Noname manuscript No. (will be inserted by the editor) Can Li Ignacio E. Grossmann

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1 Noname manuscript No. (wi be inserted by the editor) A finite ɛ-convergence agorithm for two-stage convex 0-1 mixed-integer noninear stochastic programs with mixed-integer first and second stage variabes Can Li Ignacio E. Grossmann Received: date / Accepted: date Abstract In this paper, we propose a generaized Benders decompositionbased branch and bound agorithm, GBDBAB, to sove two-stage convex 0-1 mixed-integer noninear stochastic programs with mixed-integer variabes in both first and second stage decisions. In order to construct the convex hu of the MINLP subprobem for each scenario in cosed-form, we first represent each MINLP subprobem as a generaized disjunctive program (GDP) in conjunctive norma form (CNF). Second, we appy basic steps to convert the CNF of the MINLP subprobem into disjunctive norma form (DNF) to obtain the convex hu of the MINLP subprobem. We prove that GBD is abe to converge for the probems with pure binary variabes given that the convex hu of each subprobem is constructed in cosed-form. However, for probems with mixed-integer first and second stage variabes, we propose an agorithm, GBDBAB, where we may have to branch and bound on the continuous first stage variabes to obtain the optima soution. We prove that the agorithm GBDBAB can converge to ɛ-optimaity in a finite number of steps. Since constructing the convex hu can be expensive, we propose a sequentia convexification scheme that progressivey appies basic steps to the CNF. Computationa resuts demonstrate the effectiveness of the agorithm. Keywords Stochastic programming Integer recourse Generaized Benders decomposition Branch and bound 1 Introduction Stochastic programming provides a mathematica framework to mode decisionmaking under uncertainty [4]. The uncertain parameters are usuay repre- Ignacio E. Grossmann Department of Chemica Engineering, Carnegie Meon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA E-mai: grossmann@cmu.edu

2 2 Can Li, Ignacio E. Grossmann sented by discrete probabiity distributions, which are assumed to be known a priori. Different reaizations of the uncertain parameters are represented by the set of scenarios ω Ω, each of which has an associated probabiity τ ω. Two-stage stochastic programming probem, a specia case of stochastic programming, can be computationay expensive when the number of scenarios becomes arge. Therefore, decomposition agorithms, ike (generaized) Benders decomposition [15; 33] and Lagrangean decomposition [16], and progressive hedging [26] have been proposed to sove the two-stage stochastic programs by decomposing the fuspace probem into scenarios. However, for twostage mixed-integer stochastic programs with mixed-integer recourse, Benders decomposition cannot be appied directy given the nonconvex nature of the subprobems. Athough Lagrangean decomposition [18] and progressive hedging [13; 36; 37] have been used to sove stochastic programs with mixed-integer recourse, they are not guaranteed to obtain the optima soution because of the duaity gap. Given that there are engineering appications that invove mixed-integer second stage decisions, such as unit commitment probem [17] and panning probems with discrete transportation costs [5], decomposition agorithms that can dea with mixed-integer recourse are of interest. In the past two decades, some advances have been made in the agorithms to sove stochastic mixed-integer inear programs with (mixed) integer recourse. Laporte and Louveaux [20] propose a Benders-ike agorithm with optimaity cuts to sove two-stage stochastic programs with mixed-integer recourse and pure binary first stage variabes. Ahmed et a. [1] propose a branch-andbound agorithm for two-stage stochastic integer programs with mixed-integer first stage variabes and pure integer second stage variabes. Carøe and Schutz [8] propose a dua decomposition agorithm with branch-and-bound that is abe to cose the duaity gap of Lagrangean reaxation. In order to appy Benders decomposition to sove the probems with mixed-integer recourse, decomposition agorithms with convexification schemes, such as ift-and-project cuts [1; 25; 29; 30], reformuation inearization technique (RLT) [31; 32], Gomory cuts [14], have aso been proposed. For a more comprehensive review of the agorithmic advances in stochastic mixed-integer inear programs, we refer to Küçükyavuz and Sen [19]. For mixed-integer noninear stochastic programs, reativey itte work has been done. Mijangos [24] proposes an agorithm based on Branch-and-Fix Coordination method for convex probems with 0-1 mixed-integer variabes in the first stage and ony continuous variabes in the second stage. Atakan and Sen [2] propose a progressive hedging-based branch-and-bound agorithm for convex mixed-integer stochastic programs, which works we in the case of pure binary first stage variabes and continuous recourse. Li et a. [22] propose a nonconvex generaized Benders decomposition (NGBD) agorithm for mixedinteger noninear stochastic programs with pure binary first stage variabes. In this paper, we propose a GBD-based agorithm that can sove two-stage convex 0-1 mixed-integer noninear stochastic programs with mixed-integer first and second stage variabes with finite ɛ-convergence. This work is inspired by Sherai and Zhu [32] who propose a Benders decomposition agorithm to

