A One-Dimensional Local Tuning Algorithm for Solving GO Problems with Partially Defined Constraints

A One-Dimen sional Local T uning Algorithm for Solving GO Problem s with Partially Defined Constraint s ∗ Y aroslav D. Ser geye v 1 , 2 † Dmitri E. Kva sov 1 , 2 Falah M.H. Khalaf 3 1 DEIS, University of Calabria, V ia P . Bucci, 42C 87036 – Rende (CS), Italy , 2 Software Department, N.I. Lobatche vsky S tate University Nizhni Novgorod, R ussia, 3 Department of Mathematics, Univ ersity of Ca labria, Italy , yaro@si.de is.unical. it kvadim@si. deis.unica l.it falah@mat. unical.it Abstract Lipschitz one-dim ensional constrained global optimization (GO) prob- lems where both the ob jectiv e function and constraints can be multiextremal and non-d ifferentiable are considered in this paper . Problem s, where the con- straints are verified in an a priori giv en order fixed by the nature of the prob- lem are studied . Moreover , if a constraint is not satisfied at a point, then the remain ing constraints and the o bjective function can be un defined at this point. The co nstrained pro blem is redu ced to a d iscontinuou s unconstrained problem by the index scheme without introd ucing additional parameters or variables. A ne w g eometric method u sing adapti ve estimate s of lo cal L ips- chitz constants is introduced. The estimates are calculated by using the local tuning tech nique p roposed recen tly . Numerical exper iments show quite a satisfactory pe rforman ce of th e new metho d in co mparison with the penalty approa ch and a method using a prio ri gi ven Lipschitz constants. Key W ords :Global optimizati on, multiext remal constraints, geomet ric algorithms, inde x scheme, local tuning. ∗ This research was supported by the follo wing grants: FIRB RBNE01WBBB, FIRB RB A U01JYPN, PRIN 200 5017083 -002, and RFBR 04-01-00455-a. The authors wou ld like to thank anon ymous referees for their subtle suggestions. † Corresponding author 1 1 Introd uction It hap pens often in enginee ring o ptimizatio n p roblems (see [6, 12, 15]) that the object iv e function and constr aints can be multiextre mal, non-d if ferentia ble, and partial ly defined. The latte r means that the constrai nts are v erified in a priori gi ven order fixed by the nature of the problem and if a constrai nt is not satisfied at a point, then the remaining co nstrain ts and th e obj ecti ve function ca n be undefine d at this p oint. T his kind of pr oblems i s d if ficult t o solve eve n i n the on e-dimensi onal case (see [10, 11, 1 2, 1 4, 15]). Formall y , s upposi ng that b oth th e o bjecti ve functi on f ( x ) an d constr aints g j ( x ) , 1 ≤ j ≤ m, satisfy th e Lipschitz con dition and the feasib le re gion is no t empty , this problem can be formulated as follo w s. It is necessa ry to find a po int x ∗ and the corresp onding v alue g ∗ m +1 such that g ∗ m +1 = g m +1 ( x ∗ ) = min { g m +1 ( x ) : x ∈ Q m +1 } , (1) where, in order to unify the description process, the de signati on g m +1 ( x ) , f ( x ) has been used and regio ns Q j , 1 ≤ j ≤ m + 1 , are defined by the rules Q 1 = [ a, b ] , Q j +1 = { x ∈ Q j : g j ( x ) ≤ 0 } , 1 ≤ j ≤ m, (2) Q 1 ⊇ Q 2 ⊇ . . . ⊇ Q m ⊇ Q m +1 . Note that since the constraints g j ( x ) , 1 ≤ j ≤ m , are multiextre mal, the admissible reg ion Q m +1 and re gions Q j , 1 ≤ j ≤ m, can be co llection s of sev eral dis joint subre gions . W e suppose hereafter that all of them con sist of inte rv als of a fi nite length . W e assume also that the functio ns g j ( x ) , 1 ≤ j ≤ m + 1 , satisfy the corre- spond ing Lipsc hitz condition s | g