is Z0. solution to (CESP2 is given y (6 (7 (8 with \ Z0. The optiml The proof of Theorem 6 is omitte ecuse it repets in prt the proof of Theorems n 5. Thus with the use of Theorems 5 n 6 we cn epress components of the optiml solutions to prolems (CSE n (CESP without the m m sme s me necessity of constructing the epressions e n e with Computtionl spects Algorithms n 2 re lso pplicle in cses when for some n/or for some. However if we use the computer vlues of n t the first step of the lgorithms to chec whether the corresponing fesile region is empty or nonempty n t Step 3 in the epressions m sme n e with

23 MINIMIZATION OF A CONVEX SEPARABLE 29 n/or y mens of which we construct ine sets this coul sometimes le to rithmetic overflow. If we use other vlues of n with smller solute vlues thn those of the computer vlues of n this woul le to inconvenience n epenence on the t of the prticulr prolems. To voi these ifficulties n to te into ccount the ove iscussion it is convenient to o the following Construct the following ine sets SVN { \ Z0 > < } SV { \ Z0 > } (25 SV 2 { \ Z0 < } SV { \ Z0 }. It is ovious tht Z 0 SV SV SV 2 SVN tht is the set \ Z0 is prtitione into the four susets SVN SV SV 2 SV efine ove. When progrmming the lgorithms we use computer vlues of n for constructing the sets SVN SV SV 2 SV. In orer to construct the sets without the necessity of clculting the vlues m sme (for prolem (CSE with or ecept for the sets Z0 SV SV SV 2 SVN we nee some susiiry sets efine s follows For SVN SVN SVN m sme < < m sme

24 30 SVN SVN m sme SVN SVN m sme ; for SV SV SV < m s me (26 SV SV m s me ; for SV2 SV 2 SV 2 > m s me SV 2 SV 2 m sme ; for SV SV SV. Then SVN SV SV 2 SV SVN SV (27 SVN SV 2. We use the sets (27 s the corresponing sets with the sme nmes in Algorithms n 2.

25 MINIMIZATION OF A CONVEX SEPARABLE 3 With the use of results of this section Steps n 3 of Algorithm cn e moifie s follows respectively Aout Algorithm. Step. (Initiliztion { } ( 0 ( 0 n 0 α α n n ( 0 0 / 0. / Construct the set Z 0. If Z0 then. Set ( 0 \ 0 ( 0 Z n n Z0. Construct the sets SVN SV SV 2 SV. If SVN then if α then go to Step 2 else go to Step 9 (fesile region X is empty else if SV SVN then if α then go to Step 2 else go to Step 9 (fesile region X is empty else if SV 2 SVN then if α then go to Step 2 else go to Step 9 (fesile region X is empty else if SV 0/ then go to Step 2 (fesile region is lwys nonempty. SVN SVN SVN SV SV Step 3. Construct the sets SV 2 SV 2 SV (with ( inste of.

26 32 Construct the sets ( ( ( y using (27 n fin their crinl numers Go to Step 4. ( ( ( respectively. Similrly we cn efine susiiry ine sets of the form (26 for prolem (CESP s well n moify Steps n 3 of Algorithm 2. Moifictions of the lgorithms connecte with theoreticl n computtionl spects o not influence upon their computtionl compleity iscusse in Section 3 ecuse these moifictions o not ffect the itertive steps of lgorithms. 5. Computtionl Eperiments In this section we present results of some numericl eperiments otine y pplying lgorithms suggeste in this pper to prolems uner consiertion. The computtions were performe on n Intel Pentium (R Dul-Core CPU E GHz/2.00GB using RZTools interctive system. Ech prolem ws run 30 times. Coefficients s > 0 m > 0 > 0 for prolem (CSE n > 0 > 0 for prolem (CESP were rnomly generte. Prolem (CSE (CESP Numer of vriles N 200 n 500 n 200 n 500 Averge numer of itertions Averge run time (in secons The effectiveness of lgorithms for prolems (CSE n (CESP hs een teste y mny other emples. As we cn oserve the (verge numer of itertions is much less thn the numer of vriles n for lrge vlues of n.

27 MINIMIZATION OF A CONVEX SEPARABLE 33 We provie elow the solution of two simple prticulr prolems of the form (CSE n (CESP respectively otine y using the pproch suggeste in this pper. The results re roune to the fourth igit. Emple. min { ( 2( ( 2 } 2 c e e suect to The optiml solution otine y Algorithm is ( ( c min c( Numer of itertions 2. Emple 2. 2 { ( min c e e 2 } suect to The optiml solution otine y Algorithm 2 is Numer of itertions. ( ( c c( min

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