A Geometric Characterization of the Power of Finite Adaptability in Multi-stage Stochastic and Adaptive Optimization

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1 MATHEMATICS OF OPERATIONS RESEARCH Vol. xx, No. x, Xxxxxxx 200x, pp. xxx xxx ISSN X EISSN x xx0x 0xxx inform DOI 0.287/moor.xxxx.xxxx c 200x INFORMS A Geometric Characterization of the Power of Finite Adaptability in Multi-tage Stochatic and Adaptive Optimization Dimitri Bertima Sloan School of Management and Operation Reearch Center, Maachuett Intitute of Technology, Cambridge MA dbertim@mit.edu Vineet Goyal Dept of Indutrial Engineering and Operation Reearch, Columbia Univerity, New York NY vgoyal@ieor.columbia.edu Xu Andy Sun Operation Reearch Center, Maachuett Intitute of Technology, Cambridge MA unx@mit.edu In thi paper, we how a ignificant role that geometric propertie of uncertainty et, uch a ymmetry, play in determining the power of robut and finitely adaptable olution in multi-tage tochatic and adaptive optimization problem. We conider a fairly general cla of multi-tage mixed integer tochatic and adaptive optimization problem and propoe a good approximate olution policy with performance guarantee that depend on the geometric propertie of the uncertainty et. In particular, we how that a cla of finitely adaptable olution i a good approximation for both the multi-tage tochatic a well a the adaptive optimization problem. A finitely adaptable olution generalize the notion of a tatic robut olution and pecifie a mall et of olution for each tage and the olution policy implement the bet olution from the given et depending on the realization of the uncertain parameter in pat tage. Therefore, it i a tractable approximation to a fully-adaptable olution for the multi-tage problem. To the bet of our knowledge, thee are the firt approximation reult for the multi-tage problem in uch generality. Moreover, the reult and the proof technique are quite general and alo extend to include important contraint uch a integrality and linear conic contraint. Key word: robut optimization ; multi-tage tochatic optimization ; adaptive optimization; finite adaptability MSC2000 Subject Claification: Primary: 90C5, 90C47 OR/MS ubject claification: Primary: Robut Optimization/ Stochatic Optimization Hitory: Received: ; Revied:.. Introduction. In mot real world problem, everal parameter are uncertain at the optimization phae and deciion are required to be made in the face of thee uncertaintie. Determinitic optimization i often not ueful for uch problem a olution obtained through uch an approach might be enitive to even mall perturbation in the problem parameter. Stochatic optimization wa introduced by Dantzig [5] and Beale [], and ince then ha been extenively tudied in the literature. A tochatic optimization approach aume a probability ditribution over the uncertain parameter and eek to optimize the expected value of the objective function typically. We refer the reader to everal textbook including Infanger [22], Kall and Wallace [24], Prékopa [26], Shapiro [28], Shapiro et al. [29] and the reference therein for a comprehenive review of tochatic optimization. While the tochatic optimization approach ha it merit and there ha been reaonable progre in the field, there are two ignificant drawback of the approach. i) In a typical application, only hitorical data i available rather than probability ditribution. Modeling uncertainty uing a probability ditribution i a choice we make, and it i not a primitive quantity fundamentally linked to the application. ii) More importantly, the tochatic optimization approach i by and large computationally intractable. Shapiro and Nemirovki [3] give hardne reult for two-tage and multi-tage

2 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 2 tochatic optimization problem where they how the multi-tage tochatic optimization i computationally intractable even if approximate olution are deired. Dyer and Stougie [6] how that a multi-tage tochatic optimization problem where the ditribution of uncertain parameter in any tage alo depend on the deciion in pat tage i PSPACE-hard. To olve a two-tage tochatic optimization problem, Shapiro and Nemirovki [3] how that a ampling baed algorithm provide approximate olution given that a ufficiently large number of cenario are ampled from the aumed ditribution and the problem ha a relatively complete recoure. A twotage tochatic optimization problem i aid to have a relatively complete recoure if for every firt tage deciion there i a feaible econd tage recoure olution almot everywhere, i.e., with probability one ee the book by Shapiro et al. [29]). Shmoy and Swamy [32, 33] and Gupta et al. [20, 2] conider the two-tage and multi-tage tochatic et covering problem under certain retriction on the objective coefficient, and propoe ampling baed algorithm that ue a mall number of cenario ample to contruct a good firt-tage deciion. However, the tochatic et covering problem admit a complete recoure, i.e., for any firt-tage deciion there i a feaible recoure deciion in each cenario, and the ampling baed algorithm only work for problem with complete or relatively complete recoure. More recently, the robut optimization approach ha been conidered to addre optimization under uncertainty and ha been tudied extenively ee Ben-Tal and Nemirovki [6, 7, 8], El Ghaoui and Lebret [7], Goldfarb and Iyengar [9], Bertima and Sim [3], Bertima and Sim [4]). In a robut optimization approach, the uncertain parameter are aumed to belong to ome uncertainty et and the goal i to contruct a ingle tatic) olution that i feaible for all poible realization of the uncertain parameter from the et and optimize the wort-cae objective. We point the reader to the urvey by Bertima et al. [9] and the book by Ben-Tal et al. [4] and the reference therein for an extenive review of the literature in robut optimization. Thi approach i ignificantly more tractable a compared to a tochatic optimization approach and the robut problem i equivalent to the correponding determinitic problem in computational complexity for a large cla of problem and uncertainty et [9]. However, the robut optimization approach ha the following drawback. Since it optimize over the wort-cae realization of the uncertain parameter, it may produce highly conervative olution that may not perform well in the expected cae. Moreover, the robut approach compute a ingle tatic) olution even for a multi-tage problem with everal tage of deciion-making a oppoed a fully-adaptable olution where deciion in each tage depend on the actual realization of the parameter in pat tage. Thi may further add to the conervativene. Another approach i to conider olution that are fully-adaptable in each tage and depend on the realization of the parameter in pat tage and optimize over the wort cae. Such olution approache have been conidered in the literature and referred to a adjutable robut policie ee Ben-Tal et al. [5] and the book by Ben-Tal et al. [4] for a detailed dicuion of thee policie). Unfortunately, the adjutable robut problem i computationally intractable and Ben-Tal et al. [5] introduce an affinely adjutable robut olution approach to approximate the adjutable robut problem. Affine olution approache or jut affine policie) were introduced in the context of tochatic programming ee Gartka and Wet [8] and Rockafeller and Wet [27]) and have been extenively tudied in control theory ee the urvey by Bemporad and Morari [3]). Affine policie are ueful due to their computational tractability and trong empirical performance. Recently Bertima et al. [2] how that affine policie are optimal for a ingle dimenion multi-tage problem with box contraint and box uncertainty et. Bertima and Goyal [] conider affine policie for the two-tage adaptive or adjutable robut) problem and give a tight bound of O dimu)) on the performance of affine policie with repect to a fully-adaptable olution where dimu) denote the dimenion of the uncertainty et. In thi paper, we conider a cla of olution called finitely adaptable olution that were introduced by Bertima and Caramani [0]. In thi cla of olution, the deciion-maker apriori compute a mall number of olution intead of jut a ingle tatic) olution uch that for every poible realization of the uncertain parameter, at leat one of them i feaible and in each tage, the deciion-maker implement the bet olution from the given et of olution. Therefore, a finitely adaptable olution policy i a generalization of the tatic robut olution and i a middle ground between the tatic olution policy and the fully-adaptable policy. It can be thought of a a pecial cae of a piecewie-affine policy where the each piece i a tatic olution intead of an affine olution. A compared to a fully-adaptable olution, which precribe a olution for all poible cenario poibly an uncountable et), a finitely adaptable

