Incremental and Encoding Formulations for Mixed Integer Programming

Transcription

1 Incremental and Encodng Formulatons for Mxed Integer Programmng Sercan Yıldız a, Juan Pablo Velma b,c, a Tepper School of Busness, Carnege Mellon Unversty, 5000 Forbes Ave, Pttsburgh, PA 523, Unted States b Sloan School of Management, Massachusetts Insttute of Technology, 77 Massachusetts Ave., Cambrdge, MA 0239, Unted States c Department of Industral Engneerng, Unversty of Pttsburgh, 3700 O Hara Street, Pttsburgh, PA 526, Unted States Abstract The standard way to represent a choce between n alternatves n Mxed Integer Programmng s through n bnary varables that add up to one. Unfortunately, ths approach commonly leads to unbalanced branch-and-bound trees and dmnshed solver performance. In ths paper, we present an encodng formulaton framework that encompasses and expands exstng approaches to mtgate ths behavor. Through ths framework, we generalze the ncremental formulaton for pecewse lnear functons to any fnte unon of polyhedra wth dentcal recesson cones. Keywords: Mxed Integer Programmng, Dsunctve Programmng, Formulatons. Introducton A textbook approach to selectng among n dscrete alternatves n a Mxed Integer Programmng (MIP) problem s to utlze n bnary varables {y } n wth the addtonal requrement that they add up to one. In partcular, ths s captured by the set D n := n \ {0, } n where n := y 2 R n + : P n y =. For nstance, gven a fnte set {a,...,a n } R, we may model the constrant x 2 {a,...,a n } Correspondng author. Tel.: Emal addresses: syldz@andrew.cmu.edu (Sercan Yıldız), velma@mt.edu (Juan Pablo Velma) Preprnt submtted to Elsever September 3, 203

2 va the MIP formulaton gven by x = a y, y 2 D n. () Ths approach s qute smple and, as we dscuss n Secton 3, t can be easly extended to more general settngs. Unfortunately, although the approach s usually qute e ectve, t has some unfavorable propertes that can slow down branch-andbound based MIP solvers. These propertes stem from the way the formulaton s a ected by standard varable branchng n a branch-and-bound algorthm. Whle fxng a gven bnary varable y to one (usually denoted up-branchng) fxes all other varables to zero, fxng the same varable to zero (usually denoted downbranchng) does not mpose any constrants on the other varables (unless n = 2). It s well-known that ths behavor leads to unbalanced branch-and-bound trees and may result n long soluton tmes. To ths date, there have been three central approaches to dealng wth the branchng ssues that arse n formulatons based on D n. The frst approach replaces the usual varable branchng strategy wth a more sophstcated constrant branchng scheme []. For nstance, Beale and Tomln [2] ntroduced SOS branchng as a very e ectve branchng scheme for D n and other related sets. Unfortunately, f SOS branchng s not mplemented usng modern technques, t can be slower than varable branchng n state-of-the-art solvers [3]. One way to crcumvent ths problem s to use bnary varables to smulate a specalzed constrant branchng scheme [4, 5]. The second approach s normally employed n schedulng problems where selectng opton corresponds to executng a certan actvty n perod. In ths settng, we can use an alternatve set of bnary varables that nstead ndcate whether the actvty s executed n perod or before. These new varables are sometmes called by-varables and lead to balanced branch-and-bound trees that can sgnfcantly reduce soluton tmes [6 2]. The thrd approach proposes to model n alternatves wth a number of bnary varables that s logarthmc n n [5, 3 7]. Ths approach tends to generate more balanced branch-and-bound trees, but, as we wll see n Secton 2, they are not mmune to mbalance ssues ether. Nevertheless, formulatons that are bult usng a logarthmc number of bnary varables usually provde a sgnfcant computatonal advantage. The encodng formulatons that we descrbe n Secton 2 provde a general framework n whch all three approaches can be vewed. In Secton 3, we use ths framework to come up wth an ncremental formulaton for the fnte unon 2