j ( x ′ ) − g j ( x ′′ ) | ≤ L j | x ′ − x ′′ | , x ′ , x ′′ ∈ Q j , 1 ≤ j ≤ m + 1 , (3) 0 < L j < ∞ , 1 ≤ j ≤ m + 1 . (4) In order to illustrate the probl em under consideratio n and to highlight its dif- ferenc e with resp ect to problems wher e constrain ts and the objecti ve fu nction are defined ov er the whole search region, let us consid er an ex ample – test pro blem number 6 fro m [2] sho wn in Figu re 1. T he pr oblem has two mult iext remal con- straint s an d is formulated as follows f ∗ = f ( x ∗ ) = min { f ( x ) : g 1 ( x ) ≤ 0 , g 2 ( x ) ≤ 0 , x ∈ [0 , 1 . 5 π ] } , (5) where g 3 ( x ) , f ( x ) =                  1 3  100 9 π 2 x 2 + 1 2  , x ≤ 3 π 10 , 5 3 sin  20 3 x  + 1 2 , 3 π 10 < x ≤ 9 π 10 , 1 3  100 9 π 2 x 2 − 80 3 π x + 33 2  , x > 9 π 10 , (6) 2 Figure 1: Problem number 6 from [2] where functions f ( x ) and g 1 ( x ) , g 2 ( x ) are defined ov er the whole sear ch region [0 , 1 . 5 π ] g 1 ( x ) = 7 10 −   sin 3 (3 x ) + cos 3 ( x )   , (7) g 2 ( x ) = −     ( x − π ) 3 100     + | cos(2( x − π )) | − 1 2 . (8) The admissible region of problem (5)–(8) consists of two disjoint subreg ions shown in Figure 1 at the line f ( x ) = 0 ; the global minimizer is x ∗ = 3 . 76 984 . Problem of the typ e (1)–(4) con sidered in this paper and using the same func- tions g 1 ( x ) – g 3 ( x ) fr om (6)–(8) is sh own in Figure 2. It has the same globa l mini- mizer x ∗ = 3 . 76 984 an d is formulated as follo w s Q 1 = [0 , 1 . 5 π ] , Q 2 = { x ∈ Q 1 : g 1 ( x ) ≤ 0 } , (9) Q 3 = { x ∈ Q 2 : g 2 ( x ) ≤ 0 } , (10) g ∗ 3 = g 3 ( x ∗ ) = min { g 3 ( x ) : x ∈ Q 3 } . (11) It can be seen from Figur e 2 that both g 2 ( x ) a nd f ( x ) are part ially defined : g 2 ( x ) is defined only o ver Q 2 and the objec ti ve function f ( x ) is defined on ly over Q 3 which coinci des with th e admissible region of problem (5)–(8). It is not easy to fi nd a traditio nal alg orithm for sol ving probl em (1)–(4). For exa mple, the penalty approach requires that f ( x ) and g i ( x ) , 1 ≤ i ≤ m, are defined over the w hole search interv al [ a, b ] . At first glance it seems that at the reg ions where a functio n is not defined it can be simply filled in with either a 3 Figure 2: G raphic al representati on of problem (9)–(11) big n umber o r th e fun ction v alue a t th e nea rest feasi ble poin t. Unfortu nately , in the conte xt o f L ipschi tz algo rithms, inco rporati ng such ideas ca n l ead t o infinitely hi gh Lipschitz constants, causing degenerat ion of the methods and non-app licabili ty of the penalty appro ach. A promising approac h called the index scheme has been prop osed in [13] (see also [11, 14 , 15]) in combin ation with information stochastic Bayesi an algo rithms for solv ing problem (1)–(4). An i mportant adv antag e of t he inde x scheme is that it does n ot int roduce a ddition al var iables and/o r para meters as t radition al approach es do (see, e.g, [1, 3, 4]). It ha s b een re cently sho wn in [1 0] tha t the index scheme can be also succes sfully used in combination with the Branch-and-Bou nd approac h if the Lipschitz consta nts L j , 1 ≤ j ≤ m + 1 , from (3), (4) are kno wn a priori. Ho wev er , in practica l application s (see, e.g. [6]) the Lipschitz constants L j , 1 ≤ j ≤ m + 1 , are very ofte n unkno wn. Thus, th e probl em of their esti mating arises ine vitab ly . If there exists an ad ditiona l information all o wing us to ob tain a priori fixed co nstants K j , 1 ≤ j ≤ m + 1 , su ch that L j < K j < ∞ , 1 ≤ j ≤ m + 1 , then the algori thm IBB A from [10] can be used. In this pa per , the case where the