3 Tite Suppressed Due to Excessive Length 3 sove two-stage mixed-integer inear stochastic programs having mixed-integer first and second stage variabes. The authors use reformuation inearization technique (RLT) to convexify the origina MILP subprobems. In this paper, we try to generaize the agorithm proposed by Sherai and Zhu [32] to convex noninear probems. Due to recent advances in generaized disjunctive programming (GDP), especiay the work of Ruiz and Grossmann [27], we now have a procedure to obtain the convex hu of the feasibe region of a convex MINLP. By appying the convexification scheme proposed in [27], we are abe to deveop a GBD-based agorithm with finite ɛ-convergence. 2 Background In this section, we provide some background on disjunctive convex programming and basic steps [27], which are used to construct the convex hu of the MINLP subprobem in cosed-form. A convex set C can be defined as C={x IR n φ(x) 0}, where φ(x) : IR n IR 1 is a convex function. Given a coection of sets such that C j ={x IR n φ j (x) 0}, j M, the union of the convex sets, which is an eementary disjunctive set, is defined as: H = C j = {x IR n φ j (x) 0} (1) j M j M and the intersection of convex sets can be expressed as: P = C j = {x IR n φ j (x) 0} (2) j M j M A disjunctive set can be expressed in different forms. Two extreme cases are the conjunctive norma form (CNF) and the disjunctive norma form (DNF). The CNF is the intersection of some eementary disjunctive sets: F CNF = i T H i (3) where each H i is an eementary disjunctive set. The DNF is the union of some convex sets: F DNF = P i (4) i D where each set P i is a convex set defined by convex functions φ j, j M i. P i can aso be represented in a more succinct form P i = {x IR n g i (x) 0}, where g i (x): IR n IR m. A reguar form (RF) between the CNF and DNF can be represented as: F RF = S k (5) k K where S k = i Dk P i. Each S k is caed a conjunct, which is a disjunction over some convex sets P i, i D k. Baas [3] proposes an operation caed basic step, which when appied to the reguar form can reduce the number of conjuncts

4 4 Can Li, Ignacio E. Grossmann by one in the inear case. The basic step is extended by Ruiz and Grossmann [27] to convex noninear case. Here, we rewrite Theorem 2.1 in [27], which is an extension of the theorem in Baas [3] to show how a CNF can be transformed into a DNF. Theorem 1 Let F RF be a disjunctive set in reguar form. Then F RF can be brought to DNF by K 1 recursive appications of the foowing basic step which preserves reguarity: For some r, s K, bring S r S s to DNF by repacing it with: S rs = (P i P j ) i D r,j D s Theorem 1 wi be used to construct the convex hu the MINLP subprobem in our proposed agorithm. 3 Overview of basic ideas The goa of this paper is to design a decomposition agorithm that has finite ɛ-convergence for two-stage convex 0-1 mixed-integer noninear stochastic programs with mixed-integer first and second stage variabes. Whie GBD cannot be appied to probems with mixed-integer recourse directy, it has finite ɛ-convergence for probems with convex constraints and continuous recourse variabes as proved by Geoffrion [15]. Therefore, we propose to construct the convex hu for the integer-constrained subprobems by appying basic steps so that GBD can be appied to sove the reaxed probem. However, the reaxed probem is not equivaent to the origina probem if we have mixed-integer first stage variabes. Therefore, we first prove that the reaxed probem is equivaent to the origina fuspace probem for the specia case of pure binary first stage variabes. Based on the theoretica resuts of the probems with pure binary variabes, we propose a GBD-based branch and bound agorithm for the genera probems with mixed-integer first and second stage variabes where we may branch on the continuous first stage variabes. The detais of the agorithm are described in section 4. However, constructing the convex hu by appying a the possibe basic steps for each subprobem can be expensive, especiay when we have a arge number of binary variabes in the second stage. Therefore, in section 5, we describe a sequentia convexification scheme, which is a more imited way for convexifying the subprobem to avoid appying a the basic steps when certain sufficient conditions are satisfied. Computationa resuts of the proposed GB- DBAB agorithm are shown in section 6 with one sma iustrative exampe and two process systems engineering appications. 4 GBD-based branch and bound agorithm In this paper, we consider two-stage convex 0-1 mixed-integer noninear stochastic programs with mixed-integer first and second stage variabes. The probem

5 Tite Suppressed Due to Excessive Length 5 is formay defined by (6)-(10). (P ) min c T x + ω Ω τ ω d T ω y ω (6) A 0 x b 0, g 0 (x) 0 (7) A 1,ω x + g 1,ω (y ω ) b 1,ω ω Ω (8) x X, X = { x : x i {0, 1}, i I 1, 0 x x ub} (9) y ω Y ω Ω, Y = { y : y j {0, 1}, j J 1, 0 y y ub} (10) Here, x represents the first stage decisions. y ω represents the second stage decisions in scenario ω. g 0, g 1,ω, are smooth convex functions. Both the first and the second stage decisions are mixed-integer. Let I = {1, 2,, n} be the index set of a the first stage variabes. I 1 I is the subset for indices of the binary first stage variabes. Let J = {1, 2,, m} be the index set of a the second stage variabes. J 1 J is the subset for the indices of the binary second stage variabes. x ub is a vector that represents the upper bound of a the first stage variabes. y ub is a vector that represents the upper bound of a the second stage variabes. In this paper, we assume probem (P ) has reativey compete recourse, i.e., any soution x that satisfies the first stage constraints has feasibe recourse decisions in the second stage. The feasibe region of (P ) is assumed to be compact. As (P ) has mixed-integer recourse, generaized Benders decomposition (GBD) [15] cannot be appied to sove it directy as in the case of convex continuous recourse. Therefore, we construct convex reaxations of (P ) so that GBD can be appied to sove the probem. We first define the set, S ω = { (x, y ω A 1,ω x + g 1,ω (y ω ) b 1,ω, y ω Y, 0 x x ub} (11) where the second stage constraints and the upper and ower bound constraints for the first stage decisions are incuded for each scenario ω. The fuspace probem with (x, y ω ) in the convex hu of S ω for a ω Ω is defined as (P C): (P C) min c T x + ω Ω τ ω d T ω y ω (12) A 0 x b 0, g 0 (x) 0 (13) x X (14) (x, y ω ) conv(s ω ) ω Ω (15) Since the integraity constraints in the second stage are reaxed, (P C) is a reaxation of (P ). Specificay, conv(s ω ) can be characterized by appying the hierarchy of reaxations for noninear convex generaized disjunctive programs proposed by Ruiz and Grossmann [27].

