Global optimization of water distribution networks through a reduced space branch-and-bound search

Global optimization of water distribution networks through a reduced space branch-and-bound search
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  WATER RESOURCES RESEARCH, VOL. 37, NO. 4, PAGES 1083-1090, APRIL 2001 Global optimization of water distribution networks through a reduced space branch-and-bound search Andr6 L. H. Costa, Jos6 Luiz de Medeiros, and Fernando L. P. Pessoa Department of Chemical Engineering, scola de Quimica, Federal University of Rio de Janeiro Rio de Janeiro, Brazil Abstract. A branch-and-bound pproach o the problem of optimal design of water distribution networks s presented. Global optimum s reached hrough he generation of convergent equences f upper and lower bounds. The relaxations esponsible or the lower bounds correspond o linear programming roblems ormulated hrough he enlargement of the srcinal easible egion by outer approximations f the constraints. Although formulated within an arc-based ramework, he proposed scheme does not apply the branching process o all flow variables; by utilizing he mass conservation rinciple, only a reduced set of variables s assigned or branching. he algorithm was applied o three variants of a classical roblem from the literature. Comparisons with previous esults indicates a faster convergence o the optimum (fewer linear programming roblems solved) n several situations. 1. Introduction Water distribution networks are key systems n modern ur- ban and industrial infrastructure. The whole issue of water distribution system design covers a long list of subjects de- manding examination: apital and operating costs, ayout de- sign, reliability, water quality constraints, tc. Among these several aspects elated o network design, he least cost design problem LCD) has been frequently studied n the literature. The LCD involves he design of network elements determi- nation of pipe diameters, eservoir levations, tc.) associated with the lowest network cost n order to supply specified low rates at demand sites subject o hydraulic head bounds. Many works approached he LCD employing ocal optimi- zation echniques e.g.,Hansen t al., 1991]. However, he LCD formulation presents onlinear erms hat may occasion mul- tiple local optima. For this reason some papers n recent years have focused on the determination of the global optimum. Global optimization methods an be divided nto stochastic and deterministic. Stochastic methods nclude random steps that allow movements mong different ocal optima. However, this class of methods does not indicate how far from the global optimum he achieved olution s. Simulated nnealing Loga- nathan et al., 1995] and genetic algorithms Simpson t al., 1994; Dandy et al., 1996; Savic and Walters, 997] are stochastic techniques hat have already been applied o the optimization of water distribution networks. Deterministic methods or global optimization ind a solu- tion according o a previously established olerance from global optimality. These methods, ifferent rom the stochastic ones, explore he particularities f the problem's mathematical structure. Branch-and-bound method is the main deterministic method employed or pipe network optimization. Eiger et al. [1994] applied a branch-and-bound cheme utilizing a non- smooth optimization method or the upper bound update and a linear programming LP) problem ormulated with the du- Copyright 2001 by the American Geophysical nion. Paper number 2000WR900267. 0043-1397/01/2000WR 900267 $ 09.00 ality theory for the lower bound update. Sherali and Smith [1997] used he reformulation-linearization echnique o create the necessary elaxations o evaluate ower bounds of the prob- lem solution for the branch-and-bound method. In Sherali et al. [1998], ower bounds re obtained by outer approximations f the set of nonlinear constraints n the problem. This work approaches he LCD with a new global optimiza- tion algorithm based on the branch-and-bound method utiliz- ing a linear relaxation. Starting with a network model built with unidirectional low variables, he relaxation s formulated by enlarging he feasible egion of the problem through he re- placement of the energy conservation onstraints y a set