An Exact Algorithm for Two-stage Robust Optimization with Mixed Integer Recourse Problems

Transcription

1 An Exac Algorihm for Two-sage Robus Opimizaion wih Mixed Ineger Recourse Problems Long Zhao and Bo Zeng Deparmen of Indusrial and Managemen Sysems Engineering Universiy of Souh Florida January, 2012 Absrac In his paper, we consider a linear wo-sage robus opimizaion model wih a mixed ineger recourse problem. Currenly, his ype of wo-sage robus opimizaion model does no have any exac soluion algorihm available. We firs presen a se of sufficien condiions under which he exisence of an opimal soluion is guaraneed. Then, we presen a nesed column-and-consrain generaion algorihm ha can derive an exac soluion in finie seps. The algorihm developmen also yields novel soluion mehods o solve general mixed ineger bi-level programs and some fourlevel programs. Finally, he proposed framework is demonsraed hrough an applicaion of a robus rosering problem wih par-ime saff scheduling in he second sage. Key words: wo-sage robus opimizaion, mixed ineger recourse problem, ri-level program, bi-level program 1 Inroducion Robus opimizaion (RO) is a recen opimizaion approach ha deals wih daa uncerainy. Differen from sochasic programming, anoher well-known and popular modeling mehod, RO does no assume probabiliy disribuions of random parameers. Also, raher han looking for an opimal soluion wih respec o he expeced objecive value as in sochasic programming, RO pursues a soluion ha guaranees he bes performance in any of he wors cases (Ben-Tal and Nemirovski (1998, 1999, 2000), El Ghaoui e al. (1998), Bersimas and Sim (2003, 2004)). Since he soluion is expeced o hedge agains any possibiliy, a single-sage RO model ends o be over-conservaive (Aamurk and Zhang 2007). To address his issue, especially in siuaions where some recourse decisions can be made afer he uncerainy is revealed, wo-sage RO, or robus adjusable or adapable opimizaion, has been proposed and sudied (Ben-Tal e al. 2004, Aamurk and Zhang 2007, Bersimas e al. 2011a). Equaion (1) presens he mos popular form of wo-sage RO, where vecor y denoes he firs-sage decision variables, u is a poin in he uncerainy se U, and vecor x denoes he recourse decision variables in he second sage. min max min y Y u U x F(y,u) h(y, u, x). (1) The uncerainy se, which is beyond he conrol of he decision maker, ofen akes he form of a mixed ineger se, e.g., a finie discree se or a polyope, or a nonlinear se, e.g., an ellipsoid. In very recen years, wo-sage RO has gained populariy in solving pracical problems where he randomness is significan bu is probabiliy disribuion is difficul o characerize, or implemening an infeasible soluion is exremely cosly. Those applicaions include nework/ransporaion sysems (Aamurk and Zhang 2007, Ordonez and Zhao 2007, Gabrel e al. 2011), porfolio opimizaion (Takeda e al. 2008), and power sysems conrol and scheduling problems (Zhao and Zeng 2010, 1

2 Jiang e al. 2011, Bersimas e al. 2011b). However, differen from sochasic programming, which is in a monolihic form, he wo-sage RO is acually a ri-level opimizaion model ha is very challenging o compue. Even a simple formulaion wih linear programming (LP) problems in boh sages could be NP-hard (Ben-Tal e al. 2004). To solve his complex problem, Ben-Tal e al. (2004) assume ha recourse decision variables ake values according o some affine funcions of he value of uncerainy. Then, a wo-sage RO can be reduced o a formulaion close o a single-sage RO model and could be solved by fas algorihms. This approximaion sraegy is called he affine rule mehod. Differen from ha approximaion mehod, a maser-subproblem framework, in wo varians, recenly has been developed and implemened o derive he soluion of a few wo-sage RO problems (Thiele e al. 2010, Zhao and Zeng 2010, Bersimas e al. 2011b, Jiang e al. 2011, Gabrel e al. 2011). The firs ype of maser-subproblem procedure is developed in a spiri similar o Benders decomposiion, in which cuing planes supplied o he maser problem are defined wih he dual informaion of he recourse problem and revealed uncerainy informaion (Thiele e al. 2010, Zhao and Zeng 2010, Bersimas e al. 2011b, Jiang e al. 2011, Gabrel e al. 2011). The second algorihm generaes recourse variables and associaed consrains for significan scenarios in he uncerainy se and supplies hem o he maser problem as cuing planes, which is a column-and-consrain generaion procedure (Zhao and Zeng 2010, Zeng and Zhao 2011). I can be proven ha when he recourse problem is LP, boh algorihms converge o an opimal soluion in finie ieraions (Thiele e al. 2010, Zhao and Zeng 2010, Bersimas e al. 2011b, Jiang e al. 2011, Gabrel e al. 2011, Zeng and Zhao 2011). Neverheless, hey perform very differenly in he compuaional aspec. As shown in (Zhao and Zeng 2010, Zeng and Zhao 2011), for robus power sysem scheduling problems and robus locaion-ransporaion problems, he column-and-consrain generaion algorihm performs an order of magniude faser han an implemenaion of Benders ype algorihms. Alhough mos exising research focuses on wo-sage RO wih LP recourse problems in he second sage, we observe ha many real problems require discree recourse decisions. Over he las 20 years, sochasic programming wih ineger or binary recourse variables has received remendous research aenion in boh applicaion and algorihmic aspecs (Ahmed e al. 2004, Carøe and Tind 1998, Lapore and Louveaux 1993, Sen and Sherali 2006, Sen 2003). As anoher opimizaion approach o deal wih uncerainy, i is worh developing efficien algorihms for wo-sage RO wih discree recourse problems and applying hem o solve pracical issues. Neverheless, compared wih wosage RO wih LP recourse problems, solving wo-sage RO models wih mixed ineger programming (MIP) recourse problems is more challenging. To he bes of our knowledge, very limied research has been done o solve his ype of problem. The only approach of which we are aware is he revised affine rule based approximaion algorihms o deal wih discree recourse variables (Bersimas e al. 2011a), which could derive reasonable and compuaionally-effecive soluions in many applicaions. Noe ha discree variables, especially binary variables, ofen are used o capure he complicaed logic requiremens, where i is difficul o idenify some simple decision rules o deermine variable values. If his is he case wih he recourse problem, affine rule based approximaion mehods may no be able o produce good soluions. Also, given ha srong dualiy does no hold in general MIP problems, i is difficul o exend curren Benders-ype algorihms o generae cuing planes (Thiele e al. 2010, Zhao and Zeng 2010, Bersimas e al. 2011b, Jiang e al. 2011, Gabrel e al. 2011, Zeng and Zhao 2011). As menioned in (Zeng and Zhao 2011), he column-and-consrain generaion algorihm, which simply creaes cuing planes wih primal decision variables, may provide an effecive framework o solve hose difficul problems. In fac, when he uncerainy se is a finie discree se, i is clear ha we can exend he column-and-consrain generaion sraegy o deal wih MIP recourse problems. However, a few issues remain unsolved for general cases. Among hem, he criical one is how o quickly idenify he significan scenarios oher han enumeraion. In his paper, we address hose issues by deriving srucural insighs and developing compuaion mehods o exacly solve wo-sage RO wih an MIP recourse problem. We firs provide a sufficien condiion o guaranee he exisence of an opimal soluion. Then, we describe he applicaion of a column-and-consrain generaion algorihm o solve he mixed ineger bi-level programs, which serves o idenify significan scenarios. This algorihm can be embedded ino an ouer-level column-and-consrain generaion scheme o obain a complee algorihm for wo-sage RO wih a MIP recourse problem. We refer o i as nesed column-and-consrain generaion algorihm. Finally, we demonsrae he srengh of our algorihm on a simple wo-sage robus employee scheduling model. Our conribuion includes: (i) he properies o ensure he exisence of opimal soluions o wo-sage RO wih an MIP recourse 2