6 6 Can Li, Ignacio E. Grossmann As a subset of the second stage variabes y ω are binary, i.e., (y ω ) j {0, 1}, j J 1, the set S ω is equivaent to the disjunctions in (16). A 1,ω x + g 1,ω (y ω ) b 1,ω A 1,ω x + g 1,ω (y ω ) b 1,ω 0 x x ub 0 y ω y ub 0 x x ub 0 y ω y ub j J 1 (16) (y ω ) j = 1 (y ω ) j = 0 We represent (16) as Sωj 1 S0 ωj, j J 1. (16) is a conjunctive norma form (CNF) representation of S ω where S ω is represented as a conjunction over the disjunctions, i.e., S ω = j J1 (Sωj 1 S0 ωj ). The hu reaxation of the CNF, h re ( j J1 (Sωj 1 S0 ωj )), is equivaent to j J1 conv(sωj 1 S0 ωj ). We know that j J1 conv(sωj 1 S0 ωj ) conv( j J 1 Sωj 1 S0 ωj ) = ω). Therefore, the hu reaxation of the CNF is a reaxation of conv(s ω ). In order to construct conv(s ω ), we appy the basic step proposed by Baas [3] to the CNF (16). Based on the Theorem 1, the CNF can be converted to the foowing DNF shown in (17), A 1,ω x + g 1,ω (y ω ) b 1,ω r R 0 x x ub 0 y ω y ub (17) (y ω ) j = e rj j J 1 where R is the set that corresponds to a the possibe combinations of the binary variabes (y ω ) j, j J 1. For each r R, e rj can be either zero or one. There are 2 J1 components in the set R. Ceria and Soares [9] proved that the convex hu of the DNF (17) can be expressed as the projection of a higher dimensiona convex set where (x, y ω ) has to satisfy the foowing constraints: x = r R u r ω y ω = r R v r ω γω r = 1, 0 γω r 1, r R r R A 1,ω u r ω + γωg r 1,ω (vω/γ r ω) r b 1,ω γω, r 0 u r ω x ub γω, r r R 0 vω r y ub γω, r r R (v ω ) j = e rj γω r j J 1, r R r R (18) According to Ruiz and Grossmann [27], (x, y ω ) is in conv(s ω ) if and ony if it satisfies the constraints in (18). Since the perspective function of g 1,ω used in (18) can give rise to numerica difficuties at γ r ω = 0, the approximation proposed by Furman et a. [12] is used in our impementation, γ r ωg 1,ω (v r ω/γ r ω) ( (1 κ)γ r ω +κ ) g 1,ω ( v r ω (1 κ)γ r ω + κ ) κg 1,ω (0)(1 γ r ω) (19)

7 Tite Suppressed Due to Excessive Length 7 where κ is a sma number ike We shoud note that this approximation is exact at γ r ω = 0 and γ r ω = 1. As we are abe to construct conv(s ω ) for a ω Ω in cosed-form, GBD can be appied to sove (P C) to ɛ-optimaity. We briefy describe how GBD can be used to sove (P C), the reader can refer to Geoffrion [15] for detais. The basic idea of GBD is to decompose (P ) into a Benders master probem which ony contains first stage decisions and Benders subprobems each of which ony contains second stage decisions for a given scenario. We first define the Benders master probem (MB h ) at iteration h of the GBD agorithm. (22) are Benders cuts that are generated up unti iteration h, which are derived by soving the Benders subprobems (SB h ω). Benders master probem is a reaxation of (P C) projected on the x space. Therefore, a ower bound of (P C) is obtained by soving the MINLP Benders master probem: (MB h ) : min z h MB = ω η ω (20) s.t. A 0 x b 0, g 0 (x) 0 (21) η ω z h SB,ω + (λ h ω ) T (x x h ) + τ ω c T x ω Ω, h h (22) x X, η ω IR 1, ω Ω (23) Assume that the soution to the Benders master probem at iteration h is x h, x is fixed at x h in (P C). The rest of the probem can be decomposed into Benders subprobems (SB h ω). (SB h ω) : min z h SB,ω = τ ω d T ω y ω (24) A Benders cut can be generated at iteration h: s.t. x = x h (25) (x, y ω ) conv(s ω ) (26) η ω z h SB,ω + (λ h ω) T (x x h ) + τ ω c T x (27) where λ h ω are the optima dua mutipiers for constraints (25), zsb,ω h is the optima objective vaue of (SBω). h A feasibe soution to (P C) is obtained at the optima soution of the Benders subprobem. After soving the Benders subprobems at iteration h, GBD wi add the newy generated Benders cuts to the Benders master probem and keep iterating between the Benders master probem and the Benders subprobems unti (P C) is soved to ɛ-optimaity. Athough GBD can be appied to sove (P C), for the probem with both binary and continuous first stage variabes, Sherai and Zhu [32] showed that in the MILP setting (P C) is not aways equivaent to (P ), since for fixed first stage decision x, the extreme points of conv(s ω ) {(x, y) : x = x} wi not aways have binary vaues for y j, j J 1. As convex MINLP is a generaization of MILP, it is straightforward to show that (P C) is not aways equivaent to (P ).