3 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 3 olution ha only a mall finite number of olution. Therefore, the deciion pace i much parer and it i ignificantly more tractable than the fully-adaptable olution. Furthermore, for each poible cenario, at leat one of the finitely adaptable olution i feaible. Thi make it different from ampling baed approache where a mall number of cenario are ampled and an optimal deciion i computed for only the ampled cenario, while the ret of the cenario are ignored. We believe that finitely adaptable olution are conitent with how deciion are made in mot real world problem. Unlike dynamic programming that precribe an optimal deciion for each poibly uncountable) future tate of the world, we make deciion for only a few tate of the world. We aim to analyze the performance of tatic robut and finitely adaptable olution policie for the two-tage and multi-tage tochatic optimization problem, repectively. We how that the performance of thee olution approache a compared to the optimal fully-adaptable tochatic olution depend on fundamental geometric propertie of the uncertainty et including ymmetry for a fairly general cla of model. Bertima and Goyal [] analyze the performance of tatic robut olution in two-tage tochatic problem for perfectly ymmetric uncertainty et uch a ellipoid, and norm-ball. We conider a generalized notion of ymmetry of a convex et introduced in Minkowki [25], where the ymmetry of a convex et i a number between 0 and. The ymmetry of a et being equal to one implie that it i perfectly ymmetric uch a an ellipoid). We how that the performance of the tatic robut and the finitely adaptable olution for the correponding two-tage and multi-tage tochatic optimization problem depend on the ymmetry of the uncertainty et. Thi i a two-fold generalization of the reult in []. We extend the reult in [] of performance of a tatic robut olution in twotage tochatic optimization problem, for general convex uncertainty et uing a generalized notion of ymmetry. Furthermore, we alo generalize the tatic robut olution policy to a finitely adaptable olution policy for the multi-tage tochatic optimization problem and give a imilar bound on it performance that i related to the ymmetry of the uncertainty et. The reult are quite general and extend to important cae uch a integrality contraint on deciion variable and linear conic inequalitie. To the bet of our knowledge, there were no approximation bound for the multi-tage problem in uch generality. 2. Model and preliminarie. In thi ection, we firt etup the model for two-tage and multitage tochatic, robut, and adaptive optimization problem conidered in thi paper. We alo dicu the olution concept of finitely adaptability for multi-tage problem. Then we introduce the geometric quantity of ymmetry of a convex compact et and ome propertie that will be ued in the paper. Latly, we define a quantity called tranlation factor of a convex et. 2. Two-tage optimization model. Throughout the paper, we denote vector and matrice uing boldface letter. For intance, x denote a vector, while α denote a calar. A two-tage tochatic optimization problem, Π 2 Stoch, i defined a, z Stoch := min x,yb) c T x + E µ [d T yb)] 2.).t. Ax + Byb) b, µ-a.e. b U R m +, x R p R n p +, yb) R p2 R n2 p2 +, b U. Here, the firt-tage deciion variable i denoted a x; the econd-tage deciion variable i yb) for b U, where b i the uncertain right-hand ide with the uncertainty et denoted a U. The optimization in 2.) for the econd tage deciion y ) i performed over the pace of piecewie affine function, ince there are finitely many bae of the ytem of linear inequalitie Ax + Byb) b. Note that ome of the deciion variable in both the firt and econd tage are free, i.e., not contrained to be non-negative. A probability meaure µ i defined on U. Both A and B are certain, and there i no retriction on the coefficient in A, B or on the objective coefficient c, d. The linear contraint Ax + Byb) b hold for µ almot everywhere on U, i.e., the et of b U for which the linear contraint are not atified ha meaure zero. We ue the notation µ-a.e. b U to denote thi. The key aumption in the above problem i that the uncertainty et U i contained in the nonnegative orthant. In addition, we make the following two technical aumption.