re is no any additional info rmation abou t the Lipschitz consta nts is co nsidere d. A new GO Algo rithm with Local T uning (AL T ) adapti vely es timating the local Lipschitz co nstants during the searc h is proposed . The local tuning techniq ue introduced in [7, 8 , 9] for solving unco nstrain ed prob - lems a llo ws o ne to accelerat e the se arch sig nificantly in comparison with the meth- 4 ods using estima tes of the global Lipschitz constan t. The new me thod AL T unifies this ap proach wit h t he ind ex s cheme an d geo metric idea s allo wing one to const ruct auxili ary fun ctions similar to minora nts used in the IBB A (see [10 ]). In a seri es of numerica l expe riments it is sho wn that usage of a dapti ve local es timates calc ulated during the search instead of a priori giv en estimates of global Lipsc hitz constan ts accele rates th e search significantly . 2 A New Geometric Index Algorithm with Local T unin g Let us assoc iate with e very point of the interv al [ a, b ] an ind e x ν = ν ( x ) , 1 ≤ ν ≤ M , which is defined by the condi tions g j ( x ) ≤ 0 , 1 ≤ j ≤ ν − 1 , g ν ( x ) > 0 , (12) where for ν = m + 1 the last inequality is omitted. W e shall call a trial the operat ion of e valu ation of the fun ctions g j ( x ) , 1 ≤ j ≤ ν ( x ) , at a point x . Thus, the in dex scheme considers c onstrai nts one at a ti me at e very point whe re it has been decided to try to calculate the objecti ve function g m +1 ( x ) . Each con- straint g i ( x ) is ev aluated at a point x only if al l the inequalitie s g j ( x ) ≤ 0 , 1 ≤ j < i, ha ve been satisfied at this point. I n its turn the objecti ve function g m +1 ( x ) is compute d on ly for those points where all the constraint s ha ve been satis fied. Suppose no w that k + 1 , k ≥ 1 , trials ha ve been ex ecute d at s ome points a = x 0 < x 1 < . . . < x i < . . . < x k = b (13) and the index ν i = ν ( x i ) , 0 ≤ i ≤ k , hav e been calculated follo wing ( 12 ) . Due to the inde x scheme, the estimate z ∗ k = m in { g M k ( x i ) : 0 ≤ i ≤ k, ν ( x i ) = M k } (14) of the minimal v alue of the function g M k ( x ) found afte r k iter ations can be calcu- lated and the v alues z i = g ν ( x i ) ( x i ) −  0 , if ν ( x i ) < M k z ∗ k , if ν ( x i ) = M k (15) can be assoc iated with the poi nts x i from (13). In the ne w alg orithm AL T , we propos e to adapt iv ely est imate at each iterat ion the local Lipsc hitz consta nts ov er subinterv als [ x i − 1 , x i ] ⊂ [ a, b ] , 1 ≤ i ≤ k , by using the in formation obtained from ex ecuting trials at the points x i , 0 ≤ i ≤ k , from (13). Partic ularly , at each point x i , 0 ≤ i ≤ k , ha ving the index ν i we 5 calcul ate a loca l estimate η i of the Lipschitz constan t L ν i at a neigh borhoo d of the point x i as follo w s η i = max { λ i , γ i , ξ } , 0 ≤ i ≤ k . (16) Here, ξ > 0 is a small n umber reflecting our suppositio n that t he o bjecti ve functi on and constr aints are no t just constants ove r [ a, b ] , i.e., L j ≥ ξ , 1 ≤ j ≤ m + 1 . The v alues λ i are calc ulated as fo llo ws λ i =                                                            max {| z j − z j − 1 | ( x j − x j − 1 ) − 1 : j = i, i + 1 } , if ν i − 1 = ν i = ν i +1 max {| z i − z i − 1 | ( x i − x i − 1 ) − 1 , z i ( x i +1 − x i ) − 1 } , if ν i − 1 = ν i < ν i +1 max {| z i +1 − z i | ( x i +1 − x i ) − 1 , z i ( x i − x i − 1 ) − 1 } , if ν i − 1 > ν i = ν i +1 max { z i ( x i − x i − 1 ) − 1 , z i ( x i +1 − x i ) − 1 } , if ν i < ν i − 1 , ν i < ν i +1 z i ( x i − x i − 1 ) − 1 , if ν i − 1 > ν i > ν i +1 z i ( x i +1 − x i ) − 1 , if ν i − 1 < ν i < ν i +1 | z i − z i − 1 | ( x i − x i − 1 ) − 1 , if ν i − 1 = ν i > ν i +1 | z i +1 − z i | ( x i +1 − x i ) − 1 , if ν i − 1 < ν i = ν i +1 0 , otherwise (17) where z i , 0 ≤ i ≤ k , ar e from (15). Naturally , when i = 0 or i = k , only one of the two e xpres sions in the first four cases are defined and are used to calc ulate λ i . The v alues γ i , 0 ≤ i ≤ k , are calculated in the follo wing way: γ i = Λ ν i max { x i − x i − 1 , x i +1 − x i } /X max ν i , (18) Λ ν i = Λ ν i ( k ) = max { Λ ν i ( k − 1) , max { λ j : ν j = ν i , 0 ≤ j ≤ k }} , (19) where Λ ν i are adap ti ve estimates of the global Lipschitz constants L ν i and X max ν i = max { x j − x j − 1 : ν j = ν i or ν j − 1 = ν i , 1 ≤ j ≤ k } . (20) The v alues λ i and γ i reflect the influence on η i of the local and global in- formatio n obtain ed during the pre vious iteration s. When both interv als [ x i − 1 , x i ] and [ x i , x i +1 ] are small, then γ i is small too (see (18 )) and, du e to (16), the local informat ion repres ented by λ i has major importance. The value λ i is calcula ted by con siderin g the interv als [ x i − 2 , x i − 1 ] , [ x i − 1 , x i ] , and [ x i , x i +1 ] (see (17)) as those which hav e the stro ngest influence on the local estimat e at the poin t x i and, in general, at the interv al [ x i − 1 , x i ] . When at least one of th e inte rv als [ x i − 1 , x i ] , [ x i , x i +1 ] is v ery wide, the local information is not reli able and, due to (1 6), the global information repr esented by γ i has the major influence on η i . Thus, local 6 and globa l informat ion are balan ced in the v alues η i , 0 ≤ i ≤ k . Note that the method uses the local informat ion ov er the whole search re gion [ a, b ] during the global search both for the objecti ve function and constraints . W e are ready no w to describ e the ne w algo rithm AL T . Step 0 (Initializa tion). Suppose that k + 1 , k ≥ 1 , trai ls ha ve been al ready e xe - cuted in a way at poin ts x 0 = a, x 1 = b, x 2 , x 3 , ..., x i , ...x k − 1 , x k (21) and their index es and the v alue M k = max { ν ( x i ) : 0 ≤ i ≤ k } (22) ha ve bee n calculate d. The v alue M k defined in (22) is th e maximal inde x obtain ed du ring the search after k + 1 trials. The choice of the point x k +1 , k ≥ 1 , w here the next trial will be exe cuted is determined by the rules presen ted bel o w . Step 1. Renumb er the points x 0 , ...., x k of the pre vious k iteration s by sub scripts 1 in order to form the sequen ce (13). Step 2. Recal culate the estimate z ∗ k of the minimal v alue of the functio n g M k ( x ) found afte r k itera tions and the v alues z i by usin g formulae (14) and (15), respec ti vely . For each tria l poin t x i ha ving th e ind ex ν i , 0 ≤ i ≤ k , calculat e estimate η i from (16). Step 3. Fo r each in terv al [ x i − 1 , x i ] , 1 ≤ i ≤ k , calculate the char acteris tic of the interv al R i =    ( 1 η i + η i − 1 ) [ η i z i − 1 + η i − 1 z i + rη i − 1 η i ( x i − 1 − x i )] , ν i − 1 = ν i z i − rη i ( x i − x i − 1 − z i − 1 /r η i − 1 ) , ν i − 1 < ν i z i − 1 − rη i − 1 ( x i − x i − 1 − z i /r η i ) , ν i − 1 > ν i (23) where r > 1 is th e rel iability parameter of the method (this ki nd of par ame- ters is quite traditional in Lipschitz glob al optimization ; discussio ns related to its choice and meaning can be found in [6, 12, 15]). Step 4. Find an int erv al t co rrespon ding to the minimal charac teristic, i.e., t = arg min { R i : 1 ≤ i ≤ k } . (24) If the minimal valu e of the characterist ic is attained for sev eral subinterv als, then the minimal inte ger satisfying (24) is accepted as t . 