3 problem; (ii) an algorihm o exacly solve his ype of problems; (iii) a preliminary compuaional sudy showing he effeciveness of he proposed algorihm on a wo-sage robus rosering problem. Our algorihm developmen also yields algorihms o solve general mixed ineger bi-level program and some four-level programs. The remainder of his paper is organized as follows. Two-sage robus opimizaion wih an MIP recourse is analyzed in Secion 2. The soluion mehod is described in Secion 3. An applicaion of solving he wo-sage rosering problem is presened in Secion 4. Finally, Secion 5 concludes he paper. 2 Two-sage Robus Opimizaion wih MIP Recourse In his secion, he general form of linear wo-sage RO formulaion wih an MIP recourse is analyzed, and some condiions under which an exac soluion exiss are idenified. Considered firs is he general wo-sage robus model wih an MIP recourse (wo-sage RO(MIP)) as follows: inf cy + sup y Y u U inf dx + gz, (2) z,x F(y,u) where Y = {y R m + Z m + : Ay b}, F(y, u) = {(z, x) Z n + R p + : Ex + Gz f Ru Dy, Tz v}, and he uncerainy se U is a bounded mixed ineger se in he form of U = {u Z q + Rq + : Hu a}, wih all coefficiens/parameers raional in hose expressions. Wihou loss of generaliy, one can assume ha P roj zf(y, u) Φ = {z Z n + : Tz v} for any y Y and u U. Noe in (2) ha he firs-sage decision y akes ino accoun all possible fuure uncerain siuaions, which are capured by U wihou probabiliy informaion, and he recourse decision variables z and x represen he recourse acions ha can be implemened afer he firs-sage decision is made and he uncerainy is revealed. Clearly, such a framework allows us o model a sequenial decision-making environmen where (discree) recourse acions can be made in a wai-and-see fashion. Because he recourse problem is a linear MIP problem, i can be reaed simply as a regular minimizaion problem, i.e., he infimum can be replaced by is minimum. We are now ineresed in condiions under which (2) is equivalen o min y Y cy + max u U min z,x F(y,u) dx + gz. (3) I is obvious ha he reducion from (2) o (3) is valid if U is a finie discree se. In fac, if ha is he case, he ri-level formulaion in (2) can be furher reformulaed ino a monolihic mixed ineger opimizaion problem. Assume ha U = {u i } I i=1. The following formulaion is equivalen o (2) and (3): MIP Equivalen : min cy + η (4) s. Ay b (5) η dx i + gz i, i = 1,..., I (6) Ex i + Gz i f Ru i Dy, Tz i v, i = 1,..., I (7) y R m + Z m +, z i Z n +, x i R p +, i = 1,..., I. (8) Neverheless, such a reducion may no be feasible for a general muli-level formulaion. For example, consider he following bi-level problem: sup 0 d 1 min {x 1 + x 2 : x 1 d, x 2 1 d, x 1 Z x 1,x +, x 2 0}. 2 If d = 0, solving he above bi-level formulaion will yield x 1 = 0, x 2 = 1, and x 1 + x 2 = 1. When d = δ (0, 1], we have x 1 = 1, x 2 = 1 δ and x 1 + x 2 = 2 δ. Clearly, no opimal d can be derived o achieve he supremum, which is 2. Therefore, his bi-level formulaion canno be reduced o a max min problem. Alhough he aforemenioned reducion is no valid in general for muli-level programs, i can be proven ha (3) is equivalen o (2) under some mild condiions. Nex, we presen such a sufficien condiion adaped from (Hoang 1998, Takeda e al. 2008). 3