3 of polyhedra wth dentcal recesson cones. We end the paper wth a note on the selecton of encodngs n Secton 4. In the remander, we let [n] := {,...,n} wth the understandng that [0] = ;. We use 0 n 2 R n to refer to the n-dmensonal vector of all zeros and e,n 2 R n to refer to the th n-dmensonal standard unt vector. 2. Encodng Formulatons and Branch-and-Bound The bass for our analyss wll be a reformulaton of D n whch has been part of the mathematcal programmng folklore snce at least 972 [3] and whch has recently been re-dscovered by several authors [5, 5, 7]. Ths formulaton requres a set of vectors n b o n {0, }m such that b, b for any,. For any such set, t models y 2 D n as y =, u = b y, u 2 {0, } m, y 2 R n +. (2) A key theoretcal strength of (2) s the tghtness of ts LP relaxaton. In ths regard, a MIP formulaton has the strongest possble property when all extreme ponts of ts LP relaxaton obey the correspondng ntegralty requrements. Such formulatons are referred to as locally deal by Padberg [8] and Padberg and Ral [9]. We note here that (2) s locally deal for any choce of n b o n {0, }m as long as b, b for any, [6, 7]. Formulaton (2) may seem wasteful as t contans more varables and constrants n o than the orgnal defnton of D n. However, dependng on the choce of n b, (2) can present a computatonal advantage by sgnfcantly reducng the number of bnary varables or by leadng to more balanced branch-and-bound trees. One reason for ths advantage stems from the fact that varable branchng on u nduces a constrant branchng scheme on y n the sprt of [4, 5]. More specfcally, we have that up-branchng on the varable u fxes y = 0 for all such that b = 0 whle down-branchng on u fxes y = 0 for all such that b =. The e ectveness of ths nduced branchng scheme wll of course depend on the choce of n b o n. We now descrbe three such choces and analyze the branchng schemes they nduce. We wll refer to the resultng formulatons as encodng formulatons snce one can thnk of n b o n as an encodng of a selecton among the alternatves n the sense that u = b f and only f alternatve s selected. 3

4 We obtan the smplest encodng when we set m = n and b = e,n. In ths case, (2) reduces to y =, u = y, 8 2 [n], (3) u 2 {0, } n, y 2 R n +, whch s nothng more than a redundant reformulaton of D n. We can easly see that the nduced branchng scheme s ust the standard varable branchng on y and, hence, we do not obtan any benefts. We refer to ths choce as the unary encodng formulaton. A more meanngful choce appears n the development of logarthmc-szed formulatons. For ease of exposton, let us assume that n = 2 k for some postve nteger k 2 Z + although the approach can easly be adapted to more general cases as well. In ths settng, we can smply let m = k and n b o n = {0, }m. Ths choce transforms (2) nto a formulaton wth a logarthmc number of bnary varables whch was the orgnal case studed n [5, 3, 5, 7]. The nduced constrant branchng scheme s not qute clear, but we can easly see that t creates balanced branch-and-bound trees by observng that we have n 2 [n] :b 0 = 0, b = u, 8 2 J o = n 2 [n] :b 0 =, b = u, 8 2 J o for all 0 2 [m], J [m] \ { 0 } and u 2 {0, } m. Here, J can be consdered as the subset of n u o m whch has been fxed to some value at a partcular node of = the branch-and-bound tree. Together wth the reduced number of varables, ths property can lead to a sgnfcant computatonal advantage. We refer to ths choce as the bnary encodng formulaton. Whle the bnary encodng formulaton nduces a non-trval branchng scheme, as noted n [5], t does not nduce the tradtonal SOS constrant branchng scheme. To construct a formulaton that nduces SOS branchng on y, we can set m = n, b = 0 m and b = P = e,m for all 2 {2,...,n}. In ths case, we have that up-branchng on the varable u fxes y = 0 for all apple whle downbranchng on u fxes y = 0 for all >. Thus, we recover precsely the SOS branchng scheme of Beale and Tomln [2]. We refer to ths choce as the ncremental encodng formulaton because t can be used to construct a formulaton for pecewse-lnear functons that s known as the ncremental model n the lterature. 4