1 Thus, two numerations are used during the work of the algorithm. The record x i from (21) means that this point has be en generated d uring the i -th iteration of the AL T . The record x i indicates the place of t he point in the ro w (13 ). Of course, the second enumeration is changed du ring every iteration. 7 Step 5. If fo r the interv al [ x t − 1 , x t ] , where t is from (24), the stoping rule x t − x t − 1 ≤ ε ( b − a ) , (25) where a and b are from (1) –(4), is satisfied for a preset accuracy ε > 0 , then Stop – th e required acc urac y has been rea ched. In the opposite cas e, go to Step 6. Step 6. Ex ecute the ( k + 1) -th tria l at the point x k +1 =  ( 1 r η t + r η t − 1 ) [ z t − 1 − z t + r η t − 1 x t − 1 + rη t x t ] , if ν i − 1 = ν i 0 . 5( x t − 1 + x t ) , if ν i − 1 6 = ν i , (26) and e v aluate its index ν ( x k +1 ) . Step 7. This st ep consists of the follo wing alternati ves: Case 1. If ν ( x k +1 ) > M k , then perf orm tw o additio nal tria ls at the points x k +2 = 0 . 5( x t − 1 + x k +1 ) , x k +3 = 0 . 5( x k +1 + x t ) , (27) calcul ate thei r index es, set k = k + 3 , and go to Step 8. Case 2. If ν ( x k +1 ) < M k and among the poin ts (13) there exists only one point x T with the maximal index M k , i.e., ν T = M k , then execut e two additi onal trials at the poi nts x k +2 = 0 . 5( x T − 1 + x T ) , (28) x k +3 = 0 . 5( x T + x T +1 ) , (29) if 0 < T < k , ca lculate the ir ind ex es, set k = k + 3 , and go to Step 8. If T = 0 then t he trial is execu ted on ly a t the poin t ( 29). Analogously , if T = k then the trial is e xec uted only at the poi nt (28 ). In these two cases, calculate the inde x of the additional point, set k = k + 2 and go to Step 8 . Case 3. In all th e remaining cases set k = k + 1 and go to Step 8 . Step 8. Calcu late M k and go to Step 1. Global con ver gence condi tions of the AL T are described by the foll owin g two theore ms giv en, due to the lack of space , without pro ofs that can be deriv ed using Theorems 2 and 3 from [8] and Theorem 2 from [10]. Theor em 2.1 Let th e feas ible r e gion Q m +1 6 = ∅ c onsists of intervals having fi nite length s, x ∗ be any solut ion to pr oblem (1)–(4), and j = j ( k ) be the numb er of an interval [ x j − 1 , x j ] containing th is point during the k -th iteration . Then, if for k ≥ k ∗ the foll owing con ditions r Λ ν j − 1 > C j − 1 , r Λ ν j > C j , (30) 8 C j − 1 = z j − 1 / ( x ∗ − x j − 1 ) , C j = z j / ( x j − x ∗ ) . (31) tak e place, then the point x ∗ will be a limit point of the tri al sequence { x k } gene r - ated by the ALT . Theor em 2.2 F or any pr oblem (1)– (4) ther e e xists a value r ∗ suc h that condition s (30) ar e sa tisfied for all paramet ers r > r ∗ , wher e r is fr om (23) and (26). 3 Numerical Comparison The ne w algorit hm has bee n numerically compared w ith the follo wing metho ds: – The method proposed by Pijavsk ii (see [3, 5]) combined with the penal ty ap- proach used to redu ce the constrain ed proble m to an u nconstr ained one ; this method i s indicated hereaf ter as PEN. The Lipschitz constant of the ob tained uncon strained problem is supposed to b e kno wn as it is requir ed by Pija vskii algori thm. – The meth od IBBA from [10] using the inde x scheme in combination with the Branch-a nd-Bound a pproac h and the kno wn Lipschitz constant s L j , 1 ≤ j ≤ m + 1 , from (3), (4). T en non -dif ferentiable test prob lems intro duced in [2] hav e been use d in the exp eriments (since there were sev eral mispri nts in the original paper [2], the ac- curate ly ver ified