4 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 4 i) z Stoch i finite. Thi aumption implie that z Stoch 0. Since if z Stoch < 0, then it mut be unbounded from below a we can cale the olution by an arbitrary poitive factor and till get a feaible olution. ii) For any firt-tage olution x, and a econd-tage olution y ) that i feaible for µ almot everywhere on U, E µ [yb)] exit. Thee technical condition are aumed to hold for all the multi-tage tochatic model conidered in the paper a well. We would like to note that ince A i not necearily equal to B, our model doe not admit relatively complete recoure. A two-tage adaptive optimization problem, Π 2 Adapt, i given a, z Adapt := min x,yb) c T x + max b U dt yb).t. Ax + Byb) b, b U R m +, 2.2) x R p R n p +, yb) R p2 R n2 p2 +, b U. Note that the above problem i alo referred to a an adjutable robut problem in the literature ee Ben-Tal et al. [5] and the book by Ben-Tal et al. [4]). The correponding tatic robut optimization problem, Π Rob, i defined a, z Rob := min x,y c T x + d T y.t. Ax + By b, b U R m +, 2.3) x R p R n p +, y R p2 R n2 p2 +. Note that any feaible olution to the robut problem 2.3) i alo feaible for the adaptive problem 2.2), and thu z Adapt z Rob. Moreover, any feaible olution to the adaptive problem 2.2) i alo feaible for the tochatic problem 2.) and in addition, we have c T x + E µ [d T yb)] c T x + max b U dt yb), leading to z Stoch z Adapt. Hence, we have z Stoch z Adapt z Rob. We would like to note that when the left hand ide of the contraint, i.e., A and B are uncertain, even a two-tage, two-dimenional problem can have an unbounded gap between the tatic robut olution and the tochatic olution. 2.2 Multi-tage optimization model. For multi-tage problem, we conider a fairly general model where the evolution of the multi-tage uncertainty i given by a directed acyclic network G = N, A), where N i the et of node correponding to different uncertainty et, and A i the et of arc that decribe the evolution of uncertainty from Stage k to k + ) for all k = 2,..., K. In each Stage k + ) for k =,..., K, the uncertain parameter u k belong to one of the N k uncertainty et, U k,..., UN k k R m +. We alo aume that the probability ditribution of u k conditioned on the fact that u k Uj k for all k = 2,..., K, j =,..., N k i known. In our notation, the multi-tage uncertainty network tart from Stage 2 and not Stage ) and we refer to the uncertainty et and uncertain parameter in Stage k + ) uing index k. Therefore, in a K-tage problem, the index k {,..., K }. In Stage 2, there i a ingle node U, which we refer to a the root node. Therefore, N =. For any k =,..., K, uppoe the uncertain parameter u k in Stage k + ), belong to Uj k for ome j =,..., N k. Then for any edge from Uj k k+ to Uj in the directed network, u k+ U k+ j with probability p k j,j, which i an obervable event in Stage k + 2). In other word, in Stage k + 2), we can oberve which tate tranition happened in tage k + ). Therefore, at every tage, we know the realization of the uncertain parameter in pat tage a well a the path of the uncertainty evolution in G and the deciion in each tage depend on both of thee. Note that ince we alo oberve the path of the uncertainty evolution i.e. what edge in the directed network were realized in each tage), the uncertainty et in a given tage need not be dijoint. Thi model of uncertainty i a generalization of the cenario tree

5 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 5 model often ued in tochatic programming ee Shapiro et al. [30]) where the multi-tage uncertainty i decribed by a tree. If the directed acyclic network G in our model i a tree and each uncertainty et i a ingleton, our model reduce to a cenario tree model decribed in [30]. The evolution of the multi-tage uncertainty i illutrated in Figure. We would like to note that thi i a very general model of multi-tage uncertainty. For intance, conider a multi-period inventory management problem, where the demand i uncertain. In each tage, we oberve the realized demand and alo a ignal from the market about the next period demand uch a the weekly retail ale index. In our model, the oberved market ignal correpond to the path in the multi-tage uncertainty network and the oberved demand i the actual realization of the uncertain parameter. A another example, conider a multi-period aet management problem with uncertain aet return. In each period, we oberve the aet return and alo a market ignal uch a S&P 500 or NASDAQ indice. Again, the market ignal correpond to the path in the multi-tage uncertainty network in our model and oberved aet return are the realization of uncertain parameter in the model. p 2, U 3 p, U 2 U p 2,2 p 2 2,2 U 3 2 p,2 U 2 2 p 2 2,3 U 3 3 Figure : Illutration of the evolution of uncertainty in a multi-tage problem. Let P denote the et of directed path in G from the root node, U, to node U j, for all j =,..., N. We denote any path P P a an ordered equence, j,..., j ) of the indice of the uncertainty et in each tage that occur on P. Let ΩP ) denote the et of poible realization of the multi-tage uncertainty from uncertainty et in P. For any P = j,..., j ) P, ΩP ) = { ω = b,..., b ) bk U k j k, k =,..., K }. 2.4) For any P = j,..., j ) P, and k =,..., K, let P [k] denote the index equence of Path P from Stage 2 to Stage k + ), i.e., P [k] = j,..., j k ). Let P[k] = {P [k] P P}. Alo, for any ω = b,..., b ) ΩP ), let ω[k] denote the ubequence of firt k element of ω, i.e., ω[k] = b,..., b k ). We define a probability meaure for the multi-tage uncertainty a follow. For any k =,..., K, j =,..., N k, let µ k j be a probability meaure defined on U j k that i independent of the probability meaure over other uncertainty et in the network. Therefore, for any P = j,..., j ) P, the probability meaure µ P over the et ΩP ) i the product meaure of µ k j k, k =,..., K. Moreover, the probability that the uncertain parameter realize from path P i given by K 2 pk j k,j k+. Thi define a probability meaure on the et of realization of the multi-tage uncertainty, P P ΩP ).