4 Proposiion 1. Assume ha he uncerainy se U is a polyope, i.e, q = 0. If θ(u) := min z,x F(ŷ,u) dx + gz is a quasiconvex funcion over U for any given ŷ, hen (2) reduces o (3). Furhermore, boh of hem are equivalen o min y Y cy + max min u U z,x F(y,u) where U = {u j } J j=1 is he se of all exreme poins of U. dx + gz, (9) I is easy o generalize his resul o he case where U is a bounded mixed ineger se. Under his siuaion, noe ha U is he union of a collecion of polyopes, i.e. U = U 1... U K, and each of hem corresponds o a paricular assignmen o he se of ineger variables. Corollary 1. If θ(u) is quasiconvex over U k, k = 1,..., K for any given ŷ, hen (2) reduces o (3). Furhermore, boh of hem are equivalen o min y Y cy + max min u Û z,x F(y,u) where Û = {U k } K k=1 and U k is he se of all exreme poins of U k. dx + gz, (10) We menion ha he quasiconvex requiremen is no resricive because a broad range of funcions is quasiconvex. The nex corollary liss a few ypical funcions ha are quasiconvex. Corollary 2. θ(u) is quasiconvex if i is (i) a convex funcion; or (ii) a non-increasing funcion; or (iii) a non-decreasing funcion. To he bes of our knowledge, in all exising applicaions of wo-sage RO, including (Thiele e al. 2010, Zhao and Zeng 2010, Bersimas e al. 2011b, Jiang e al. 2011, Gabrel e al. 2011), θ(u) is eiher non-decreasing or non-increasing in u because u represens random facor(s) ha lead o eiher a posiive or negaive impac o he decision maker. So, unless explicily menioned, we always assume ha θ(u) is a quasiconvex funcion in he remainder of his paper. Noe ha when he recourse problem is an LP problem, i is no necessary o carry his assumpion, as he equivalence among (2), (3), and (10) is always valid (Zeng and Zhao 2011). I is worh poining ou ha he equivalence among (2), (3), and (10) is non-rivial, as (10) is a compuaionally convenien formulaion. Similar o he case where he uncerainy se is a finie discree se, we can develop he corresponding MIP-Equivalen reformulaion in he form of (4-8) o solve (10). However, enumeraing all possible scenarios (or he exreme poins in Û) and lising all corresponding variables and consrains is no realisic. In fac, we would like o highligh ha he consrains in (6) indicae ha no all scenarios (and heir variables and consrains) in Û are necessary in defining he opimal soluion and he wors-case objecive value. Tha is, probably only a small subse of Û and he associaed variables and consrains play a significan role. Such an observaion moivaes he developmen of he column-and-consrain generaion algorihm (Zeng and Zhao 2011, Zhao and Zeng 2010), a soluion procedure ha creaes exra recourse variables and relaed consrains for significan uncerainy scenarios ha are generaed on he fly. This procedure has been successfully applied o solve a few wo-sage RO problems wih LP recourse problems (Zeng and Zhao 2011, Zhao and Zeng 2010) and has exhibied a superior compuaional performance over Benders-ype algorihms. In he nex secion, we firs briefly review he column-and-consrain generaion algorihm in is basic form and hen presen is exension o solve wo-sage RO wih an MIP recourse problem. 3 Solving wo-sage Robus Problem wih an MIP Recourse Problem As menioned, he column-and-consrain generaion procedure creaes recourse decision variables (and relaed consrains) for dynamically-idenified significan uncerainy scenarios. Because hose variables and he corresponding consrains are primal o he decision maker, no dual informaion is necessary in generaing hose variables and consrains, which is differen from Benders cuing plane soluion procedures. Hence, we believe ha i could evolve ino a more capable algorihm o solve wo-sage RO wih an MIP in he second sage, for which no exac algorihm has been repored. We firs briefly review he column-and-consrain generaion algorihm in is basic form, along wih some resuls on is complexiy. Then, we sudy an exension of he basic form, he nesed 4

5 column-and-consrain generaion algorihm, o solve wo-sage RO wih an MIP recourse problem. Our soluion procedure also provides a novel approach o solve he challenging bi-level max min problem wih discree variables in he second level. Remark 1. An opimal soluion o he problem in (3) is jus an opimal y ha performs bes, wih he help of recourse acions, in any wors siuaions. The acual recourse decisions will be made afer u is revealed. To engage in a meaningful discussion, we assume ha here exiss an opimal y Y o problem (3) wih a bounded objecive value. 3.1 A Review of he Column-and-Consrain Generaion Algorihm The column-and-consrain generaion procedure is implemened in a wo-level maser-subproblem framework. We firs assume ha an oracle is available o solve he following bi-level max min problem, i.e., he subproblem in his procedure: SP : Q(ŷ) = max u U min dx + gz z,x s.. Ex + Gz f Ru Dŷ (11) Tz v z Z n +, x R p +. This oracle can eiher derive an opimal scenario u U wih a finie opimal value or idenify some u for which he recourse problem is infeasible, i.e., Q(ŷ) = + by convenion. Noe ha any feasible soluion o (3) provides an upper bound. Also, solving a relaxaion of is monolihic equivalen form, i.e., he formulaion in he form of MIP Equivalen wih respec o a subse of Û, yields a lower bound. By making use of upper and lower bounds, we can develop an exac algorihm o solve (3). Le LB and UB be he lower and upper bounds, respecively, and ɛ be he opimaliy olerance. We have Column-and-Consrain Generaion Algorihm 1. Se LB =, UB = + and k = Solve he following maser problem. MP : min y,η,x s.. cy + η Ay b η dx l + gz l, 1 l k (12) Dy + Ex l + Gz l f Ru l, 1 l k Tz l v, 1 l k y Y, η R, z l Z n +, x l R p +, 1 l k Derive an opimal soluion (y k+1, η k+1, z 1,..., z k, x 1,..., x k ) and updae LB = cy k+1 + η k+1. If UB LB ɛ, reurn y k+1 and erminae. 3. Call he oracle o solve subproblem Q(y k+1) in (11) and updae If UB LB ɛ, reurn y k+1 and erminae. UB = min{ub, cy k+1 + Q(y k+1)}. 4. Creae variables (z k+1, x k+1 ) and add he following consrains: η dx k+1 + gz k+1 Dy + Ex k+1 + Gz k+1 f Ru k+1, Tz k+1 v z k+1 Z n +, x k+1 R p + o MP where u k+1 is he scenario solving Q(y k+1). Updae k = k + 1 and go o Sep 2. 5