of outer inequality constraints. The branching process s re- stricted to a reduced set of flow rate variables, here called the set of branching ariables SBV), since he search egion or all flow rate variables can be uniquely projected onto the SBV subspace hrough he material conservation quations or the entire network. The algorithm was applied n the optimization of three variants of the classical lperovits nd Shamir 1977] problem. Results ndicate hat n several ituations he solution can be reached equiring ewer explored subregions nd solv- ing fewer LP problems when compared with a previous ap- proach. The remainder of this paper s organized s ollows. Section 2 presents he variables, bjective unction, and constraints f the optimization model. Section describes he formulation of the relaxation used o calculate he lower bounds of the opti- mum. Section 4 demonstrates how a reduced set of flow vari- ables SBV) can define he search egion or all flow variables, thus educing he number of branching ariables n the branch- and-bound procedure. Section 5 describes he branch-and- bound algorithm or network optimization. Section 6 presents the computational esults. Finally, section 7 reports conclu- sions and suggestions or further research. 2. Mathematical Model Consider a water distribution network formed by a set of supply nodes Vs) and a set of demand nodes Vd) intercon- nected by pipelines links). Supply odes must provide water to 1083  1084 COSTA ET AL.: GLOBAL OPTIMIZATION OF WATER DISTRIBUTION NETWORKS demand nodes according o fixed-demand low rates under head specifications. etwork nodes are identified by the index t = 1, ..., N. Elevation of node is represented y z t. Links are identified by the index k - 1, ..., S and partitioned nto two sets: ndependent inks (K•), composed f S - N + 1 links and dependent inks (Kr,), composed f the remaining N- 1 links. 2.1. Variables Flow in pipe k is modeled with a pair of nonnegative ari- ables q•2)k and (q2•)k. Variable (q•2)• is the flow rate along a predefined direction associated ith link k, while corresponds o the flow rate going n the opposite direction. External flow rates of supply and demand nodes are network parameters epresented y w (wt > 0 for t G Vs and w -< 0 for t G Va). The set of external low rates must obey he mass conservation rinciple, .e., 5; w = 0. The hydraulic ead at node t is described y Ht. The cost per unit head at supply node G Vs is represented y ct. In each ink k the heads at the terminal nodes are represented y H•,• for the starting node and H•,2 for the ending node, according o the assigned link direction. Each ink k is formed rom a set of pipe seg- ments ndexed with n = 1, ..., Q such hat each pipe seg- ment has a defined commercial diameter D•,n with cost per unit length c/,,•. The design of a link k is made by the decision variable X•,n which corresponds o the length of pipe segment n present n link k. The sum of pipe segment engths n link k must be equal to the link length L 2.2. Optimization Problem According o the variables resented n section .1, the LCD is equivalent o problem P s Q Minimize Z C nXkn - • ctHt, k=l n=l t•Vs subject o • [(q•2)k- q2•)k] • [(q•2)•- (q20•] wt--- k•Q•t k•Q? t t=l,...,N-1, Hicl Hic2 K( =l kbX--•k•) (q12)•-- q21)•] Ztcl - tc2 - k=l .... ,S, H• nin--< t-< Hi TM t • Va, 0 --< Ht--< Hi TM t • Vs, (la) (lb) (•c) (ld) (le) Q Z Xkn-- mk k = 1 .... S, (if) rt=l Xkn 0 Vk, Vn, (lg) 0 -< (q•n)• _< q•2)• -< q•2ax)• k • Ki, (lh) max k • r,. (li) in• (q21) < (q21)k <-- q2• //,- /,- The objective unction la) is the total cost of the network, represented y the sum of the costs of all pipe segments nd the costs f the heads at supply nodes. Constraints lb) corre- spond o the mass conservation quations escribed sing he pair of flow rates associated ith each ink. Links connected o node and oriented oward t are grouped n the set Q}n, while links belonging o the set Q•Ut are connected o t but are leaving t, according o the assigned ink direction. Constraints (lc) correspond o the energy conservation quation section 2.3). Constraints ld) guarantee hydraulic heads at demand nodes under specified ounds. Constraints le) establish he head bounds at supply nodes. Constraints if) and (lg) de- scribe each ink as a series of pipe segments. ounds on flow rates, corresponding o constraints lh) and (li), must be at- tributed by the designer. These constraints erve as starting points for the branch-and-bound earch. Alternative defini- tions of initial bounds are given by Eiger et al. [1994] and Sherali nd Smith 1997]. These bounds define he set o which corresponds o the search egion n the context of variables participating n the branching rocess SBV) (set KI). 