5 To llustrate the potental advantage of the ncremental encodng formulaton, we study the behavor of a smple branch-and-bound algorthm on the d erent formulatons of the followng smple problem. For a gven a 2 (Z \ {0}) n such that a < a + for all 2 [n ], consder the problem mn x s.t. x 2 {a,...,a n }. (4a) (4b) We can use (2) to model the dscrete alternatves constrant (4b) and usual lnear programmng trcks to lnearze the obectve functon. Applyng these technques leads us to the MIP formulaton of (4) gven by mn t s.t. (5a) x apple t, x apple t (5b) x = a y, y =, y 2 R n +, (5c) u = b y, u 2 {0, } m. (5d) Note that, n (5), we have modeled the dscrete alternatves constrant and lnearzed the obectve functon ndependently. Whle t s possble to construct a stronger formulaton for ths problem by consderng both aspects at the same tme, we refran from ths because our ntent here s to evaluate the e ectveness of the formulatons as they would be used n practce where d erent portons of an optmzaton problem are modeled ndependently and then combned. The followng proposton shows that the ncremental encodng formulaton never requres branchng on more than a sngle varable to solve ths problem. In contrast, there are choces of n and a for whch the unary and bnary encodng formulatons may need branchng on up to n/2 and log 2 n varables, respectvely. Proposton 2... For any n and choce of a 2 (Z \ {0}) n, the ncremental encodng verson of (5) can be solved by a branch-and-bound algorthm by branchng on at most one varable. 5

6 2. If n s even, a < 0 for all apple n/2 and a > 0 for all > n/2, then a branchand-bound algorthm solvng the unary encodng verson of (5) requres branchng on at least n/2 varables. 3. If n s a power of two, a < 0 < a n for all apple n, then a branch-and-bound algorthm solvng the bnary encodng verson of (5) requres branchng on at least log 2 n varables. Proof. Pror to any branchng, the LP relaxaton of (5) equals the nterval [a, a n ] n the x-space. When a n < 0 or a > 0, the optmal soluton of the LP relaxaton occurs at one of the endponts of ths nterval and satsfes (4b). Therefore, no branchng s necessary. Otherwse, there exsts 2 [n] such that a < 0 < a +. It s clear that the optmal soluton to (4) s ether x = a or x = a +. Recall that, n the ncremental encodng formulaton, we have b = 0 for all apple and b = for all >. Hence, up-branchng on the sngle varable u fxes y = 0 for all apple whle down-branchng on u fxes y = 0 for all >. When proected on to the x-space, the LP relaxaton of (5) n the frst branch corresponds to the nterval [a +, a n ] and has the optmal soluton x = a +. Smlarly, the LP relaxaton n the second branch proects down to [a, a ] and has the optmal soluton x = a. Ths proves Clam. Now, suppose a branch-and-bound algorthm can solve the unary encodng verson of (5) by branchng on the varables {u } 2K where K [n] and K < n/2. As we have observed before, up-branchng on the varable u fxes y = and y = 0 for all, n the unary encodng formulaton. Hence, the optmal value of (5) at any leaf reached by up-branchng on a varable u must equal a > 0. On the other hand, the LP relaxaton of (5) at the leaf reached by down-branchng on all {u } 2K has the optmal value 0 snce there exst dstnct 0, 0 2 [n] \ K such that 0 apple n/2 and 0 > n/2 and the proecton of the LP relaxaton on to the x-space contans 0. Ths leaf could have been pruned by nether ntegralty nor bound and must stll be actve. Ths contradcton proves Clam 2. The proof of Clam 3 follows an outlne smlar to that of Clam 2. Let l := log 2 n and suppose a branch-and-bound algorthm can solve the bnary encodng verson of (5) by branchng on the varables {u } 2K where K [l]. At termnaton, the branch-and-bound tree has a leaf at whch u = b n s a feasble assgnment. Let 0 2 [l] \ K. Let 0 2 [n] be such that b 0 = b n 0 and b 0 0 = b n for all, 0. It follows that u = b 0 s a feasble assgnment at ths leaf as well. The fact that a 0 < 0 mples that the proecton of the LP relaxaton of (5) on to the x-space at ths leaf contans 0. Agan, we conclude that ths leaf could have been pruned by nether ntegralty nor bound. Ths completes the proof of Clam 3. 6