6 Nex, we presen a resul ha is adaped from (Zeng and Zhao 2011) on he finie convergence of his algorihm. This resul holds in wo-sage models wih boh LP recourse and MIP recourse problems, as i is up o only he descripion of he uncerainy se. Proposiion 2. (Zeng and Zhao 2011) Given ha an oracle can find a wors-case u for any given y, he column-and-consrain generaion algorihm converges an opimal soluion in a finie number of ieraions. The number of ieraions is bounded by Û. The above column-and-consrain generaion algorihm dynamically convers a wo-sage RO problem ino a monolihic MIP formulaion, MP in (12), which acually is a parial enumeraion of he MIP Equivalen form. In fac, MP has a srucure ha is very close o he deerminisic equivalen form of wo-sage sochasic programming. Hence, i probably builds a connecion o sochasic programming and, herefore, exising compuaional echniques for wo-sage sochasic programming could be useful o solve MP efficienly. Then comes he criical sep for successfully implemening he column-and-consrain generaion procedure: solving SP and idenifying a significan scenario u. As SP is a bi-level program, general exising algorihms o bi-level programs can be adoped o solve i. Neverheless, he majoriy of exising sudy on bi-level programs focuses on formulaions wih an LP as he lower-level problem. A very limied sudy is available on solving hose wih an MIP in he lower level (Shimizu e al. 1997). Such a siuaion may be explained by he fac ha he srong dualiy does no hold, and less srucural informaion can be used in algorihm developmen. As a resul, his ype of problem remains a challenging one in boh heoreical and compuaional aspecs. However, o solve a wosage RO wih an MIP recourse problem, i is necessary o incorporae an efficien algorihm o solve SP wihin he column-and-consrain generaion procedure. Towards his end, we propose o solve he bi-level SP wih a MIP recourse problem hrough is ri-level srucure, which allows us o make use of he idea behind he column-and-consrain generaion mehod. 3.2 Solving Bi-level SP by Is Tri-level Form Differen from he mainsream idea ha direcly reduces a bi-level program ino a monolihic opimizaion problem, our sraegy is o firs expand i ino a ri-level problem in a srucure similar o wo-sage RO. Then, by using he column-and-consrain generaion approach, we can dynamically conver he ri-level problem ino an (equivalen) monolihic form. The feasible se of discree recourse variables can be considered as he uncerainy se, while he opimal soluion we are looking for is in erms of U. To focus on he main resuls and algorihm developmens in his paper, we make a few very mild assumpions. Firs, we assume p 1, i.e., he MIP recourse problem has a leas one coninuous recourse variable. Second, we assume ha he LP problem, obained by fixing y, u and z o heir any possible values, is always feasible and bounded, which is referred o as he exended relaively complee recourse propery. Finally, we assume ha he feasible se of discree recourse variables, i.e., Φ, is bounded, which is no resricive for pracical problems. Observe ha he formulaion of SP defined in (11) is equivalen o he following ri-level formulaion obained by separaing discree variables from coninuous variables: Q(ŷ) = max u U min gz + z Φ min dx x s. Ex f Ru Dŷ Gz (13) x R p + Denoe Φ = {z r } R r=1. By making use of he counabiliy of Φ, we have Bi/Tri Equivalen I : Q(ŷ) = max θ θ gz r + min{dx r : Ex r f Ru Dŷ Gz r, x R p +} (14) r = 1,..., R u U. In he above formulaion, x r are he corresponding decision variables for z r, a paricular assignmen of z. Clearly, he whole Bi/Tri Equivalen I shares a grea similariy o MIP Equivalen, which moivaes us o invesigae he idea of column-and-consrain mehod. 6

7 We firs presen he derivaion of is monolihic formulaion. Noe ha he minimizaion problem in each consrain is an LP model. If he exended relaively complee recourse assumpion holds, we can apply he classical Karush-Kuhn-Tucker (KKT) condiion o conver his minimizaion problem ino a feasibiliy problem. Specifically, le π r be he dual variables of he minimizaion problem in he r h consrain of Bi/Tri Equivalen I wih a compaible dimension p. Then, deriving an opimal x r of ha minimizaion problem is equivalen o obaining a feasible soluion (x r, π r ) ha saisfies he following consrains: Ex r f Ru Dŷ Gz r (15) E π r d (16) x r (d E π r ) = 0 (17) π r (Ex r f + Ru + Dŷ + Gz r ) = 0 (18) x r R p +, π r R p +. (19) The las wo consrains are complemenary slackness condiions, which ensure ha wo feasible soluions are opimal o primal and dual problems, respecively. In fac, hose complemenary consrains can be linearized by inroducing binary variables and making use of big-m, i.e. a sufficienly large number. For example, a consrain in (17) can be reformulaed as x r j Mδ r j, (d E π r ) j M(1 δ r j ), δ r j {0, 1}, j = 1,..., p. Hence, he above feasibiliy problem can be convered ino a binary MIP formulaion. Consequenly, using consrains (15-19) o idenify feasible (opimal) soluions, we now can reformulae Bi/Tri Equivalen I as a monolihic model. To simplify our exposiion, consrains of nonlinear complemenary slackness condiions are kep in he model while he big-m mehod can be called o linearize hem. Proposiion 3. The bi-level program, SP defined in (11), is equivalen o he following formulaion: Q(ŷ) = max θ θ gz r + dx r, r = 1,..., R Ex r f Ru Dŷ Gz r, r = 1,..., R E π r d, r = 1,..., R (20) x r (d E π r ) = 0, r = 1,..., R π r (Ex r f + Ru + Dŷ + Gz r ) = 0, r = 1,..., R u U, x r R p +, π r R p +, r = 1,..., R. Using an argumen similar o ha for he column-and-consrain generaion algorihm, we anicipae ha i is no necessary o enumerae variables and consrains for all he possible z r s and o consruc he complee equivalen model in (20) o solve he bi/ri-level SP. Probably only a small number of z r s and heir associaed (x r, π r ) and consrains are sufficien. So, we propose o re-apply he column-and-consrain generaion sraegy o solve SP. To disinguish from he previously-described column-and-consrain generaion procedure, we denoe he one described following as he inner-level column-and-consrain generaion procedure and denoe he previous one as he ouer-level column-and-consrain generaion procedure. The Inner-Level Column-and-Consrain Generaion Algorihm 1. Se LB =, UB = + and k = 0. 7