2.3. Energy Conservation Equation The energy conservation quation (lc) evaluates he link head oss by the Hazen-Williams quation Savic and Walters, 1997] applied o each pipe segment, here he total head oss in a link is the sum of the head osses n each pipe segment belonging o it. Constant K is to/C a, where C is the Hazen- Williams coefficient and to is a conversion constant. The adopted values, n SI units, for constants o, a, and b are 10.6688, 1.852, and 4.87, respectively. The mathematical orm of (lc) is able o model he pipe low in both directions. Because of the structure of P, its solution will be a feasible point where, for each ink k, two flow con- figurations an occur: q 12)/, >- 0 and (q 2 )/, = 0, meaning that flow direction corresponds o the assigned ink direction, or (q • 2)•, = 0 and (q 2 )/, >- 0, meaning hat flow is in the opposite direction. As justified below, flow configurations (q • 2)• • 0 and (q2 )• • 0 (which do not have physical meaning), even though mathematically easible over P, they cannot present ower costs. Consider o solutions f P, I and II, assigning net flow rate q through a pipe be•een locations and 2. Representing the flow by (lc), we analyze hese o feasible low configura- tions: , where q • 2 = q and q2 = 0 and II, where q • 2 = q + $ and q2• = •, with $ > 0. •t a function of flow rates be denoted by F(q•2 , q2•) = (q•2) a - (q2•) a. Thus (lc) is H• - H2- K F(q•2, q21) + z• - z2 = 0. n=l The expression f F for configurations and I is F x = qa and F I= (q + t•) a - t• , respectively, here > 1. Since II> m , configuration I presents arger ead oss or he same et flow rate transported, s can be seen n (lc), or alternatively, configuration I corresponds o larger diameters or an equiv- alent head oss. Therefore, even hough mathematically easi- ble, solution II cannot exhibit lower costs relative to solution I. 2.4. Remark Feasible points where both flow rates related to a link are not equal o zero (i.e., (q•2)• -• 0 and (q2•)k -• 0 for any k) are not considered s possible olutions f P.  COSTA ET AL.: GLOBAL OPTIMIZATION OF WATER DISTRIBUTION NETWORKS 1085 3. Relaxation The energy conservation quality constraints contain the nonlinearities esponsible or the nonconvex tructure of P (indices elated o links are omitted or simplification): H1 - H2- K [(q12) -- (q21) ] + Zl- Z2-- 0. n=l Since low rates are nonnegative constraints lh) and (li)) and the flow rate exponent n the Hazen-Williams equation s not <1, functions q12) a and (q21) a are convex. Thus, for these functions a linear interpolation with two points is an upper bound and a tangential pproximation s a lower bound. [ _ min The following nequalities re then valid n the intervals q 12, maxl and r-min 12 .• tq21 , q•"]' (qmax•a min•a 2 -- (q 12 _minx _ _minx (2a) q 12) • (q 12 a "Jr- _max min (q 12 q 12 , (q 12) • (q 12) d- a (q 12) - (q 12 - q 12), (2b) _ maxx a _ minx a (q21) -- (q21) _ minx a _ minx (q21) < tq21 J + _max min (q21 , (3a) --q21 ) q21 -- q21 (q21) • (q21) + a(q21)a-l(q21- 21), (3b) where he tangential points are chosen nside he intervals: 12 • [qmin [_min max] n constraint lc) the 2 , q•X] and q21 G tq21 q21 l- substitution f (q12) a by lower bound (2b) and (q21) a by upper bound (3a) yields, after rearrangement f terms, the following nequality: H1 - H2- K •-s Xn d- • Xnq12- •-• Xnq21 n=lDn Dn Dn d- Z 1 -- Z 2 • 0, (4) where A' = (q12)a a (q12) ... in a - (q 21 d- (q•lax) ... in, 6/21 max _ min q min 21 , B' = a(q12) -l, -- (q21) C' (q•lax)a rnin, max min 21 m q21 Similarly, he replacement f (q12) a in constraint lc) by up- per bound 2a) and of (q21 by lower bound 3b) gives a Hi - H2 - K • Xn d- • Xnq 2 - '• Xnq21 n=iDn Dn Dn d- Z 1 - Z 2 • 0, (5) where min•a A" = -(q21) a + a(q21) + (q12 - _ maxxa minh a (q12 -- (q12 J max _ min q12 - q12 _ 12 [q12 ) B" (qmax•a t_rninxa