7 3. Incremental Formulatons An nterestng behavor of the ncremental encodng formulaton s that t naturally nduces the order u u 2... u n. Ths orderng condton s also shared by the by-varable formulatons [6 2], whch explans why they lead to balanced branch-and-bound trees. Moreover, t s explctly mposed n other more complcated ad-hoc formulatons for pecewse-lnear functons [20 23] and probablstc constrants [24, 25]. We now show that all these formulatons can be obtaned through a generc procedure. The frst step n ths procedure s to note that we can use D n to formulate more general dscrete alternatves by usng the followng formulaton ntroduced by Lowe [26] and Jeroslow and Lowe [27]. Proposton 3. ([26, 27]). Let {P } n be a fnte famly of polyhedra such that there exsts n r ko R k=0 Rd and n o v V =0 Rd for 2 [n] such that P := conv n o v V + =0 cone n r ko R for all 2 [n]. Then a MIP formulaton of x 2 S n k=0 P s gven by x = VX VX =0 =0 XR v + k=0 = y, 8 2 [n], µ k r k, (6a) (6b) µ 0, 0, 8 2 [n], y 2 D n. (6c) Formulaton (6) s locally deal [26 28]. We can combne ths standard formulaton wth the generc encodng formulaton (2) of D n to obtan the followng generalzaton of a formulaton for pecewse-lnear functons ntroduced n [6]. Corollary 3.2. Let {P } n be a fnte famly of polyhedra such that there exsts n o R r k R d and n o v V k=0 R d for 2 [n] such that P := conv n o v V =0 + =0 cone n r ko R for all 2 [n]. Then a MIP formulaton of x 2 S n k=0 P s gven by 7

8 x = VX =0 VX =0 XR v + =, u = k=0 µ k r k, (7a) VX =0 b, (7b) µ 0, 0, 8 2 [n], u 2 {0, } m. (7c) The fact that (7) s locally deal follows drectly from the smlar result on (2). For the specfc case of the ncremental encodng, we can use smple algebrac manpulatons to obtan the followng formulaton whch explctly consders the order property among the u varables. Proposton 3.3. Let {P } n be a fnte famly of polyhedra such that there exsts n r ko R k=0 Rd and n o v V R d for 2 [n] such that P := conv n o v V =0 + =0 cone n r ko R for all 2 [n]. Then a MIP formulaton of x 2 S n k=0 P s gven by Xn x = v 0 + u v 0 + v V + VX = v v 0 XR + µ k r k, k=0 (8a) u apple V, 8 2 [n ], (8b) VX + = + apple u, 8 2 [n ], (8c) VX = apple, (8d) µ 0, 0, 8 2 [n], u 2 {0, } n. (8e) Proof. We exhbt a becton whch preserves the values of the x varables between ponts satsfyng (8) and the ncremental encodng verson of (7). The clam then follows from Corollary 3.2. In the rest of the proof, we refer to the ncremental encodng verson of (7) as smply (7). Let (x, u,, µ) be a soluton to (7). Defne u 0 := and u n := 0 and observe that (7b) can be rewrtten as u = P n l=+ 8 P Vl =0 l

9 for all 2 {0,,...,n } when the vectors n b o n are chosen accordng to ncremental encodng. Let := for all 2 [V 2] and V := u + V for all 2 [n]. The valdty of (8b) and (8e) for (x, u,, µ) follows from the defnton of and the non-negatvty of. Smlarly, (8c) and (8d) are satsfed because the nonnegatvty of mples P V + = + = u + + P V + = + apple u + + P V + =0 + = u for all 2 [n ] and P V = = u + P V = apple u + P V =0 =. To verfy that (8a) also holds, let r = P R k=0 µ k r k and observe VX x = v + r = = =0 00 VX VX B@ B@ u u CA v0 + v CA + r = = 0 VX B@ (u u )v 0 + (v v 0 ) CA + r Xn = v 0 + u (v 0 + Xn = v 0 + u (v 0 + v 0 ) + 0 VX 2 B@ = = VX v 0 ) + = (v v 0 ) + r (v v 0 ) + ( V u )(v V v 0 ) CA + r Xn = v 0 + u (v 0 + v V ) + VX = (v v 0 ) + r. To prove the reverse ncluson, let ( x, ũ,, µ) be a soluton to (8). Defne ũ 0 := and ũ n := 0 and let := for all 2 [V 2], V := V ũ and 0 P := ũ V = for all 2 [n]. It s not d cult to see that these defntons mply ũ = P n P Vl l=+ =0 l for all 2 {0,,...,n }. Furthermore, the equaltes n the frst part of the proof contnue to hold. Thus, ( x, ũ,, µ) satsfes (7a)-(7b). To complete the proof, t s enough to show that s non-negatve. However, ths follows drectly from ts defnton and (8b)-(8c). Formulaton (8) s locally deal. To see ths, observe that the lnear mappng 9