8 8 Can Li, Ignacio E. Grossmann However, it aso shown in Sherai and Zhu [32] that in the MILP setting, under the restriction of pure binary first stage variabes, (P C) and (P ) are equivaent. Therefore, before we consider the proposed GBD-based agorithm to sove probem (P ) with both binary and continuous first stage variabes, the equivaence of (P ) and the corresponding (P C) with pure binary first stage variabes is proved in Proposition 1. The convergence of the GBD-based agorithm for the probem with mixed-integer first stage variabes wi be proved ater based on Proposition 1 and Coroary 1. Proposition 1 Consider a specia case of (P ) where the first stage variabes are a binary. We assume that in the corresponding probem (P C), the convex hu of S ω is expressed in the form of (18). Then (P ) and (P C) are equivaent in the sense that they have the same optima objective vaue and the optima soution of (P ) can aways be obtained based on the optima soution of (P C). Proof Note that (P C) is a reaxation of (P ). Hence the optima objective vaue of (P C) shoud be ess than or equa to (P ). Let (x, yω 1, yω 2,, yω Ω ) be the optima soution of (P C). Now if we seect any scenario ω, we have (x, yω) conv(s ω ). For this scenario ω, we have x = r R ur ω and yω = r R vr ω. As we have assumed that x are pure binary variabes and we have 0 u r ω eγω r, for any (x ) i = 1, we must have (u r ω ) i = γω r, otherwise r R (ur ω ) i wi not sum up to 1. For any (x ) i = 0, (u r ω ) i must be zero for a r. Therefore, for a γω r > 0, we have u r ω /γω r = x. Furthermore, for a γω r > 0, d T vω r /γω r = d T yω. Otherwise, there must exist r and r such that d T v r ω /γ r ω > d T yω and d T v r ω /γ r ω < d T yω. Then v r ω /γ r ω satisfies a the constraints of (P C) and eads to a ower objective vaue than yω, which contradicts with the fact that yω is the optima soution. Therefore, we can seect any γω r > 0, and et y ω = vω r /γω r to construct a feasibe soution that yieds the same cost in the objective as yω. We can foow the same procedure to construct the optima soution of every scenario ω Ω. Note that the soution that we construct satisfies a the constraints of (P ) and yieds the same objective vaue as the optima soution to (P C). In that sense, (P C) and (P ) are equivaent. The foowing theorem can be proved based on Proposition 1. Theorem 2 GBD can be appied to obtain the ɛ-optima soution of probem (P ) with pure binary first stage variabes by soving its reaxation (P C). Proof As we have assumed that the feasibe region of (P ) is nonempty compact and the probem has reativey compete recourse, conv(s ω ) is nonempty compact convex set for any fixed x, s.t. x X, A 0 x b 0, g 0 (x) 0, for every ω Ω. It foows directy from Theorem 2.5 of Geoffrion [15] that GBD has finite ɛ-convergence when appied to sove (P C). It is proved in Proposition 1 that (P ) and (P C) are equivaent for the probem with pure binary first stage variabes. Therefore, GBD can be appied to obtain the ɛ-optima soution of probem (P ) with pure binary first stage variabes by soving its reaxation (P C).