6 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 6 We can now formulate the K-tage tochatic optimization problem Π K Stoch, where the right-hand ide of the contraint i uncertain, a follow. [ ]] zstoch K = min c T x + E P P E µp [d T k y k ω[k], P [k]).t. P P, µ P -a.e. ω = b,..., b ) ΩP ) Ax + x R p R n p +, B k y k ω[k], P [k]) y k ω[k], P [k]) R p k R n k p k +, k =,..., K, There i no retriction on the coefficient in the contraint matrice A, B k or on the objective coefficient c, d k, k =,..., K. However, we require that each uncertainty et Uj k Rm + for all j =,..., N k, k =,..., K. A mentioned before, we aume that zstoch K i finite and the expectation of every feaible multi-tage olution exit. We alo formulate the K-tage adaptive optimization problem Π K Adapt a follow. z K Adapt = min c T x + max P P,ω ΩP ) b k, min y k ω[k],p [k]),,...,.t. P P, ω = b,..., b ) ΩP ) Ax + x R p R n p +, B k y k ω[k], P [k]) b k, y k ω[k], P [k]) R p k R n k p k +, k =,..., K. d T k y k ω[k], P [k]) We alo formulate the K-tage tochatic optimization problem Π K Stochb,d), where both the right-hand ide and the objective coefficient are uncertain, a follow. The ubcript Stochb, d) in the ubcript of the problem name denote that both the right hand ide b and the objective coefficient d are uncertain. Here, ω denote the equence of uncertain right-hand ide and objective coefficient realization and for any k =,..., K, ω[k] denote the ubequence of firt k element. Alo, for any P P, the meaure µ P i the product meaure of the meaure on the uncertainty et in Path P. zstochb,d) K = min [ ]] ct x + E P P E µp [d T k y k ω[k], P [k]).t. P P, µ P -a.e. ω = b, d ),..., b, d ) ) ΩP ) Ax + x R p R n p +, B k y k ω[k], P [k]) b k, y k ω[k], P [k]) R p k R n k p k +, k =,..., K. Alo, we can formulate the K-tage adaptive problem Π K Adaptb,d), 2.5) 2.6) 2.7) z K Adaptb,d) = min ct x + max P P,ω ΩP ) min y k ω[k],p [k]),,...,.t. P P, ω = b, d ),..., b, d ) ) Ω K K Ax + B k y k ω[k], P [k]) b k, x R p R n p +, y k ω[k], P [k]) R p k R n k p k +, k =,..., K. d T k y k ω[k], P [k]) 2.8)

7 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 7 In Section 7, we conider an extenion to the cae where the contraint are general linear conic inequalitie and the right hand ide uncertainty et i a convex and compact ubet of the underlying cone. Furthermore, in Section 8, we alo conider extenion of the above two-tage and multi-tage model where ome deciion variable in each tage are integer. 2.3 Example. In thi ection, we how two claical problem that can be formulated in our framework to illutrate the applicability of our model. Multi-period inventory management problem. We how that we can model the claical multiperiod inventory management problem a a pecial cae of 2.5). In a claical ingle-item inventory management problem, the goal in each period i to decide on the quantity of the item to order under an uncertain future period demand. In each period, each unit of exce inventory incur a holding cot and each unit of backlogged demand incur a per-unit penalty cot and the goal i to make ordering deciion uch that the um of expected holding and backorder-penalty cot i minimized. A an example, we model a 3-tage problem in our framework. Let Ω denote the et of demand cenario with a probability meaure µ, x denote the initial inventory, y k b,..., b k ) denote the backlog and z k b,..., b k ) denote the order quantity in Stage k+) when the firt k-period demand i b,..., b k ). Let h k denote the per unit holding cot and p k denote the per-unit backlog penalty in Stage k + ). We model the 3-tage inventory management problem a follow where the deciion variable are x, y b ), y 2 b, b 2 ) and z b ) for all b, b 2 ) U. min E b [ h x + y b ) b ) + p y b ) + E b2 b [ h2 x + z b ) + y 2 b, b 2 ) b + b 2 ) ) + p 2 y 2 b, b 2 ) ]].t. x + y b ) b, µ-a.e. b, b 2 ) Ω x + z b ) + y 2 b, b 2 ) b + b 2, µ-a.e. b, b 2 ) Ω x, y b ), y 2 b, b 2 ), z b ) 0. The above formulation can be generalized to a multi-period problem in a traightforward manner. We can alo generalize to multi-item variant of the problem. However, we would like to note that we can not model a capacity contraint on the order quantity, ince we require that the contraint are of greater than or equal to form with nonnegative right-hand ide. Neverthele, the formulation i fairly general even with thi retriction. Capacity planning under demand uncertainty. In many important application, we encounter the problem of planning for capacity to erve an uncertain future demand. For intance, in a call center, taffing capacity deciion need to be made well in advance of the realization of the uncertain demand. In electricity grid operation, generator unit-commitment deciion are required to be made at leat a day ahead of the realization of the uncertain electricity demand becaue of the large tartup time of the generator. In a facility location problem with uncertain future demand, the facilitie need to be opened well in advance of the realized demand. Therefore, the capacity planning problem under demand uncertainty i important and widely applicable. We how that thi can be modeled in our framework uing the example of the facility location problem. For illutration, we ue a 2-tage problem. Let F denote the et of facilitie and D denote the et of demand point. For each facility i F, let x i be an integer deciion variable that denote the capacity for facility i. For each point j D, let b j denote the uncertain future demand and let y ij b) denote the amount of demand of j aigned to facility i when the demand vector i b. Alo, let d ij denote the cot of aigning a unit demand from point j to facility i and let c i denote the per-unit capacity cot of facility i. Therefore, we can formulate the problem a follow. Let U be the uncertainty et for the demand and