8 2. Solve he following maser problem (in is linearized form) of SP MP S : Q(ŷ) = max θ θ gz r + dx r, 1 r k Ex r f Ru Dŷ Gz r, 1 r k E π r d, 1 r k (21) x r (d E π r ) = 0, 1 r k π r (Ex r f + Ru + Dŷ + Gz r ) = 0, 1 r k u U, x r R p +, π r R p +, 1 r k. Obain an opimal u, denoed i by u k+1, and updae UB = Q(ŷ). If UB LB ɛ, reurn u k+1 and erminae. 3. Solve he subproblem of SP SP S : min dx + gz. z,x F(ŷ,u ) k+1 Obain an opimal soluion (z k+1, x k+1 ) and updae LB = max{lb, gz k+1 + dx k+1 }. If UB LB ɛ, reurn u k+1 and erminae. 4. Creae variables (x k+1, π k+1 ) and add he following consrains Ex k+1 f Ru Dŷ Gz k+1 E π k+1 d x k+1 (d E π k+1 ) = 0 π k+1 (Ex k+1 f + Ru + Dŷ + Gz k+1 ) = 0 o MP S. Updae k = k + 1 and go o Sep 2. x k+1 R p +, π k+1 R p + Nex, we show ha he inner-level column-and-consrain procedure converges finiely. Proposiion 4. For any given ŷ Y, he inner-level column-and-consrain generaion procedure is finiely convergen, and he number of ieraions is bounded by R. Proof. Claim: Any repeaed z in his procedure implies he opimaliy, i.e., LB = UB. Then, he proposiion follows immediaely due o he fac ha Φ is a finie se. Suppose a ieraion k a wors-case uncerainy u is obained by solving MP S, and i leads o opimal z and x o SP S. I follows ha UB LB gz + dx. If z has been idenified in a previous ieraion k wih 1 k k 1, hen u will be an opimal soluion o MP S a ieraion k + 1 as MP S does no change from ieraion k o k + 1. Given ha x is opimal when u = u and z = z, i follows from he firs consrain in (20) (also he firs consrain in (14)) ha UB = Q(ŷ) gz + dx in ieraion k + 1. Therefore, we have LB = UB. We menion ha he srong dualiy propery of a linear program can also be used o build an alernaive ri-level formulaion of SP, which allows us o develop anoher monolihic equivalen form and a column-and-consrain generaion varian. Specifically, by dualizing he innermos minimizaion problem in (13), SP is equivalen o he following ri-level problem: max u U min gz + max (f Ru Dy z Φ π Π Gz) π, (22) where Π = {π R p + : E π d }. By enumeraing all possible assignmens of z, we can obain is monolihic equivalen form as follows: Bi/Tri Equivalen II : Q(ŷ) = max θ θ gz r + (f Ru Dŷ Gz r ) π r, r = 1,..., R (23) E π r d, u U, π r 0, r = 1,..., R. r = 1,..., R 8

9 Clearly, a column-and-consrain generaion procedure, which generaes only dual variables π and consrains in (23), can be used o solve he ri-level formulaion in (22). Neverheless, as he firs consrains in (23) are quadraic consrains, an efficien algorihm for a quadraicallyconsrained quadraic program (QCQP) is necessary. However, when U can be represened as a binary se, such as cardinaliy consrained uncerainy se (Bersimas and Sim 2003), consrains in (23) can be linearized by big-m mehod. Then, an off-he-shelf MIP solver is sufficien o suppor his column-and-consrain generaion procedure. As can easily be developed in he same spiri as he aforemenioned, deails of his varian are omied in his paper. Remark 2. (i) If he exended relaively complee recourse assumpion does no hold, he KKT condiion based reformulaion in (20) is no valid, and he corresponding column-and-consrain generaion algorihm is no applicable. However, he srong dualiy based reformulaion in (23) is valid. Noe ha Π is independen of y, u, and z. Given ha wo-sage RO (MIP) has a feasible soluion (in y), i follows ha Π is no empy (oherwise, i conradics he exisence of a feasible soluion). Therefore, he corresponding column-and-consrain generaion varian is applicable. (ii) If for any fixed u and z, he remaining linear programming problem has a unique soluion, boh KKT condiion and srong dualiy based column-and-consrain generaion procedures perform he same number of ieraions. (iii) To he bes of our knowledge, he developed column-and-consrain generaion procedure(s) for SP is a novel algorihm o solve general mixed ineger bi-level programs. I has been successfully applied o solve a power grid inerdicion problem wih line swiching decisions in he lower level, for which i significanly ouperforms exising (heurisic) procedures (Zhao and Zeng 2011). 3.3 The Nesed Column-and-consrain Generaion Algorihm Wih he aforemenioned ouer and inner level column-and-consrain generaion algorihms, we hen inegrae hem ino a unified soluion procedure o solve wo-sage RO wih an MIP recourse problem. As he column-and-consrain generaion mehod is implemened in boh he ouer and inner levels, i.e., in a nesed fashion, we call he whole procedure he nesed column-and-consrain generaion algorihm o solve wo-sage RO wih an MIP recourse problem. Corollary 3. The nesed column-and-consrain generaion algorihm converges o an opimal soluion of wo-sage RO (MIP) wih a finie number of column-and-consrain generaion ieraions. Remark 3. The nesed column-and-consrain generaion algorihm acually is also a soluion procedure for he four-level min max min max program if he las maximizaion problem is a linear program. Using KKT condiion, he las min max pair can be convered ino a mixed ineger minimizaion program. Hence, we obain an opimizaion problem in he min max min form, which renders iself suiable for our nesed column-and-consrain generaion algorihm where he inner level one can be direcly applied o solve he max min max par. We believe ha implemening he column-and-consrain generaion mehod in a nesed fashion could be a soluion sraegy o solve a broad range of muli-level programs. 4 A Numerical Sudy: Two-Sage Robus Rosering Problem In his secion, he soluion capabiliy of he nesed column-and-consrain generaion algorihm is demonsraed. Raher han focusing on developing a mulifaceed and deailed model wih many pracical facors of real sysems and providing comprehensive analysis, our inenion is o describe he implemenaion procedure of his algorihm on a simple model and presen is compuaional behavior. Wih his moivaion, he sudy was performed on a simplified personnel scheduling (rosering) problem wih uncerain demands. We firs buil a basic deerminisic rosering model and presen is wo-sage robus opimizaion counerpar. Then, we describe specific forms of generaed variables and consrains, using KKT condiion and srong dualiy, respecively. Finally, we presen he compuaional resuls on wo-sage robus rosering model wih wo differen uncerainy ses, i.e., a cardinaliy se and a polyope. 9