9 Tite Suppressed Due to Excessive Length 9 Now we go back to focus on the probem (P ) with both binary and continuous first stage variabes. A coroary can be derived based on Proposition 1. Coroary 1 For (PC) with both binary and continuous first stage variabes, if the optima first stage variabes x to (P C) are a at their upper or ower bound, i.e., (x ) i = 0 or (x ) i = (x ub ) i, i I, then (P C) and its corresponding (P) are equivaent in the sense that they have the same optima objective vaue. Proof The proof is simiar to the case where we have pure binary first stage variabes. As (x ) i = 0 or (x ) i = (x ub ) i, i, we must have u r ω /γω r = x for a γω r > 0. Therefore, we can construct a feasibe soution to (P ) by etting y ω = vω r /γω r where γω r > 0 for a ω Ω. The feasibe soution yieds the same objective vaue as the optima soution to (P C). If the condition of Coroary 1 does not hod, the equivaence of (P C) and (P ) cannot be guaranteed as we have discussed. In order to understand why (P C) and (P ) are not equivaent in the genera case, we can determine if the proof of Proposition 1 sti hods for the probem with mixed-integer first stage variabes. As we construct the convex hu of each scenario separatey, if we take u r ω /γω r for some γω r > 0 for each scenario ω as we did in Proposition 1 and Coroary 1, it is possibe that we fai to obtain a unique first stage decisions for a the scenarios if some (x ) i is not at its ower or upper bound. The reason why this occurs is that for component i of x such that (x) i does not ie at its upper or ower bound, there may exist one scenario ω for which we have (x ) i = r R (ur ω ) i but there exist r, r with γ r ω > 0, γ r ω > 0, (u r ω /γ r ω ) i < (x ) i, (u r ω /γ r ω ) i > (x ) i. As (u r ω /γω r ) i, γω r > 0, is not aways equa to (x ) i, the uniqueness of x obtained by taking u r ω /γω r for some γω r > 0 for each scenario ω cannot be guaranteed. However, if we branch on the continuous first stage variabes x, for exampe, by forcing (x) i x i at a given branch-and-bound node and furthermore in the construction of convex hu (u r ω) i (x) i γr ω are added, the soution where (u r ω /γ r ω ) i < (x ) i wi be cut off. Inspired by the work of Sherai and Zhu [32], we propose a spatia branchand-bound agorithm where we branch on the continuous first stage variabes. At each node q of the branch-and-bound tree. The foowing probem (P CBAB q ) is soved, (P CBAB q ) min c T x + ω Ω τ ω d T ω y ω (28) A 0 x b 0, g 0 (x) 0 (29) x b q x x ub q (30) x X (31) (x, y ω ) conv(sω) q (32) Sω q = { (x, y ω A 1,ω x + g 1,ω (y ω ) b 1,ω, y ω Y, x b q x x ub } q (33)

10 10 Can Li, Ignacio E. Grossmann Note that the ony difference between Sω q and S ω is that the first stage decisions x is constrained in [x b q, x ub q ] in Sω. q As we assume that probem (P ) has reativey compete recourse, conv(sω) q is a nonempty compact convex set for any fixed x, s.t. x X, x b q x x ub q, A 0 x b 0, g 0 (x) 0, for every ω Ω. Therefore, at each node q of the branch-and-bound tree, GBD can be appied to sove (P CBAB q ) to ɛ-optimaity in the same way that we sove (P C). We denote the optima objective of (P CBAB q ) as v(p CBAB q ). The optima soution to (P CBAB q ) is (x q, yqω 1, yqω 2,, yqω Ω ). A heuristic agorithm can be appied to find a feasibe soution to probem (P ), whie x has to satisfy the upper and ower bound constraints at node q. Heuristic 1: Fix the first stage decisions at the optima soution of (P CBAB q ), x q. Sove the subprobems with integraity constraints in parae with some MINLP sovers, such as DICOPT [34], Pajarito [23], SBB [6], AphaECP [38]. Note that we assume reativey compete recourse. Therefore, a feasibe soution to (P ) can be obtained at node q, which is denoted as (x f q, yqω f 1,yqω f 2,,yqω f Ω ). We denote the objective of the feasibe soution found at node q as V (P CBAB q ). Note that v(p CBAB q ) and V (P CBAB q ) are vaid ower and upper bounds for the origina probem (P ), respectivey, if the domain of x is restricted to [x b q, x ub q ]. In the GBD-based branch and bound agorithm, if V (P CBAB q ) v(p CBAB q ) + ɛ, node q can be fathomed by optimaity. We denote the goba ower bound as v, the goba upper bound as V. If v(p CBAB q ) V, node q is fathomed by bound. Otherwise, we seect one component of the continuous first stage variabes to branch on. The foowing branching rue is used to seect one component of the continuous first stage variabe to branch on. Branching rue A: At node q, cacuate the distance of each component of the optima continuous first stage variabes to its bounds at node q. σ i = min{(x q) i (x b q ) i, (x ub q ) i (x q) i }, i I\I 1 (34) The variabe with maximum distance to its bounds is seected to branch on. p = arg max σ i (35) i I\I 1 The node q is partitioned into two new nodes q 1, q 2. Constraints (x b q ) p (x) p (x q) p and (x q) p (x) p (x ub q ) p are added to node q 1 and q 2 respectivey. The steps of the GBD-based branch-and-bound agorithm are outined as foows. Agorithm GBDBAB Step 0: Initiaization Step. Initiaize the goba upper bound V = +, the goba ower bound v =, the iteration counter k = 1, the ist of active nodes L k = {q 0 }. Set x b q = 0, x ub 0 q = x ub. Let ɛ 0 be a seected optimaity 0 toerance. Use GBD to sove the probem at the root node q 0 to ɛ-optimaity. Let v = v(p CBAB q 0). Appy a heuristic agorithm ike heuristic 1 to obtain a feasibe soution to (P ) denoted as (x f q 0, y f q 0 ). Let the objective vaue of the