8 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 8 let µ be a probability meaure defined on U. min c i x i + E µ d ij y ij b) j D.t. i F y ij b) b j, µ-a.e. b U, j D i F x i j D y ij b) 0, µ-a.e. b U, i F x i Z +, i F y ij b) R +, i F, j D, µ-a.e. b U. 2.4 Finitely adaptable olution. We conider a finitely adaptable cla of olution for the multitage tochatic and adaptive optimization problem decribed above. Thi cla of olution wa introduced by Bertima and Caramani [0] where the deciion-maker compute a mall et of olution in each tage apriori. A tatic robut olution policy pecifie a ingle olution that i feaible for all poible realization of the uncertain parameter. On the other extreme, a fully-adaptable olution policy pecifie a olution for each poible realization of the uncertain parameter in pat tage. Typically, the et of poible realization of the uncertain parameter i uncountable which implie that an optimal fully-adaptable olution policy i a function from an uncountable et of cenario to optimal deciion for each cenario and often uffer from the cure of dimenionality. A finitely adaptable olution i a tractable approach that bridge the gap between a tatic robut olution and a fully-adaptable olution. In a general finitely adaptable olution policy, intead of computing an optimal deciion for each cenario, we partition the cenario into a mall number of et and compute a ingle olution for each poible et. The partitioning of the et of cenario i problem pecific and i choen by the deciion-maker. A finitely adaptable olution policy i a pecial cae of a piecewie affine olution where the olution in each piece i a tatic olution. For the multi-tage tochatic and adaptive problem 2.5) and 2.6), we conider the partition of the et of cenario baed on the realized path in the multi-tage uncertainty network. In particular, in each Stage k + ), the deciion y k depend on the path of the uncertainty realization until Stage k + ). Therefore, there are P[k] different olution for each Stage k + ), k =,..., K ; one correponding to each directed path from the root node to a node in Stage k+) in the multi-tage uncertainty network. For any realization of uncertain parameter and the path in the uncertainty network, the olution policy implement the olution correponding to the realized path. Figure 2 illutrate the number of olution in the finitely adaptable olution for each tage and each uncertainty et in a multi-tage uncertainty network. In the example in Figure 2, there are following four directed path from Stage 2 to Stage K K = 4): P =,, ), P 2 =,, 2), P 3 =, 2, 2), P 4 =, 2, 3). We know that P j [] = ) for all j =,..., 4. Alo, P [2] = P 2 [2] =, ) and P 3 [2] = P 4 [2] =, 2). Therefore, in a finitely adaptable olution, we have the following deciion variable apart from x. Stage 2 : y ) Stage 3 : y 2, ), y 2, 2) Stage 4 : y 3,, ), y 3,, 2), y 3, 2, 2), y 3, 2, 3). 2.5 The ymmetry of a convex et. Given a nonempty compact convex et U R m and a point u U, we define the ymmetry of u with repect to U a follow. ymu, U) := max{α 0 : u + αu u ) U, u U}. 2.9) In order to develop a geometric intuition on ymu, U), we firt how the following reult. dicuion, aume U i full-dimenional. For thi Lemma 2. Let L be the et of line in R m paing through u. For any line l L, let u l and u l be the point of interection of l with the boundary of U, denoted δu) thee exit a U i full-dimenional and compact). Then,

9 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 9 {y 3,, )} {y 2, )} U 3 {y )} U U 2 {y 2, 2)} {y 3,, 2), y 3, 2, 2)} U 3 2 U 2 2 {y 3, 2, 3)} U 3 3 Figure 2: A Finitely Adaptable Solution for a 4-tage problem. For each uncertainty et, we pecify the et of correponding olution in the finitely adaptable olution policy. u u l a) ymu, U) min u u l, u u l u u u u b) ymu, U) = min min l l L l ), u u l, u u l u u l Proof. Let ymu, U) = α. Conider any l L and conider u l δu) a illutrated in Figure 3. Let u l,r = u + α u u l ). From 2.9), we know that u l,r U. Note that u l,r i a caled reflection of u l about u by a factor α. Furthermore, u l,r lie on l. Therefore, u l,r U implie that which in turn implie that Uing a imilar argument tarting with u l To prove the econd part, let ). u u l,r u u l α u u l u u l, ymu, U) = α u u l u u l. intead of u l, we obtain that ymu, U) u u l u u l. u u α 2 = min min l l L u u l, u u l u u From the above argument, we know that ymu, U) α 2. Conider any u U. We how that u + α 2 u u )) U. Let l denote the line joining u and u. Clearly, l L. Without lo of generality, uppoe u belong to the line egment between u and u l. Therefore, u u = γu u l ) for ome 0 γ and We know that l ). u u u u l. 2.0) α 2 u u l u u l α 2 u u l u u l α 2 u u u u l,

10 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 0 Figure 3: Geometric illutration of ymu, U). where the lat inequality follow from 2.0). Therefore, u + α 2 u u belong to the line egment between u and u l and thu, belong to U. Therefore, ymu, U) α 2. The ymmetry of et U i defined a ymu) := max{ymu, U) u U}. 2.) An optimizer u 0 of 2.) i called a point of ymmetry of U. Thi definition of ymmetry can be traced!" back to Minkowki [25] and i the firt and the mot widely ued ymmetry meaure. Refer to Belloni and Freund [2] for a broad invetigation of the propertie of the ymmetry meaure defined in 2.9) and 2.). Note that the above definition generalize the notion of perfect ymmetry conidered by Bertima and Goyal []. In [], the author define that a et U i ymmetric if there exit u 0 U uch that, for any z R m, u 0 +z) U u 0 z) U. Equivalently, u U 2u 0 u) U. According to the definition in 2.), ymu) = for uch a et. Figure 4 illutrate ymmetrie of everal intereting convex et. Figure 4: The figure on the left i a ymmetric polytope with ymmetry. The middle figure illutrate a tandard implex in R m with ymmetry /m. The right figure how the interection of a Euclidean ball with R m +, which ha ymmetry / m. Lemma 2.2 Belloni and Freund [2]) For any nonempty convex compact et U R m, the ymmetry of U atifie, ymu). m

11 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS The ymmetry of a convex et i at mot, which i achieved for a perfectly ymmetric et; and at leat /m, which i achieved by a tandard implex defined a = {x R m + m i= x i }. The lower bound follow from Löwner-John Theorem [23] ee Belloni and Freund [2]). The following lemma i ued later in the paper. Lemma 2.3 Let U R m + be a convex and compact et uch that u 0 i the point of ymmetry of U. Then, ) + u 0 u, u U. ymu) Proof. From the definition of ymmetry in 2.), we have that for any u U, u 0 + ymu)u 0 u) = ymu) + )u 0 ymu)u U, which implie that ymu) + )u 0 ymu)u 0 ince U R m The tranlation factor ρu, U). For a convex compact et U R m +, we define a tranlation factor ρu, U), the tranlation factor of u U with repect to U, a follow. ρu, U) = min{α R + U α) u R m + }. In other word, U := U ρ)u i the maximum poible tranlation of U in the direction u uch that U R m +. Figure 5 give a geometric picture. Note that for α =, U α) u = U R m +. Therefore, 0 < ρ. And ρ approache 0, when the et U move away from the origin. If there exit u U uch that u i at the boundary of R m +, then ρ =. We denote ρu) := ρu 0, U), where u 0 i the ymmetry point of the et U. The following lemma i ued later in the paper. Figure 5: Geometry of the tranlation factor. Lemma 2.4 Let U R m + be a convex and compact et uch that u 0 i the point of ymmetry of U. Let = ymu) = ymu 0, U) and ρ = ρu) = ρu 0, U). Then, + ρ ) u 0 u, u U.