q•2ax _min 12 min q12 , C" = a(q21) -1. The substitution f equality lc) by the pair of inequalities 4) and (5) can enlarge he feasible domain, elaxing P. However, the presence f bilinear terms Xnq12 and xnq21 still implies a nonconvex structure. The bilinear term Xnq12 can be bounded by the following linear forms [McCormick, 1976; Al-Khayyal, 1992]: ma max (6a) nq 2 -• Lq 12 + q 12 On - Lq 12 _min. (6b) nq 12 q 12 /•n, _min. -- m min (6c) nq 2 q 12 Xn d- Lq 12 q 12 Xnq 2 -• q n•a'scn (6d) and analogously, __ ma , _max (7a) nq21 Lq21 + q21 on •q21 , min. (7b) nq21 • q21 xn, min. __ _min (7C) nq21•-q21xn + Lq21 •q21, ma Xnq 1 -- q 21 On. (7d) The bilinear erms present n (4) and (5) can be relied using the linear appro•mations epresented n (6a)-(6d) and (7a)- (7d). Two inear orms j = 1 and = 2) are derived rom (4) through the replacement of terms x•q12 and x•q21 by their lower and upper bounds, espectively. or j = 1, x•q•2 is replaced by (6a) and x•q21 is replaced by (7c). For j = 2, Xnq12 s replaced by (6b) and Xnq21 s replaced by (7d). The resultant nequalities re H1-H2+E; • +r; • q12+G; • q21 =l n=l n=l +H +Zl-22 •0 j= 1,2, (8) n=l where max , _rnin ' -KB ' L [ = -KA'-KB'qn + KC 6121, F1 = , G1 KC'L H[ KB' _max KS .... in = , = Lql 2 -- Lq21 , min t max ' -KA'-KB'q12 +KC q21 2-- F[=G[=H[=O. The same operations re applied, n an analogous ay, o (5) through he replacement of Xnq12 and Xnq21 by their upper and ower bounds, espectively. orj = 1, Xnq 12 s replaced y (6c) andxnq21 s replaced y (7a). Forj = 2, Xnq12 s replaced by (6d) and Xnq21 s replaced by (7b). The resultant nequali- ties are H1- H2 Ey •n =l ey •nn 12 - G7 •n q21 =l n=l + H'] + Z 1 - Z 2 0 n=l j = 1, 2, (9) where tt tt tt _ max El= -KA - KB"q• n + KC q21 F'• = -KB"L G" = KC"L tt tt _ min tt _ max H i = KB Lq12 - KC Lq 21  1086 COSTA ET AL.' GLOBAL OPTIMIZATION OF WATER DISTRIBUTION NETWORKS E" = -KA .... q ...... in "= "= H" = 0 - KB max •xc q 21 F 2 G 2 2 2 ß Finally, we can build the desired relaxation or P through elimination f constraints lc) and nclusion f inequalities 8) and (9). The relaxation s convex and corresponds o a LP problem. This relaxed orm of P will be referred henceforth as R. Note that R was created without the introduction of extra variables. he formulation of R is only based on the replace- ment of each energy conservation onstraint y four appropri- ate inequality constraints hat enlarge he feasible egion. t yields a mathematical problem with fewer variables and con- straints when compared with previous similar approaches s proposed y Sherali nd Smith 1997] and Sherali t al. [1998]. 3.1. Tangential Points Problem R is formulated with any set of tangential points q •2 and • chosen nside he search egion. However, com- putational esults ndicate hat the following procedure can give better numerical erformance. et r min q•2, q•,•x] be an interval defining the subregion where the relaxation will be built, and et q* be the solution f the previous elaxation 2 over he region whose artitioning enerated min maxl f q•2,q•2 J. ß [ _rain _maxl * ' otherwise, 2 •2 • tq•2 , q•2 1, then q•2 = q •2, = 0.5 _rain- max• The same rule is used for the determination q •2 -t- q •2 •. of q2•. At the start of the optimization process he tangential points hosen n the nitial search egion re q •2 0.5' rain- tq 12 -I- max1 for all links. q•,:•x) nd q21 = 0 5(q min q- q21 • 21 3.2. Collapsed Intervals During the branch-and-bound rocedure proposed n this work, bounding ntervals of flow rates may collapse nto a min max nd/or -min •max oint, i.e., q•2 = q•2 q2• = 6/2 . In this case he relaxed constraints 8) and (9) are still applied hrough alter- ations o the parameters resented n (4) and 5): (1) If _min 12 -- qmax 12 , then he term (q•X)a _ .