11 Tite Suppressed Due to Excessive Length 11 feasibe soution at node q 0 be V q0. Set V = V q 0. If V v + ɛ, stop. Otherwise, go to step 1. Step 1: Branching Step. Seect the node q in the active node ist with the minimum ower bound v(p CBAB q ). Branch on first stage variabe (x) p according to the Branching rue A. Create two new nodes q 1, q 2 and add constraints (x b q ) p (x) p (x q) p and (x q) p (x) p (x ub q ) p to node q 1 and q 2 respectivey. Let k = k + 1. Deete node q in L k and add node q 1, q 2 to L k. Go to step 2. Step 2: Bounding Step. Sove the two probems (P CBAB q1 ) and (P CBAB q2 ) using generaized Benders decomposition to ɛ-optimaity respectivey. Update the goba ower bound v with v = min q Lk v(p CBAB q ). A heuristic agorithm is appied to obtain feasibe soutions at node q 1 and q 2 that yied objective vaues of V (P CBAB q1 ) and V (P CBAB q2 ) respectivey. Update the goba upper bound V if V (P CBAB q1 ) V or V (P CBAB q2 ) V. Go to Step 3. Step 3: Fathoming Step. For the two new nodes q i, i = 1, 2, fathom the node if V (P CBAB qi ) v(p CBAB qi ) + ɛ, i = 1, 2. Fathom any node q with v(p CBAB q ) + ɛ V. If the node ist L k becomes empty, stop. Otherwise go to step 1. In order to prove the finite ɛ-convergence of our proposed GBDBAB agorithm we make the foowing assumption. Assumption 1 ɛ > 0, δ ɛ > 0, such that for any node q with σ i δ ɛ, i I\I 1, V (P CBAB q ) v(p CBAB q ) +ɛ, i.e., the node is fathomed by optimaity if σ i δ ɛ, i I\I 1. Here σ i, i I\I 1, is defined in Branching rue A. V (P CBAB q ) is obtained by heuristic 1. As we have assumed that probem (P ) is bounded, the change of the first stage variabes can ony have finite change of vaue in the objective. As δ ɛ 0, node q can be fathomed by optimaity based on Coroary 1. Therefore, Assumption 1 aways hods for probems that are bounded. Proposition 2 If Assumption 1 hods, the agorithm GBDBAB has finite ɛ- convergence. Proof First we prove that for a given node q and any continuous first stage variabes (x) i in node q that are constrained in the interva [ ] (x b q ) i, (x ub q ) i where (x ub q ) i (x b q ) i δ ɛ, variabe i wi not be branched on again if branching rue A is used. Assuming that variabe i is branched on again when branching rue A is used, we have σ i σ i δ ɛ, i I\{I 1 i}. However, by Assumption 1, any node q with σ i δ ɛ, i I\I 1,is fathomed by optimaity. We reach a contradiction. Therefore, in the worst case, the domain of each continuous first stage variabe wi be partitioned into intervas with ength δ ɛ. There wi be a finite number of hyperrectange partitions for the domain of the continuous first stage variabes [0, x ub ], i.e., a finite number of nodes in the branch-andbound tree. In each node, GBD can converge in a finite number of iterations

12 12 Can Li, Ignacio E. Grossmann according to Geoffrion [15]. Thus, the agorithm GBDBAB is abe to converge to ɛ-optimum in a finite number of iterations. Remark 1 Here, in order to prove the convergence of the agorithm, we assume that the convex hu of S q ω is constructed by the hierarchy of reaxations proposed by Ruiz and Grossmann [27]. If there are ony a few binary variabes in the second stage, the representation of the convex hu is tractabe. However, if the number of binary variabes in the second stage is arge, it is computationay expensive to represent the convex hu since there can be 2 J1 disjuncts. Therefore, a more imited sequentia convexification approach shoud be appied, which wi be discussed in section 5. Remark 2 Note that at each branching step, the chid nodes generated are restrictions of their parent node. The Benders cuts in the master probem of the parent node can by inherited by the chid nodes, which wi make the GBD agorithm for the chid node require fewer iterations to converge compared to running it from scratch. An iustrative exampe The foowing probem is soved by GBDBAB as an iustrative exampe. min x 1 +x 2 +3x 3 +3x 4 + ( ) τ ω y1ω 12y 2ω +100y 3ω +3y 4ω 3y 5ω (36) ω=ω 1,ω2 x 1 4x 3, x 2 2x 4, (37) x 1, x 2 0 x 3, x 4 {0, 1} (38) y 1ω x 1, y 2ω x 2 ω = ω 1, ω 2 (39) (y 1ω 3) 2 + (y 2ω 2) (1 y 4ω ) ω = ω 1, ω 2 (40) (y 1ω 1) 2 + y 2 2ω y 4ω ω = ω 1, ω 2 (41) y 2 1ω + (y 2ω 1) (1 y 5ω ) ω = ω 1, ω 2 (42) (y 1ω 4) 2 + (y 2ω 1) y 5ω ω = ω 1, ω 2 (43) y 1ω + y 2ω + y 3ω d ω ω = ω 1, ω 2 (44) y 1ω, y 2ω, y 3ω 0, y 4ω, y 5ω {0, 1} ω = ω 1, ω 2 (45) where x 1, x 2, x 3, x 4 are the first stage variabes, y 1ω, y 2ω, y 3ω, y 4ω, y 5ω are the second stage variabes. Both the first and the second stage variabes are mixedinteger. τ ω1 = τ ω2 = 0.5. d ω is the uncertain parameter. d ω1 = 1.5, d ω2 = 2. Here we describe how this probem can be soved by GBDBAB. Reca that at each node q of the branch and bound tree, GBD is appied to sove probem (P CBAB q ). For this probem, three nodes are visited in order to sove the probem to a reative optimaity gap within 0.1% (see Figure 1). At each node, the ower bound v(p CBAB q ) is the ower bound obtained by the GBD agorithm, i.e., the objective vaue of the Benders master probem in the ast iteration. The upper bound V (P CBAB q ) is obtained by using Heuristic 1.