12 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 2 Proof. Let U = U ρ)u 0. Let u := u 0 ρ)u 0. Alo let z := u 0 u = ρ)u 0. Figure 5 give a geometric picture. Note that ymu ) = ymu) =. From Lemma 2.3, we know that + ) u u, u U. Adding z on both ide, we have, + ) u + z u, u U, + ) ρu 0 + ρ)u 0 u, u U, + ρ ) u 0 u, u U. 3. Our contribution. The contribution of thi paper are two-fold. We preent two ignificant generalization of the model and reult in Bertima and Goyal [], where the author characterize the performance of tatic robut olution for two-tage tochatic and adaptive optimization problem under the aumption that the uncertainty et are perfectly ymmetric. Firtly, we generalize the two-tage reult to general uncertainty et. We how that the performance of a tatic robut olution for two-tage tochatic and adaptive optimization problem depend on the general notion of ymmetry 2.) of the uncertainty et. The bound are independent of the contraint matrice and any other problem data. Our bound are alo tight for all poible value of ymmetry and reduce to the reult in [] for perfectly ymmetric et. Secondly, we conider the multi-tage extenion of the two-tage model in [] a decribed above and how that a cla of finitely adaptable olution which i a generalization of the tatic robut olution, i a good approximation for both the tochatic and the adaptive problem. The proof technique are very general and eaily extend to the cae where ome of the deciion variable are integer contrained and the cae where the contraint are linear conic inequalitie for a general convex cone. To the bet of our knowledge, thee are the firt performance bound for the multi-tage problem in uch generality. Our main contribution are ummarized below. Stochatic optimization. For the two-tage tochatic optimization problem under right-hand ide uncertainty, we how the following bound under a fairly general condition on the probability meaure, z Rob + ρ ) z Stoch, where = ymu) and ρ = ρu) are the ymmetry and the tranlation factor of the uncertainty et, repectively. Note that the above bound compare the cot of an optimal tatic olution with the expected cot of an optimal fully-adaptable two-tage tochatic olution and how that the tatic cot i at mot + ρ/) time the optimal tochatic cot. The performance of the tatic robut olution for the twotage tochatic problem can poibly be even better. For the two-tage problem, we only implement the firt-tage part of the tatic robut olution. For the econd-tage, we compute an optimal olution after the uncertain parameter right hand ide in our cae) i realized. Therefore, the expected econd-tage cot for thi olution policy i at mot the econd-tage cot of the tatic robut olution and the total expected cot i at mot z Rob + ρ/) z Stoch. Since z Rob i an upper bound on the expected cot of the olution obtained from an optimal tatic olution, the bound obtained by comparing z Rob to the optimal tochatic cot i in fact a conervative bound. For multi-tage problem, we how that a K-tage tochatic optimization problem, Π K Stoch, can be well approximated efficiently by a finitely adaptable olution. In particular, there i a finitely adaptable olution with at mot P olution that i a + ρ/)-approximation of the original problem where i the minimum ymmetry over all et in the uncertainty network and ρ i the maximum tranlation factor of any uncertainty et. Note that when all et are perfectly ymmetric, i.e., =, the finitely adaptable olution i a 2-approximation, generalizing the reult of [] to multi-tage. We alo how that the bound of + ρ/) i tight.