formula e ha ve been applied, which are a vaila ble at the W eb-site http://wwwinfo. deis.unical.it/ ∼ y aro/constraints.html ). In th is s et o f tests, pro blems 1 –3 ha ve on e constrai nt, proble ms 4–7 two constra ints, and probl ems 8–10 three con- strains . In these test proble ms, all constrain s and the objecti ve functio n are defined ov er the whole region [ a, b ] fr om (2). These test problems were used because the PEN n eeds th is ad ditiona l information fo r its work a nd is not a ble to solve problem (1)–(4). Naturally , the methods IBBA and AL T solved all the probl ems usi ng th e statemen t (1)–(4) an d did not take be nefits from the add itional info rmation gi ven (see ex amples from Figure s 1 and 2) by the statement (5)–(8) in comparis on w ith (9)–(11). In order to demonstrate the influen ce of chang ing th e search acc urac y ε on the con ver gence speed of the methods , two dif feren t v alues of ε , namely , ε = 10 − 4 and ε = 10 − 5 ha ve been used. The same v alue ξ = 10 − 6 from (16) has bee n used in all the exp eriments for all the methods. T able 1 repr esents the results for th e PEN (see [2]). The constrained probl ems were reduc ed to the unc onstrai ned ones as follo ws f P ∗ ( x ) = f ( x ) + P ∗ max { 0 , g 1 ( x ) , g 2 ( x ) , . . . , g N v ( x ) } . (32) The column “Ev al. ” in T able 1 sho ws the total number of e val uations of the objec- ti ve fu nction f ( x ) and all the constrain ts. T hus, it is equal to ( N v + 1 ) × N trial s , 9 T able 1: N umerical results obtain ed by th e PE N N P ∗ ε = 10 − 4 ε = 10 − 5 T rials E v al. T rials E v al. 1 15 247 49 4 419 838 2 15 241 482 313 626 3 15 917 1834 2127 4254 4 15 273 819 861 2583 5 20 671 2013 1097 3291 6 15 909 2727 6367 19101 7 15 199 597 221 663 8 15 365 1460 415 1660 9 15 1183 4732 4549 18 196 10 15 135 540 169 676 A v . − 514 . 0 1569 . 8 1653 . 8 5188 . 8 where N v is the number of cons traints and N trial s is the number of the trials ex e- cuted by the PEN for each prob lem. Results obtaine d by the IBBA (s ee [10]) and by the ne w method AL T with the paramete r r = 1 . 3 are summarized in T ables 2 and 3, respec ti vely . Columns i n the tables ha ve the follo wing meani ng for each value of th e search accurac y ε : − the column N indicates the problem number; − the co lumns N g 1 , N g 2 , an d N g 3 repres ent the nu mber of tr ials wher e the con- straint g i , 1 ≤ i ≤ 3 , was the last e v aluated constrain t; − the co lumn “T rials” is the total number o f trial poin ts generate d by the meth- ods; − the column “Ev al. ” is the total number of e v aluat ions of the ob jecti ve fun c- tion and the constr aints. This quantity is equal to: − N g 1 + 2 × N f , for probl ems with one cons traint; − N g 1 + 2 × N g 2 + 3 × N f , for probl ems with two c onstrain ts; − N g 1 + 2 × N g 2 + 3 × N g 3 + 4 × N f , for probl ems with three con straint s. The aster isk in T able 3 indicates that r = 1 . 3 was not suffici ent to find the global minimizer of probl em 7. The resu lts for this problem in T able 3 are ob- tained using the value r = 1 . 