(qminha _min• 2; ]/[qm•X _ pre- 2 ql2 ] min max sented n A" and B" is replaced by zero. (2) If q21 = q2• , then he erm (qg,•x)• . mi ... ax ' q21 ) ]/[q2• - qg,•,n] resented n A' and C' is replaced by zero. 4. Set of Branching Variables (SBV) The network connectivity an be represented by the inci- dence matrix M (size N x S). This matrix s defined according to the directions assigned o links: Mtk = i if link k enters node t, Mtl = - 1 if link k leaves node t, and Mtl = 0 if link k is not connected to node t. Let M' be the reduced incidence matrix (size N - 1 x S) given by the deletion of a row from M. At this point, et all link flow rates be represented y vector q. Each component q•, of vector q corresponds o the bidirec- tional flow rate in link k: If flow is along the assigned ink direction, hen q•, > 0; otherwise, •, < 0. Let w' be a vector containing nown external low rates at nodes = 1, ..., N - 1. Employing his representation, he mass conservation rin- ciple at nodes can be described y a simple equation M'q = -w'. (10) There are S variables link flow rates) and N - 1 equations n (10) or, equivalently, - N + 1 degrees f freedom; hat is, the specification f a certain set of S - N + 1 flow rates determines ll other flow rates. t is important o note that S - N + 1 is also equal o the number of fundamental ycles n the network [Mah and Shacham, 1978]. So we assume he follow- ing partitioning of matrix M' and vector q: m' = [MD ml] (11) qr= [q• qiT]. (12) This partitioning s defined according o the separation f links into a dependent and an independent et of variables: Vector qo (size N - 1 x 1) contains low rates q•, for k G KD, and vector q• (size S - N + 1 x 1) contains low rates q•, for k G K,•. The sets of dependent nd independent inks are defined such hat the rank of M•) is N - 1; thus the inverse of M•) exists. A heuristic ule to compound he independent et of links is to select one link in each fundamental cycle of the network. The introduction f (11) and (12) into (10) gives oqo + M = - (13) qI ß Solving 13) for qD gives qB = -(Mb)-•w '- (Mb)-•M[qi (14) Equation (14) allows the determination of vector qD from vector qi- In a similar way, starting rom the bounding ntervals for vector qi, it is possible, sing 14), to determine he corre- sponding ounding ntervals or vector q,. This operation an be conducted with interval arithmetic Moore, 1966]. In conclusion, he entire search egion of flow rates can be mapped rom the feasible domain of independent low rates since he bounds on dependent low rates are generated with (14). The independent low rate variables efine he aforemen- tioned set of branching ariables SBV) for the branch-and- bound search. 4.1. Generation of Bounding ntervals for Flow Rates The bounds on independent nd dependent low rates are used ogether n the optimization or creating he relaxations to evaluate ower bounds of the problem solution. However, problem P and its relaxation R are formulated with unidirec- tional flow rates, and (14) is developed or bidirectional low rates. For this reason the algorithm in section 4.2 must be applied to determine bounds on all flow rate variables. 4.2. Algorithm [ _min Consider a search egion defined by the intervals q12, q•':•x]k nd qmi ... ] for k • K . 1 , q21 Jk 4.2.1. Step 1. Generate he corresponding ntervals of bi- directional low variables or indcpcndent inks: _min q[nax]. th , Omitting ndex we have he ollowing: f _min _max q21 = q21 = 0, then q•nin q•n•n nd q•nax q•,:•x. f _min= q•n•x •2 = O, then : max 0, ax nd q•nax min f q•,•x 4:0 and q2• •nin --q2• --q21 ß then q•nin _q2ml•x nd qp•X = q•n•x. ccording o the nature of the branching rocess mployed n our branch-and-bound algorithm as stated n the section 5), the cases considered above cover all possible ombinations or the pairs of flow rates q • 2 and q 21 of a same inkß 4.2.2. Step 2. Starting with the independent low rate in- tervals _min [qI , q[nax], enerate hrough (14), the intervals of _min •aX]. ependent low rates [q•) , 4.2.3. Step 3. Determine ntervals qmin and 2, q n, x] , [qmin max] or k G KD from ntervals min •aX]. q•in qD , If > 1 , q21 lk -- 0 and q•aX > 0, then q[n2n _min ?•x = _max min D , q qD , 6/21 = 0, and _ min _max _ max = 0. If q•9 n < 0 and q•ax > 0, then q•2 = 0 q•2 21 , --  COSTA ET AL.: GLOBAL OPTIMIZATION OF WATER DISTRIBUTION NETWORKS 1087 q ... in max _q•in. If q•9 n < 0 and q•aX < O, ,q2• =O, andq2• - min __ 0 _ max 0 _ min max _ max in hen q • 2 , , •2 = q2• =-qo ,andq2• = -q• Applying his algorithm, he bounding ntervals of all flow variables, q•2)•, and (q2•)•,, for ¾ k, are determined rom the bounding ntervals of the independent low variables, q •2)•, and (q21)/, for k • K•. 