13 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 3 Note that ince 0 < ρ and /m, + ρ ) m + ), which how that, for a fixed dimenion, the performance bound for robut and finitely adaptable olution i independent of the particular data of an intance. Thi i urpriing ince when the left-hand ide ha uncertainty, i.e., A, B are uncertain, even a two-tage two-dimenional problem can have an unbounded gap between the tatic robut olution and the tochatic olution a mentioned earlier. Thi alo indicate that our aumption on the model, namely, the right-hand ide uncertainty and/or cot uncertainty) and the uncertainty et contained in the nonnegative orthant, are tight. If thee aumption are relaxed, the performance gap become unbounded even for fixed dimenional uncertainty. For the cae when both cot and right-hand ide are uncertain in Π K Stochb,d), the performance of any finitely adaptable olution can be arbitrarily wore a compared to the optimal fully-adaptable tochatic olution. Thi reult follow along the line of arbitrary bad performance of a tatic robut olution in two-tage tochatic problem when both cot and right-hand ide are uncertain a hown in Bertima and Goyal []. Adaptive optimization. We how that for a multi-tage adaptive optimization problem, Π K Adapt, where only the right-hand ide of the contraint i uncertain, the cot of a finitely adaptable olution i at mot + ρ/) time the optimal cot of a fully-adaptable multi-tage olution, where i the minimum ymmetry of all et in the uncertainty network and ρ i the maximum tranlation factor of the point of ymmetry over all the uncertainty et. Thi bound follow from the bound for the performance of a finitely adaptable olution for the multi-tage tochatic problem. Furthermore, if the uncertainty come from hypercube et, then a finitely adaptable olution with at mot P olution at each node of the uncertainty network i an optimal olution for the adaptive problem. For the cae when both cot and right-hand ide are uncertain in Π K Adaptb,d), we how that the wortcae cot a finitely adaptable olution with at mot P different olution at each node of the uncertainty network i at mot + ρ/) 2 time the cot of an optimal fully-adaptable olution. Extenion. We conider an extenion of the above multi-tage model to the cae where the contraint are linear conic inequalitie and the uncertainty et belong to the underlying cone. We alo conider the cae where ome of the deciion variable are contrained to be integer. Our proof technique are quite general and the reult extend to both thee cae. For the cae of linear conic inequalitie and the uncertainty et contained in the underlying convex cone, we how that a finitely adaptable tatic robut) olution i a + ρ/)-approximation for the multi-tage two-tage repectively) tochatic and adaptive problem with right hand ide uncertainty. The reult alo hold for the adaptive problem with both the right hand ide and objective coefficient uncertainty and the performance bound for the finitely adaptable tatic robut) olution i + ρ/) 2 for the multi-tage two-tage repectively) problem. We alo conider the cae where ome of the deciion variable are integer contrained. For the multitage two-tage) tochatic problem with right hand ide uncertainty, if ome of the firt-tage deciion variable are integer contrained, a finitely adaptable tatic robut repectively) olution i a + ρ/) approximation with repect to an optimal fully-adaptable olution. For the multi-tage adaptive problem, we can handle integer contrained deciion variable in all tage unlike the tochatic problem where we can handle integrality contraint only on the firt tage deciion variable. We how that for the multitage two-tage) adaptive problem with right hand ide uncertainty and integrality contraint on ome of the deciion variable in each tage, a finitely adaptable tatic robut repectively) olution i a + ρ/) -approximation. For the multi-tage two-tage) adaptive problem with both right hand ide and objective coefficient uncertainty and integrality contraint on variable, a finitely adaptable tatic robut repectively) olution i a + ρ/) + ρ/)-approximation. Outline. The ret of the paper i organized a follow. In Section 4, we preent the performance bound of a tatic-robut olution that depend on the ymmetry of the uncertainty et for the two-tage tochatic optimization problem. We alo how that the bound i tight and preent explicit bound for everal intereting and commonly ued uncertainty et. In Section 5, we preent the finitely adaptable olution policy for the multi-tage tochatic optimization problem and dicu it performance bound.

14 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 4 In Section 6, we dicu the reult for multi-tage adaptive optimization problem. In Section 7 and 8, we preent extenion of our reult for the model with linear conic contraint and integrality contraint repectively. 4. Two-tage tochatic optimization problem. In thi ection, we conider the two-tage tochatic optimization problem 2.) and how that the performance of a tatic robut olution depend on the ymmetry of the uncertainty et. Theorem 4. Conider the two-tage tochatic optimization problem in 2.). Let µ be the probability meaure on the uncertainty et U R m +, b 0 be the point of ymmetry of U, and ρ = ρb 0, U) be the tranlation factor of b 0 with repect to U. Denote = ymu). Aume the probability meaure µ atifie, E µ [b] b 0. 4.) Then, z Rob + ρ ) z Stoch. 4.2) Proof. From Lemma 2.4, we know that + ρ ) b 0 b, b U. 4.3) For brevity, let τ := +ρ/). Suppoe x, yb) : b U) i an optimal fully-adaptable tochatic olution. We firt how that the olution, τx, τe[yb)]), i a feaible olution to the robut problem 2.3). By the feaibility of x, yb) µ-a.e. b U), we have Taking expectation on both ide, we have Aτx) + Bτyb)) τb, µ a.e.b U. Aτx) + B τe[yb)] ) τe[b] τb 0 b, b U, where the econd inequality follow from 4.) and the lat inequality follow from 4.3). τx, τe[yb)]) i a feaible olution for the robut problem 2.3) and, Therefore, z Rob τ c T x + d T E[yb)]). 4.4) Alo, by definition, z Stoch = c T x + d T E b [yb)], which implie that z Stoch τ z Rob For brevity, we refer to 4.2) a the ymmetry bound. The following comment are in order. i) The ymmetry bound 4.2) i independent of the problem data A, B, c, d, depending only on the geometric propertie of the uncertainty et U, namely the ymmetry and the tranlation factor of U. ii) Theorem 4. can alo be tated in a more general way, removing Aumption 4.) on the probability meaure, and ue the ymmetry of the expectation point. In particular, the bound 4.2) hold for = yme µ [b], U) and ρ = ρe µ [b], U). However, we would like to note that 4.) i a mild aumption, epecially for ymmetric uncertainty et. Any ymmetric probability meaure on a ymmetric uncertainty et atifie thi aumption. It alo emphaize the role that the ymmetry of the uncertainty et play in the bound. iii) A already mentioned in Section 3, a mall relaxation from the aumption of our model would caue unbounded performance gap. In particular, if the aumption, U R m +, i relaxed, or the contraint coefficient are uncertain, the gap between z Rob and z Stoch cannot be bounded even in mall dimenional problem. The following example illutrate thi fact. a) U R m + : Conider the intance where m = and c = 0, d =, A = 0, B =, the uncertainty et U = [, ], and a uniform ditribution on U. The optimal tochatic olution ha cot z Stoch b) = 0, while z Rob b) = 2. Thu, the gap i unbounded.