9 ; the AL T with this va lue finds the solut ion. Finally , T able 4 represents the improv ement (in terms of the number of trials and e v alu- ations ) obtained by the AL T in comparis on with the other methods used in the exp eriments. As it can be seen from T ables 1–4, the algorithms IBBA and AL T construc ted in the f rame work o f th e ind ex s cheme si gnificantl y outperform the t radition al method PEN. T he AL T demo nstrates a high improv ement in ter ms of the trials performe d with respect to the IBBA as well. In part icular , the great er the dif ference between estimates o f the lo cal Lipschit z consta nts (for the ob jecti ve fu nction or for the con- 10 Figure 3: S olving by the method PEN the unconstr ained pro blem (32 ) constru cted from the probl em (5)–(8) sho wn in Figure 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −20 −15 −10 −5 0 5 10 15 f P * (x) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 200 400 600 800 Axis x Number of Trials straint s), the highe r is the speed up obtained by the AL T (see T able 4). The im- pro vement is e special ly high if the gl obal minimiz er lies insi de of a f easible subre- gion with a small (with respect to the global Lipsch itz consta nt) val ue of the local Lipschitz constant as it happe ns for exampl e for proble ms 6 and 9. The adv antage of the ne w metho d is more pronou nced when the search ac curac y ε increa ses (see T able 4). In order to illust rate per formance of the methods graphical ly , in Figures 3–5 we sho w dynamic diag rams of the search (with the accurac y ε = 10 − 4 in (25)) ex ecute d by t he PEN, the IBB A, and the AL T , respect iv ely , for probl em 6 from [2]. The upper su bplot of Figure 3 contai ns the function f P ∗ ( x ) from (32) construct ed from t he p roblem ( 5)–(8) shown in Figure 1. T he upper su bplots o f Figu res 4 and 5 contai n the inde x functio n (see [10, 15] for a d etailed dis cussion ) corres pondin g to the probl em (9)–(11) from Figure 2. Note that the local Lipschitz constant cor re- spond ing to the objecti ve function o ver this subre gion is sign ificantly smalle r tha n the global one (see Figures 2 and 5). The line of symbols ‘+ ’ located unde r th e g raph o f th e f unction (32 ) in Figu re 3 sho w s po ints at which trials ha ve be en ex ecuted by the PEN. T he lower subplots 11 T able 2: N umerical results obtain ed by the IBB A N ε = 10 − 4 ε = 10 − 5 N g 1 N g 2 N g 3 N f T rials Eval . N g 1 N g 2 N g 3 N f T rials Eva l. 1 23 − − 2 8 51 79 23 − − 34 57 91 2 18 − − 16 34 50 20 − − 22 42 64 3 171 − − 19 190 209 175 − − 21 196 217 4 136 15 − 84 235 418 170 15 − 226 411 878 5 168 91 − 24 283 422 188 101 − 26 315 468 6 16 16 − 597 629 1839 17 17 − 2685 2719 8106 7 63 18 − 39 120 216 65 19 − 43 127 232 8 29 11 3 21 64 144 29 14 3 23 69 158 9 8 86 57 183 334 1083 10 88 57 851 1006 3761 10 42 3 17 13 75 151 42 3 17 15 77 159 A v . − − − − 201 . 5 461 . 1 − − − − 501 . 9 1413 . 4 T able 3: N umerical resu lts obtained by the AL T with r = 1 . 3 N ε = 10 − 4 ε = 10 − 5 N g 1 N g 2 N g 3 N f T rials Ev al. N g 1 N g 2 N g 3 N f T rials Eva l. 1 27 − − 1 7 44 61 2 7 − − 19 46 65 2 19 − − 1 5 34 49 2 2 − − 16 38 54 3 12 − − 9 21 30 14 − − 10 24 34 4 45 11 − 37 93 178 4 5 11 − 48 104 211 5 73 44 − 15 132 206 7 6 44 − 17 137 215 6 21 11 − 42 74 169 2 1 11 − 64 96 235 7 ∗ 34 27 − 39 100 205 3 4 34 − 42 110 228 8 12 20 4 23 5 9 156 12 22 4 24 62 164 9 8 16 3 29 56 165 8 16 3 36 63 193 10 14 2 13 13 42 109 14 2 13 18 47 129 A v . − − − − 65 . 5 132 . 8 − − − − 72 . 7 152 . 8 T able 4: Improv ement ob tained by the AL T with r = 1 . 3 in comparis on with the other method s us ed in the experimen ts N ε = 10 − 4 ε = 10 − 5 T rials Eva l. Trials Eva l. PEN AL T IBB A AL T PEN AL T IBB A AL T PEN AL T IBB A AL T PEN AL T IBB A AL T 1 5 . 61 1 . 16 8 . 10 1 . 30 9 . 11 1 . 2 4 12 . 89 1 . 40 2 7 . 09 1 . 00 9 . 84 1 . 02 8 . 24 1 . 11 11 . 59 1 . 19 3 43 . 67 9 . 05 6 1 . 13 6 . 97 8 8 . 63 8 . 17 125 . 12 6 . 38 4 2 . 94 2 . 53 4 . 60 2 . 35 8 . 28 3 . 95 12 . 24 4 . 16 5 5 . 08 2 . 14 9 . 77 2 . 05 8 . 01 2 . 3 0 15 . 