10 4.1 Two-sage Robus Rosering Problem In he rosering problem, an organizaion such as a call cener or a clinic needs o allocae is saff members o shifs o mee service demands wih a minimized operaion cos and saisfy governmenal or indusrial regulaions/resricions (Erns e al. 2004). Someimes, o deal wih demand surges, overime from is regular saff or par-ime saff (or agency saff) will be called. In his paper, we consider he rosering problem in he laer siuaion where par-ime saff will be used o deal wih service demand flucuaion. Firs, we formulaed he rosering model in a deerminisic environmen such ha he service demand d in shif (= 0,..., T 1) is known. Le i (= 1,..., I) be he index of regular saff who work N hours, i.e. he lengh of he whole shif, and j (= 1,..., J) be he index of par-ime saff whose working hours are o be deermined. The cos parameers are: in shif, c i is he wage cos of regular saff i, f j is he fixed cos and h j is hourly rae of par-ime saff j, and M is he penaly cos for he unserved demand which will be los. Rosering MIP : min c ix i + (f jy j + h jz j) + M w (24) i j s.. x i + x i,+1 + x i,+2 2, i, T 3, (25) l i x i u i, i (26) y j + y j,+1 1, j, T 2 (27) a j y j b j, j (28) z j Ny j, j, (29) N i x i + j z j + w d, (30) x i {0, 1}, i, ; y j {0, 1}, j, ; z j R +, j, ; w R +,. (31) The rosering problem is o minimize he oal cos, including wage cos for regular saff, fixed and variable cos for par-ime saff, and penaly cos from he unserved demand (24), subjec o some consrains such as minimum/maximum workload and maximum consecuive working shifs (Cheang e al. 2003). Consrains in (25) ensure ha regular saff canno work hrough any hree consecuive shifs. A similar requiremen on par-ime saff is represened in (27) where hey canno work hrough any wo consecuive shifs. Consrains in (26) and (28) are lower and upper bounds on he oal number of shifs for regular and par-ime saff in he planning horizon. Consrains in (29) link binary acivaion and coninuous working-hour decisions for par-ime saff. Consrains in (30) guaranee ha he coverage of demands and (31) are individual variable resricions. Once a large service demand is anicipaed or observed, par-ime saff are ofen called and scheduled o deal wih demand flucuaion, which yields he recourse problem. Hence, he rosering problem wih uncerain demand can be formulaed by he following wo-sage robus opimizaion model. Because he opimal value of he recourse problem is non-decreasing in d, we direcly presen i in a min max min form. I is Rosering RO : min x X i x ic i + max d D min (y,z,w) F(x,d) { M w + j } (f jy j + h jz j), where D is he demand uncerainy se and F(x, d) = {(y, z, w) : (27 30), y j {0, 1}, z j, w 0, j, }. Noe from (32) ha scheduling of regular saff is made wih consideraion of he wors demand siuaions as well as he benefis from using par-ime saff, a righ-and-now decision. Then, afer he demand is revealed, he recourse rosering decision of par-ime saff will be made, a waiand-see decision. Given ha he recourse decision problem is a binary mixed ineger program, he nesed column-and-consrain generaion mehod will be used o solve Rosering RO. (32) 10

11 4.2 Solving Rosering RO by Nesed Column-and-Consrain Generaion Algorihm In his secion, he forms of generaed variables and consrains in ypical ieraions of ouer and inner level algorihms are provided. The soluion procedure wih complee deails can be obained by modifying he ouer and inner level algorihms in Secions 3.1 and 3.2 accordingly Generaed Variables and Consrains in he Ouer Level: Assume ha ˆd k is a significan uncerainy scenario obained in he k h ieraion. Then generaed variables and consrains in he ouer-level algorihm will be as follows. η (f jyj k + h jzj) k + M w k j y k j + y k j,+1 1, j, T 2 a j y k j b j, j z k j Ny k j, j, N i x i + j z k j + w k ˆd k, y k j {0, 1}, z k j, w k 0, j, Solving Mixed Ineger Bi-level SP by Inner-Level Algorihm To derive an opimal soluion o bi-level SP for a given ˆx, firs, he binary decision variables are separaed from remaining recourse variables, resuling in he following ri-level reformulaion: max min f jy j + min h jz j + M w, (33) d D y Y (z,w) Q(ˆx,d,y) j where Y = {y : (27 28); y j {0, 1}, j, } and Q(ˆx, d, y) = {(z, w) : (29 30); z j, w 0, j, }. Nex, variables and consrains (and heir linearized counerpars) generaed by KKT condiion and srong dualiy based inner-level column-and-consrain generaion algorihms are presened, respecively. (a) KKT condiion based inner-level algorihm Assume ha ŷ l is an opimal value by solving he recourse problem for a given ˆx, ˆd in l h inner-level ieraion. Le (λ, π) be dual variables of he innermos minimizaion problem of (33). Then, he generaed variables and consrains in he inner-level algorihm o solve (33) are as follows: θ (f jŷj l + h jzj) l + M w l (34) j j z l j Nŷ l j, j, (35) N i ˆx i + j z l j + w l d, (36) h j λ l j + π l, j, (37) M π l, (38) λ l j(nŷ l j z l j) = 0, j, (39) π l (N i ˆx i + j z l j + w l d ) = 0, (40) z l j(h j + λ l j π l ) = 0, j, (41) w l (M π l ) = 0, (42) z l j 0 j, ; w l 0 ; λ l j 0 j, ; π l 0. (43) 11

12 Given ha D is bounded mixed ineger se, le d be an upper bound of all possible d for = 0,..., T 1. Then, for any fixed ˆx and ŷ, we have he following bounds on primal and dual variables and consrains, which are useful o linearize nonlinear complemenary consrains in (39-42). λ j M ; Nŷ j z j N, j, π M ; N i ˆx i + j z j + w d I N, z j N; h j + λ j π h j + M, j, w d ; M π M,. Consequenly, he generaed complemenary consrains in (39-42) can be replaced by he following linear consrains wih addiional binary variables α, β and δ: λ l j M (1 α l j), j, Nŷ l j z l j Nα l j, j, π l M (1 β l ), N i ˆx i + j z l j N(1 γ l j), j, z l j + w l d I Nβ l, h j + λ l j π l (h j + M )γ l j, j, w l d (1 δ l ), M π l M δ l, α l j, γ l j {0, 1} j, ; β l, δ l {0, 1}. (b) Srong dualiy based inner-level algorihm By srong dualiy, (33) can be convered ino he following max min max problem: max min f jy j + max π (d N ˆx i) N λ jy j, d D y Y (λ,π) C j i j where Y = {y : (27 28); y j {0, 1}, j, } and C = {(λ, π) : λ j+π h j, j, ; π 0, j, }. Then, for a fixed ŷ l, he generaed variables and consrains are as follows: θ f jŷj l + π(d l N ˆx i) N λ l jŷj, l j i j λ l j + π l h j, j, ; π l M ; λ l j, π l 0, j,. M, ; λ j, π When D is a binary se, he firs nonlinear consrain can easily be linearized using he previouslymenioned upper bound informaion. Noe ha in boh nonlinear and linearized forms, fewer variables and consrains are generaed by a srong dualiy based inner algorihm compared o hose generaed by KKT condiion based inner algorihm. 4.3 Experimen Seup We consider a rosering insance of 12 full-ime and 3 par-ime employees, i.e., I = 12 and J = 3, wih 8 hours per shif, 3 shifs per day, and 7 days as he planning horizon, i.e., N = 8 and T = 21. The parameers c i are randomly generaed wihin [5,15]. The fixed cos f j and hourly rae h j are randomly generaed wihin [20,30] and [4,8], respecively. The penaly cos for unserved demand is randomly drawn from [40,50]. The lower and upper bounds on he number of working shifs for regular saff, i.e., l i and u i, are randomly seleced from [4,8] and [8,14], respecively. Similarly, bound parameers for for par-ime saff, i.e., a j and b j, are randomly generaed wihin [2,4] and [4,6], respecively. 12