10 defned n the proof of Proposton 3.3 s n fact a becton between ponts satsfyng the LP relaxatons of (8) and the ncremental encodng verson of (7). Gven any soluton to the LP relaxaton of (8) n whch the u varables have fractonal values, ths mappng assocates wth t a soluton to the LP relaxaton of (7) wth the same u values. However, ths cannot be an extreme pont soluton to the LP relaxaton of (7) by the smple fact that (7) s locally deal. Hence, t can be expressed as the convex combnaton of other solutons to the LP relaxaton of (7). Mappng these solutons back to (8) and usng the contnuty of lnear mappngs shows that the LP relaxaton of (8) cannot have any extreme pont solutons that volate the ntegralty restrctons. Formulaton (8) generalzes a formulaton ntroduced by Wlson [23] for pecewselnear functons n two ways. Frst, t presents a drect extenson from pecewselnear functons to the unon of a fnte number of arbtrary polyhedra wth dentcal recesson cones. Second, t enoys a broader scope of applcablty by elmnatng the need for a topologcal condton that was requred for the valdty of Wlson s formulaton. We may obtan other ncremental formulatons through the followng rather straghtforward lemma. Lemma 3.4. The ncremental encodng formulaton for y 2 D n s equvalent to u 2 {0, } n, u u +, 8 2 [n 2], (9a) y = u, y n = u n, (9b) y = u u, 8 2 {2,...,n 2}. (9c) Gven any formulaton that requres y 2 D n, we can replace ths constrant wth (9a) and elmnate every occurrence of y through the relatonshps (9b)-(9c). For nstance, followng ths procedure leads us to the formulaton for probablstc constrants studed n [24, 28] from a standard formulaton ntroduced n [29 3]. We refer the readers to [28] for detals on ths specfc transformaton. 4. Creatng and Selectng Encodngs In ths paper, we have concentrated on the unary, bnary and ncremental encodngs. Whle formulatons that use the bnary and ncremental encodngs usually outperform those that use the unary encodng, t can stll be hard to predct whch encodng wll perform best n a specfc class of nstances. For example, [5, 6] present computatonal results whch show that formulatons for pecewselnear functons based on the bnary encodng can sgnfcantly outperform those 0

11 based on the unary and ncremental encodngs. In contrast, n a d erent set of problems, [32] reports that the ncremental formulaton performs better than the bnary encodng formulaton. Formulaton (7) o ers a convenent way to prelmnarly evaluate the performance of these three encodngs wthout havng to hard-code the specfc formulatons derved from them (e.g., (8) for the ncremental n o encodng). Furthermore, (7) s vald for any selecton of dstnct bnary vectors n b and hence can be used to construct alternatve formulatons that could outperform all three prevously consdered encodngs. Developng technques for constructng such alternatve encodngs s beyond the scope of ths paper, but to llustrate ther potental, we present a herarchy of encodngs that combne the propertes of the ncremental and unary encodngs. Such encodngs could be useful n settngs where the ncremental encodng works best n certan portons of the problem whle the bnary encodng works best n others. To smplfy presentaton, we assume n the rest of ths secton that n s a power of 2. The herarchy we propose s ndexed by h 2 0,,...,log 2 n, whch can be consdered as the number of bts of n b o n that behave as n the bnary encodng. Let t l := n/2 l. For a gven value of h, we set m = h + t h and defne the vectors n b o n Rm n the encodng as follows: For all l 2 [h], let b l = f there exsts p 2 [d2 l e] such that 2 {2(p )t l +,...,(2p )t l } and b l = 0 otherwse. For all s 2 [t h ], let b h+s = f there exsts p 2 [d2 h e] such that 2 {2(p )t h + s +,...,(2p )t h + s} and b,h+s = 0 otherwse. Fgure shows ths herarchy of encodngs n matrx form (we present a matrx whose columns are n b o n ) for n = 8 and h 2 {0,, 2, 3}. Note that p ndexes the blocks of contguous s n each row n the defnton above. It can be seen that settng h = 0 produces the ncremental encodng whereas settng h = log 2 n (or even h = log 2 n ) produces the bnary encodng. However, usng d erent values of h yelds hybrd encodngs that share aspects of both the bnary and ncremental encodngs. For nstance, choosng h = n Fgure yelds an encodng where the selecton between the sets of optons {,...,4} and {5,...,8} s done as n the bnary encodng whle the selecton among the optons n {,...,4} or {5,...,8} s done as n the ncremental encodng. References [] D. M. Ryan, B. A. Foster, An nteger programmng approach to schedulng, n: A. Wren (Ed.), Computer Schedulng of Publc Transport Urban Passen-

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