15 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 5 b) A, B uncertain: Conider the following intance taken from Ben-Tal et al. [4], min.t. x [ 2 b ] [ ] x + yb) b 2 x, yb) 0, b [0, r], [ ], b [0, r], 0 where 0 < r <. From the contraint, we have that x 2/ r). Therefore, the optimal cot of a tatic olution i at leat 2/ r). However, the optimal tochatic cot i at mot 4. For detail, refer to [4]. When r approache, the gap tend to infinity. 4. Tightne of the bound. In thi ection, we how that the bound given in Theorem 4. i tight. In particular, we how that for any given ymmetry and tranlation factor, there exit a family of intance of the two-tage tochatic optimization problem uch that the bound in Theorem 4. hold with equality. Theorem 4.2 Given any ymmetry /m and tranlation factor, 0 < ρ, there exit a family of intance uch that z Rob = + ρ/) z Stoch. Proof. For p, let In Appendix A.2, we how that B + p = {b R m + b p }. ymb + p ) = ) p, m and the ymmetry point i, b 0 B + p ) = m /p + e, where b 0 U) denote the ymmetry point for any et U. Alo, let B + p ) := B + p + re, for ome r 0. Then, given any ymmetry /m and tranlation factor, ρ, we can find a p and r 0 uch that = ) p, ρ = m m /p + )r +. Now conider a problem intance where A = 0, B = I, c = 0, d = e, and a probability meaure whoe expectation i at the ymmetry point of B + p ). The optimal tatic robut olution i y = r + )e and the optimal tochatic olution i yb) = b for all b B + p ). Therefore, and, z Stoch = r + z Rob z Stoch = m /p + which how the bound in Theorem 4. i tight. r + r + m /p + ) m, z Rob = r + )m, ) = + ρ, 4.2 An alternative bound. In thi ection, we preent another performance bound on the robut olution for the two-tage tochatic problem, and compare it with the ymmetry bound 4.2). For an uncertainty et U, let b h a b h j := max b U b j. Alo, uppoe the probability meaure µ atifie 4.), i.e., E µ [b] b 0, where b 0 i the ymmetry point of U. Let θ := min{θ : θ b 0 b h }. 4.5) Uing an argument imilar to the proof of Theorem 4., we can how that the performance gap i at mot θ, where the ubcript tand for tochaticity, i.e., z Rob θ z Stoch. 4.6)

16 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 6 Let = ymb 0, U) and ρ = ρb 0, U). From Lemma 2.4), we alo know that + ρ ) b 0 b, b U. Therefore, θ + ρ/). So, θ i upper bounded by the ymmetry bound obtained in Theorem 4.. For brevity, we refer to 4.5) a a caling bound. A geometric picture i given in Figure 6. A hown in the figure, u i obtained from caling E µ [b] by a factor greater than one uch that u dominate all the point in U, and θ i the mallet uch factor. Figure 6: A geometric perpective on the tochaticity gap. In the following ection, we how that the caling bound can be trictly tighter than the ymmetry bound. On the other hand, the ymmetry bound relate the performance of a tatic robut olution to key geometric propertie of the uncertainty et. Furthermore, both bound are equal for everal intereting clae of uncertainty et a dicued in the following ection. In Theorem 4.2, we how that the ymmetry bound i tight for any ρ, /m for a family of uncertainty et. The ymmetry bound reveal everal key qualitative propertie of the performance gap that are difficult to ee from the caling bound. For example, for any ymmetric uncertainty et, without the need to compute b h and θ, we have a general bound of two on the tochaticity gap. Since ymmetry of any convex et i at leat /m, the bound for any convex uncertainty et i at mot m + ). Both thee bound are not obviou from the caling bound. Mot importantly, in practice the ymmetry bound give an informative guideline in deigning uncertainty et where the robut olution ha guaranteed performance. 4.3 Example: Stochaticity gap for pecific uncertainty et. In thi ubection, we give example of pecific uncertainty et and characterize their ymmetry. Both bound on the tochaticity gap, the ymmetry bound 4.2) a well the caling bound 4.5), are preented in Table for everal intereting uncertainty et. In mot of the cae, the caling bound i equal to the ymmetry bound. The proof of the ymmetry computation of variou uncertainty et are deferred to the Appendix. An L p -ball interected with the nonnegative orthant. We define, B + p := {b R m + b p }, for p. In Appendix A.2 we how the following. ) ymb + p p ) =, b0 B + p ) = e. 4.7) m m p + Therefore, if U = B + p and the probability meaure atifie condition 4.), then z Rob +m p )zstoch. The bound i tight a hown in Theorem 4.2. There are everal intereting cae for B + p uncertainty et. In particular,

17 Mathematic of Operation Reearch xxx), pp. xxx xxx, c 200x INFORMS 7 i) For p =, B + p i the tandard implex centered at the origin. The ymmetry i /m and tochaticity gap i m +. ii) For p = 2, B + p i the Euclidean ball in the nonnegative orthant. It ymmetry i / m, and the tochaticity gap i + m. iii) For p =, B + p i a hypercube centered at e/2 and touche the origin. The ymmetry i and the tochaticity gap i 2. Ellipoidal uncertainty et. An ellipoidal uncertainty et i defined a, U := {b R m + Eb b) 2 }, 4.8) where E i an invertible matrix. Since U i ymmetric, the tochaticity gap i bounded by 2 if U R m + and the probability meaure atifie 4.). In Appendix A.3, we how that the bound can be improved to the following. z Rob + max i m E ) ii z Stoch 2z Stoch. b i Interection of two L p -ball. Conider the following uncertainty et. U := {b R m + b p, b p2 r}, for 0 < r < and p < p ) Aume the following condition hold, r > rm p > m p 2, 4.0) e p2 e p No. Uncertainty et Symmetry Stochaticity gap {b : b p, b 0} + m p m /p a,b) {b : b b 2 p, b b } R m + + max 2 b /p i b) p < i m 3 {b : b b, b b } R m + + max b i a,b) i m 4 {b : b b p } R m + + max i m b i a,b) 5 {b : Eb b) 2 } R m + + max i m 6 {b : b p, b p2 r, b 0} Budgeted uncertainty et k k m) Demand uncertainty et DUµ, Γ) µ Γ) Demand uncertainty et DUµ, Γ) m Γ < µ < Γ) Demand uncertainty et DUµ, Γ) 0 µ m Γ) {b : L e T b U, b 0} 2 conv, {e}) 3 convb,..., b k ) 4 {B S m + I B } E ii b i rm /p + rm p k m + m k a,b) a,b) + Γ µ a,b) mµ + Γ m)γ m)γ mµ + Γ mµ + Γ mµ + Γ) + mµ + Γ) mµ + Γ m L U m m m a,b) a,b) a,b) m + ) L U a,b) m a,b) max b h i i m b i + m a) Table : Symmetry and correponding tochaticity gap for variou uncertainty et. Note that footnote a) or b) mean the tight bound i from the ymmetry bound 4.2) or the caling bound 4.6). b)