13 Table 1: Compuaional resuls for KKT-condiions based algorihm and D 1 Γ Time (s) Ouer Ieraion Inner Ieraion ,7,9, ,24,25,28, ,14,25,45,45,48,18,51, ,8,28,34, ,14,9,11,11,19,15, ,6,9,11,7,7,10,14,6,0 We consider wo ypes of uncerainy ses. The firs one is a cardinaliy-consrained binary se. Assume ha d independenly akes value from an inerval [d, d + ξ ] for = 0,..., T 1. Because all of hem rarely will ake upper bounds in he planning horizon, an ineger Γ [0, T 1] can be inroduced o conrol he number of upper bound values. Formally, we can describe he uncerainy se as D 1 = {d : d = d + ξ g, ; g Γ; g {0, 1}, }. Alhough he cardinaliy consrained binary se is simple, some srucured polyope uncerainy ses can be reduced o i (Gabrel e al. 2011, Bersimas and Sim 2003, Zhao and Zeng 2010). The second uncerainy se is a demonsraive general polyope wih wo overlapping budge consrains as follows. The general polyope se is paricularly useful in describing correlaed uncerainy facors. D 2 = { T 1 +1 d : d = d + ξ g, ; g ρ 1; =0 T 1 =T 1 1 } g ρ 2; g [0, 1],. In our numerical sudy, d is randomly seleced from [30,80] (hour person) for = 0,..., T 1. In boh discree and polyope uncerainy ses, ξ is se o be 0.05d. Two iniializaion sraegies were implemened in our nesed algorihm: (i) a feasible poin from he uncerainy se is supplied o he ouer-level algorihm o generae he firs-sage decision in he firs ieraion. Specifically, a random poin wih g = Γ is seleced in D1, and a poin solving max{ d : d D2 } is seleced in D 2 ; (ii) insead of saring from scrach, he procedure of he inner-level column-and-consrain generaion algorihm for SP sars wih he wors-case uncerain poin ha solves he previous SP. In our compuaional experimens, nesed column-and-consrain generaion varians were implemened in C++ wih CPLEX 12.2 on a PC deskop wih an Inel Core(TM) 2Duo 3.00GHz CPU and 3.25GB memory. The opimaliy olerance is se o be 1e 3 for boh ouer and inner algorihms. The whole procedure will be erminaed afer 600 seconds. 4.4 Compuaional Resuls The compuaional resuls for cardinaliy consrained binary uncerainy ses using KKT condiion and srong dualiy based nesed algorihms are given in Table 1 and 2, respecively. Typical convergence behavior of ouer- and inner-level algorihms is shown in Figure 1 and 2, respecively. Noe from Sep 3 of he ouer-level algorihm in Secion 3.1 ha he whole nesed algorihm could erminae in one ieraion wihou performing he inner-level procedure. If his is he case, as shown in Table 1 and 2, he las ouer-level ieraion carries zero inner ieraion. From his compuaional sudy, we observe ha hese wo mehods have comparable performance. Probably because he primal informaion is kep and he feasibiliy problem is easier, he KKT condiion based nesed algorihm compues soluions a lile bi faser. For some insances, we observe ha hese wo algorihms have a differen number of ieraions, in boh ouer- and lower-level procedures. Such differen numbers of ieraions mos likely sem from he seup of he opimaliy olerance, which is 1e 3 for all problems. Indeed, he ieraion numbers are more consisen if he olerance is reduced o 1e 4 wih a larger compuaional ime limi, which confirms our remarks on he number of ieraions using hose wo differen inner-level procedures. Table 3 shows he compuaional resuls for he polyhedral uncerainy ses using he KKT condiion based algorihm, which also suggess ha wo-sage RO wih a polyhedral uncerainy se 13

14 Table 2: Compuaional resuls for srong dualiy based algorihm and D 1 Γ Time (s) Ouer Ieraion Inner Ieraion ,11,4,12, ,25,27, ,13,27,45,45,51,18,50, ,8,28,35, ,14,9,11,12,18,15, ,6,8,10,7,8,9,8,8,10, Upper Bound Lower Bound Upper and Lower Bound Values Ieraions Figure 1: Performance of he ouer-level algorihm (wih KKT-based inner level) for D 1 and Γ = 12. is compuaionally more challenging han ha wih a discree se. For example, he final gap can be reduced only o be 0.14% afer 600 seconds running for he case of (ρ 1, ρ 2) = (0.3, 0.2). 5 Conclusion In his paper, we presened an exac algorihm, he nesed column-and-consrain generaion mehod, o solve he wo-sage robus opimizaion problem wih an MIP recourse problem. We also derived is convergence propery and demonsraed is compuaional behavior on a simple wo-sage robus rosering problem. To he bes of our knowledge, because here is no exac algorihm available o solve his ype of wo-sage robus opimizaion problem, his algorihm is he firs soluion procedure o derive opimal soluions. As a resul, he nesed column-and-consrain generaion mehod and he basic one presened in (Zeng and Zhao 2011) consiue an unified ool se o solve wo-sage robus opimizaion models. In addiion o solving robus models, he nesed column-and-consrain generaion mehod can be used o compue some four-level programs. Also, is subrouine, he inner-level column-and-consrain generaion mehod, yields an effecive approach o solve general mixed ineger bi-level programs. We would like o poin ou ha he (nesed) column-and-consrain generaion mehod in is Table 3: Compuaional resuls for KKT-condiions based algorihm and D 2 (ρ 1, ρ 2 ) Time (s) Ouer Ieraion Inner Ieraion (0.2,0.2) ,13,16,15,14,10,17,14,9,14,17,0 (0.3,0.3) ,8,20,14,19,13,11,19,0 (0.2,0.3) ,15,8,14,10,15,15,11,13,18,13,0 (0.3,0.2)* 600* 12 6,12,11,12,18,15,11,6,14,17,14,12 Case (0.3,0.2) is erminaed afer 600 seconds wih a final gap of 0.14% 14