13 Tite Suppressed Due to Excessive Length 13 v(p CBAB q ), V (P CBAB q ), the optima soution at each node are shown in Tabe 1. The Benders cuts that are generated at a the nodes are shown in Tabes 9-11 in Appendix 1. At the root node, 15 iterations are needed for the GBD agorithm to converge. However, the reative optimaity gap of the goba upper and ower bound that can be obtained at the root node is not within 0.1%. Therefore, we seect one continuous first stage variabe to branch on according to branching rue A. x 1 is branched on and two new nodes 1 and 2 are created. GBD is appied to sove the probem at nodes 1 and 2, respectivey. Both nodes 1 and 2 inherit the Benders cuts of the root node because the probems soved at this two nodes are restrictions of the root node and the Benders cuts of the root node are sti vaid. As a resut, ony 3 and 2 iterations are needed for the GBD to converge at nodes 1 and 2, respectivey (see Tabes 10, 11). The upper and the ower bounds of nodes 1 and 2 are within 0.1% optimaity gap. Nodes 1 and 2 can be fathomed by optimaity and GBDBAB terminates. An optima vaue of is obtained. However, if the hu reaxation of the CNF is used, which is a reaxation of conv(s q ω), the ower bound that can be obtained at the root node is The DNF yieds a much tighter ower bound than the CNF at the root node, which shows the impact of appying the basic step. x " x " Fig. 1 Branch and bound tree for the iustrative exampe Tabe 1 Optima soution, upper and ower bound at each node of the BAB tree Node v(p CBAB q) V (P CBAB q) x 1 x 2 x 3 x 4 Branching constraints x x A sequentia convexification scheme to sove (P CBAB q ) In soving probem (P CBAB q ), we assume that the convex hus of S q ω, ω Ω are constructed by appying the basic steps and converting each GDP from a CNF to a DNF. However, the number of disjuncts can be arge if we have a

14 14 Can Li, Ignacio E. Grossmann arge number of binary variabes in the second stage decisions. The number of variabes that are needed to represent the convex hu grows exponentiay with the number of binary variabes in the second stage decisions. Therefore, we propose a sequentia convexification scheme, which progressivey appies the basic steps to the CNF representation of S q ω. As we wi see ater, in some cases, it is not necessary to convert the CNF to DNF by appying a the basic steps. Namey, in some cases appying part of but not a the basic steps is sufficient to sove (P CBAB q ). The partia appication of a the possibe basic steps can be regarded as a sequentia convexification scheme to sove (P CBAB q ), which wi be discussed next. Simiary to (16), we define the disjunction that specifies the jth binary variabe as S q1 ωj Sq0 ωj, j J 1. The CNF of S q ω can be denoted as j J1 (S q1 ωj S q0 ωj ). In the sequentia convexification scheme to sove (P CBAB q), we start with the hu reaxation of the CNF, and progressivey appy basic steps to construct tighter reaxations if the probem (P CBAB q ) is not soved. Before we describe the detais of the sequentia convexification scheme, we first define some notation. A partia appication of basic steps to the CNF j J1 (S q1 ωj Sq0 ωj ) can be represented as Sq ω ( = t T q ω j D q (Sq1 ωt ωj Sq0 ωj )), where Tω q is the set of conjuncts, D q ωt is the set of the indices of the binary variabes specified by conjunct t. D q ωt are disjoint sets and t T q ω D q ωt = J 1, i.e., the vaue of each binary variabe j J 1 has to be specified in one and ony one of the conjuncts. For each t Tω, q j D q (Sq1 ωt ωj Sq0 ωj ) can be represented in DNF by appying the basic steps, j D q (Sq1 ωt ωj Sq0 ωj ) = r R q Sq ωt ωtr. R q ωt is the set of disjuncts for the DNF representation. In each Sωtr, q the vaues of a (y ω ) j, j Dωt, q are specified to 0 or 1. Note that the dimension of R q ωt is 2 Dq ωt, which corresponds to a possibe combinations of the binary variabes in Dωt. q For the CNF, Tω q = J 1 and Dωt q = 1. If we appy a the possibe basic steps to convert the CNF to the corresponding DNF, Tω q = 1 and Dωt q = J 1. The hu reaxation of t T q ω ( r R q Sq ωt ωtr) is given in (46)-(53), x = u r ωt, t Tω q (46) r R q ωt y ω = v r ωt, t T q ω (47) r R q ωt r R q ωt γ r ωt = 1, t T q ω (48) γ r ωt 0, t T q ω, r R q ωt (49) A 1,ω u r ωt + γ r ωtg 1,ω (v r ωt/γ r ωt) b 1,ω γ r ωt, t T q ω, r R q ωt (50) x b q γ r ωt u r ωt x ub q γ r ωt, t T q ω, r R q ωt (51) 0 v r ωt y ub γ r ωt, t T q ω, r R q ωt (52) (v r ωt) j = (e r ωt) j γ r ωt, t T q ω, r R q ωt, j D q ωt (53)