31 2 . 18 6 12 . 28 8 . 50 16 . 14 10 . 8 8 66 . 32 28 . 32 81 . 28 34 . 49 7 ∗ 1 . 99 1 . 20 2 . 91 1 . 05 2 . 01 1 . 15 2 . 91 1 . 02 8 6 . 19 1 . 08 9 . 36 0 . 92 6 . 69 1 . 11 10 . 12 0 . 96 9 21 . 13 5 . 96 28 . 68 6 . 56 72 . 21 15 . 97 94 . 28 19 . 4 9 10 3 . 21 1 . 79 4 . 95 1 . 39 3 . 60 1 . 64 5 . 24 1 . 23 A v . 10 . 92 3 . 44 15 . 55 3 . 45 27 . 31 6 . 50 37 . 10 7 . 25 12 sho w dynamic s of the search . The PEN has exec uted 909 tria ls and the numbe r of e v aluat ions was equa l to 909 × 3 = 2727 . In Figur e 4 the first line (from up to do wn) of symbo ls ‘+ ’, located under the graph of problem (9)–(1 1), rep resents the points where the first constraint ha s not been satisfied (n umber of such trials is equal to 16). Thus, du e to the decision rul e of the IBBA, th e second constrain t has not been e v aluate d at thes e po ints. The second line of symbo ls ‘+’ represents the points where the fi rst constraint has been satisfied bu t the seco nd constraint has bee n not (number of such trials is equa l to 16). At the se points both const raints hav e been e v aluate d but the ob- jecti ve fun ction ha s been not. The last line represents th e points where both con- straint s hav e been satisfied (number of such trials is 597) and, therefore, the ob- jecti ve fun ction has been ev aluate d too. The tot al nu mber of ev aluations is equal to 16 + 16 × 2 + 597 × 3 = 1839 . These e v aluati ons hav e been ex ecuted during 16 + 16 + 597 = 62 9 trials. Similarly , in Figure 5, the first line of symbols ‘+’ indicate s 21 trial points where the first constra int has not been satisfied. T he second line represents 11 points w here the first cons traint has been satisfied bu t the second constrain t has been not. The last lin e sho ws 42 points where both constraints ha ve been satisfied and the object iv e function has been e v aluated . The tota l number of ev aluations is equal to 21 + 11 × 2 + 42 × 3 = 169 . These ev aluations ha ve been ex ecuted during 21 + 11 + 42 = 74 tria ls. Refer ences [1] Bertsek as D.P . (1996 ), C onstr ained Optimization and Lagran ge Multiplie r Methods , A thena Scientific, Belmont, MA. [2] Famula ro D., Ser geye v Y a.D., and Pugliese P . (20 02), T est Problems for Lip- schitz Univ ariate Global Optimization with Multiextr emal Constraints . In: Dzemyda G., ˇ Saltenis V . , an d ˇ Zilinskas A. (Eds.). Stoc hasti c and Glob al Op- timizatio n , Kluwer Acad emic P ublish ers, Dordrecht, 93–110. 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Optim. 5 , 858–870. 13 Figure 4: S olving the prob lem (9 )–(11) by the method IBB A 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −2 −1 0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 100 200 300 400 500 600 Axis x Number of Trials Figure 5: S olving the prob lem (9 )–(11) by the method AL T with r = 1 . 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −2 −1 0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 20 30 40 50 60 70 80 Axis x Number of Trials 14 [8] Ser gey e v Y a.D. (1995b) A one-d imensiona l deterministic globa l minimiza- tion algori thm, Comput . Math. Math. Phys. 35 , 705–717. [9] Ser gey e v Y a.D. (199 8), Global one -dimensio nal optimiza tion using smooth auxili ary fun ctions, Math. Pr ogr am. , 81 , 127–1 46. [10] Ser geye v Y a.D., Famula ro D., and Pugliese P . (2001), Inde x B ranch- and- Bound Algorithm for L ipschi tz Univ ariate Global Optimization with Mul- tiex tremal Constraints, J. Glob al Optim. , 21 , 317–341. [11] Ser geye v Y a.D. and Markin D.L. 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