15 Upper Bound Lower Bound Upper and Lower Bound Values Ieraions Figure 2: Performance of KKT-based inner algorihm wihin he firs ouer level ieraion for D 1 wih Γ = 12. curren descripion provides only a basic scheme o solve general wo-sage robus opimizaion problems. Clearly, more advanced and sophisicaed echniques are needed o refine his mehod o deal wih large-scale real problems in a reasonable ime. We observe ha wo sraegies can be sudied o achieve his goal. One is o invesigae efficien algorihms o solve maser problems, i.e., MP and MP S in ouer and inner levels, respecively, which share a grea similariy o classical scenario-based sochasic programming models. So, i would be ineresing o develop algorihms based on Benders decomposiion or L-shape mehod. Noe also ha MP S has many complemenariy consrains. Provided ha U is a polyope, MP S is a linear program wih complemenariy consrains. Then, insead of inroducing binary variables and big-m o linearize hose consrains, anoher improvemen sraegy is o solve MP S by algorihms ha are specific for linear programs wih complemenariy consrains. Anoher ineresing direcion is o invesigae he hybrid sraegy, combining he (nesed) column-and-consrain generaion and he (revised) affine rule mehods, o develop algorihms ha allow users o achieve an opimal rade-off beween soluion qualiy and compuaional ime for complex applicaions. References S. Ahmed, M. Tawarmalani, and N.V. Sahinidis. A finie branch-and-bound algorihm for wo-sage sochasic ineger programs. Mahemaical Programming, 100(2): , A. Aamurk and M. Zhang. Two-sage robus nework flow and design under demand uncerainy. Operaions Research, 55(4): , A. Ben-Tal and A. Nemirovski. Robus convex opimizaion. Mahemaics of Operaions Research, 23(4): , A. Ben-Tal and A. Nemirovski. Robus soluions of uncerain linear programs. Operaions Research Leers, 25(1):1 14, A. Ben-Tal and A. Nemirovski. Robus soluions of linear programming problems conaminaed wih uncerain daa. Mahemaical Programming, 88(3): , A. Ben-Tal, A. Goryashko, E. Guslizer, and A. Nemirovski. Adjusable robus soluions of uncerain linear programs. Mahemaical Programming, 99(2): , D. Bersimas and M. Sim. Robus discree opimizaion and nework flows. Mahemaical Programming, 98(1):49 71, D. Bersimas and M. Sim. The price of robusness. Operaions Research, 52(1):35 53, D. Bersimas, D.B. Brown, and C. Caramanis. Theory and applicaions of robus opimizaion. SIAM Review, 53: , 2011a. 15

16 D. Bersimas, E. Livinov, X.A. Sun, J. Zhao, and T. Zheng. Adapive robus opimizaion for he securiy consrained uni commimen problem. Technical repor, submied o IEEE Transacions on Power Sysems, 2011b. C.C. Carøe and J. Tind. L-shaped decomposiion of wo-sage sochasic programs wih ineger recourse. Mahemaical Programming, 83(1): , B. Cheang, H. Li, A. Lim, and B. Rodrigues. Nurse rosering problems a bibliographic survey. European Journal of Operaional Research, 151(3): , L. El Ghaoui, F. Ousry, and H. Lebre. Robus soluions o uncerain semidefinie programs. SIAM Journal of Opimizaion, 9:33 52, A.T. Erns, H. Jiang, M. Krishnamoorhy, and D. Sier. Saff scheduling and rosering: A review of applicaions, mehods and models. European Journal of Operaional Research, 153(1):3 27, V. Gabrel, M. Lacroix, C. Mura, and N. Remli. Robus locaion ransporaion problems under uncerain demands. Technical repor, submied o Discree Applied Mahemaics, Lamsade, Universie Paris-Dauphine, T. Hoang. Convex Analysis and Global Opimizaion, volume 22. Springer, R. Jiang, M. Zhang, G. Li, and Y. Guan. Benders decomposiion for he wo-sage securiy consrained robus uni commimen problem. Technical repor, available in opimizaion-online, G. Lapore and F.V. Louveaux. The ineger L-shaped mehod for sochasic ineger programs wih complee recourse. Operaions Research Leers, 13(3): , F. Ordonez and J. Zhao. Robus capaciy expansion of nework flows. Neworks, 50(2): , S. Sen. Algorihms for sochasic mixed-ineger programming models. Handbook of Discree Opimizaion, pages , S. Sen and H.D. Sherali. Decomposiion wih branch-and-cu approaches for wo-sage sochasic mixed-ineger programming. Mahemaical Programming, 106(2): , K. Shimizu, Y. Ishizuka, and J.F. Bard. Nondiffereniable and Two-level Mahemaical Programming. Kluwer Academic Pub, A. Takeda, S. Taguchi, and RH Tuuncu. Adjusable robus opimizaion models for a nonlinear wo-period sysem. Journal of Opimizaion Theory and Applicaions, 136(2): , A. Thiele, T. Terry, and M. Epelman. Robus linear opimizaion wih recourse. Technical repor, available in opimizaion-online, B. Zeng and L. Zhao. Solving wo-sage robus opimizaion problems using a column-and-consrain generaion mehod. Technical repor, under revision, available in opimizaion-online, Universiy of Souh Florida, L. Zhao and B. Zeng. Robus uni commimen problem wih demand response and wind energy. Technical repor, available in opimizaion-online, Universiy of Souh Florida, L. Zhao and B. Zeng. An exac algorihm for power grid inerdicion problem wih line swiching. Technical repor, submied, available in opimizaion-online, Universiy of Souh Florida,