5. Branch-and-Bound Algorithm The branch-and-bound ethod s based on the generation of convergent sequences f upper and lower bounds of the problem solution. The process s stopped when the distance between a lower and an upper bound is less than a given tolerance, hus assuring feasible solution sufficiently lose o the global optimum. Upper bounds are generated by the ap- plication of a local minimization earch r a search or feasible points of P. In this work, an adaptation of the procedure proposed y $herali et al. [1998] s used: The flow rates of the solution of R are introduced nto P, yielding a LP problem or the other variables, whose solution can give a new feasible point. In each iteration the best alternative found so far is stored incumbent, lobal minimum candidate). he sequence of lower bounds s obtained by determining he solution of R in different subregions f the search space. f the optimal objective unction f R over a subregion s greater han he incumbent, his subregion s eliminated rom the search. Oth- erwise, his subregion will be divided nto two new, smaller subregions here further investigations ay be done. As re- laxations R are solved over successive artitions, hey become closer o P in each subregion. The steps of the branch-and- bound procedure shown n sections .1-5.7 are similar o those of Androulakis t al. [1995]. 5.1. Step 1: Initialization Separate he links nto S - N + 1 independent inks SBV) (set Ki) and N - 1 dependent inks (set Ko). Define bounds on the independent low rates (section 2.2). These bounds define he initial search egion o for P. Set the initial solution for upper and ower bounds s Z © - +oo and Z © = -% up inf respectively. nclude o in the list of regions where he solution may be found: U - {Io}. Solve R defined over o, employing the algorithm of section 4.2 for generating ounds on depen- dent low rates. f R is feasible, ts optimal objective unction s Zinf, . Initialize he iteration counter = 1. Otherwise, top, because P is infeasible. 3 5 Figure 1. Network structure. Table 1. Problem Data Node External Flow Rate, Elevation, Minimum Maximum m3/h m Head, m Head, m 1 1120 210 ...... 2 -100 150 30 60 3 -100 160 30 50 4 -120 155 30 55 5 -270 150 30 60 6 -330 165 30 45 7 -200 160 30 50 5.2. Step 2: Region Selection Select from the list of candidate regions a subregion n whose ower bound s Zinf, such hat Zinf, <-- Z•nf, m, VI m • U. Update he current ower bound o 7(i) = Z --' inf inf, n' 5.3. Step 3: Upper Bound Evaluation Apply Sherali et al.'s [1998] method using he flow rates of the solution of R over n. However, f there s (q •2)/, 4:0 and (q2•)/• 4:0 for any k in the solution of R, the upper bound evaluation s not applied and no feasible point s determined t this teration (see section 2.4). If a candidate olution s found, ts objective unction s an upper ound enoted y Zsu n- If Z[• •) > Zsupn then '--•5 and tore he new update he upper bound o/-,sup Zsup,n solution ound new ncumbent); therwise, he upper bound s z(i) = (i- 1) sup Zsup ß 5.4. Step : Convergence heck Verify if the stopping riteria adopted between upper bound Z?u)p nd ower ound 0) is reached. f it is rue, hen top; --, in f the solution is the current incumbent. 5.5. Step 5: Region Partitioning Divide the region n into two subregions, n• and In2 ac- cording o the partitioning procedure below and exclude he region n from the list: U = U - {In}. The split is applied to the largest nterval that delimits n Furthermore, because of the solution structure see 2.4) a possible contraction of the new regions is immediately checked. Let [qmin _max• r_min max for k • K be the 2, q•2 l/• and tq2• , q2• ]/• / minx intervals hat delimit n. Let (A•2)k = (q•X)k _ {,q•2 /• and (A2•)/' _ (q•X)• _ (q•n)• be he engths f these ntervals. If max (A•2)•,, k G Ki} -> max (A2•)•,, k G Ki}, then 8 = arg max {(A•2)•, k G KI} and the bounds or the new regions are given by the following ntervals with k Region n • _ min _maxq [ _min max [q•2, q2• q21 k4:8 •2 J•, , [ _min _min max (q•2 + •2)]k q•2 , 0.5 q [q•l n, q21 /• k = 8, Region _min max] [ _min _max-1 [q12 ql2 [0 5 mi .... x .... - tq•2 + [0 0]• k = 12 1, q12 Jk , ß If max {(A•2)/•, k • K ) < max {(A2•)/•, k • Ki) , then 8 = arg max { (A21)k, k • Ki) and the bounds or the new regions with k • K are,
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