15 Tite Suppressed Due to Excessive Length 15 Instead of soving (P CBAB q ) directy, we can sove a reaxation of (P CBAB q ) where (x, y ω ) h re ( t T q ω ( r R q Sq ωt ωtr) ). Probem (P CBAB q ) is soved in iterations where the number of iteration is denoted by. In the first iteration, = 1, the probem with (x, y ω ) h re( j J1 (S q1 ωj Sq0 ωj ), ω Ω, is soved. We denote the reaxation of (P CBAB q ) that is soved at iteration as (P CBABq). The hu reaxation of t T q ω ( r R q Sq ωt ωtr) at iteration for scenario ω is denoted as h re(s q ω ). (P CBAB q) is formay defined by (54)- (57). (P CBAB q) min c T x + ω Ω τ ω d T ω y ω (54) A 0 x b 0, g 0 (x) 0 (55) x b q x x ub q (56) (x, y ω ) h re(s q ω ), ω Ω (57) (P CBABq) can be soved using GBD to ɛ-optimaity in a finite number of steps. The optima vaue of (P CBABq) is denoted as v(p CBABq). The optima soution is denoted as (x q ω 1 ω 2, ω Ω ). If we can prove that (x q ω 1 ω 2, ω Ω ) satisfies a the constraints of (P CBAB q ), then (x q ω 1 ω 2, ω Ω ) soves (P CBAB q ). More specificay, if (x q ω ) conv(sω), q ω Ω, (x q ω 1 ω 2, ω Ω ) soves (P CBAB q ). However, deciding whether (x q ω ) is in conv(sq ω) is not an easy probem as we do not want to use the cosed-form representation of conv(sω), q since this representation coud be expensive when the number of second stage binary variabes is arge. Here, we provide a quick way to prove (x q ω ) conv(sq ω) when some conditions are satisfied, which is ony a sufficient condition for (x q ω ) to be in conv(sq ω). Note that it is easy to check if a point is in Sω q just by checking if this point satisfies a the constraints that define Sω. q Therefore, the basic idea of this procedure is that if we coud find that (x q ω ) can be expressed as convex combination of some points that are in Sω, q then (x q ω ) is in conv(sω). q More specificay, note that after we sove (P CBABq), we aso obtain vωt, r γωt, r ω Ω, t Tω, q r Rωt, q where y q ω = r R q vωt, r ω Ω, ωt t Tω. q We can estabish a sufficient condition on (x q ω ) conv(sq ω) by inspecting the vaues of v r ωt, γ r ωt in Proposition 3. Proposition 3 For a given scenario ω, if t Tω q such that (vωt r /γr ωt ) j, r R q ωt, γr ωt > 0, j J 1, are 0 or 1, i.e., v r ωt /γr ωt satisfy the integraity constraints in Sω, q we have (x q ω ) conv(sq ω). Proof By construction, (u r ωt /γr ωt,vr ωt /γr ωt ), r Rq ωt, γr ωt > 0 aready satis- ωt ), fies the continuous constraints that define Sω. q Therefore, (u r ωt /γr ωt,vr ωt /γr r R q ωt, γr ωt > 0 are in Sq ω. (x q ω ) can be expressed as convex combination of (u r ωt /γr ωt, vr ωt /γr ωt ), r Rq ωt, γr ωt > 0, i.e., (xq ω ) conv(sq ω).

16 16 Can Li, Ignacio E. Grossmann Note that Proposition 3 is ony a sufficient condition for (x q ω ) conv(sq ω). If we can prove that (x q ω ) conv(sq ω), ω Ω, (P CBAB q ) is soved. Otherwise, for the scenarios where we cannot prove that (x q ω ) conv(sq ω), we appy basic steps by updating Tω, q D q ωt to construct tighter reaxations unti we are abe to prove (x q ω ) conv(sq ω), ω Ω. Here, we propose two heuristics on how to appy the basic steps. Heuristic A Note that in Proposition 3, in order to prove (x q ω ) conv(sq ω) we need to find one disjunction t such that (vωt r /γr ωt ) j, j J 1, γωt r > 0, satisfy the integraity constraints. Therefore, in this heuristic, for a given scenario ω and for each disjunction t Tω, q we first identify the most fractiona (vωt/γ r ωt) r j, r Rωt, q γωt r > 0, j J 1 and denote it as f t. f t = max min{1 (vωt/γ r ωt) r j, (vωt/γ r ωt) r j } (58) r R q ωt,γr ωt >0,j J1 We aso denote the index of the most fractiona (v r ωt/γ r ωt) j as g t, g t = arg max min{1 (vωt/γ r ωt) r j, (vωt/γ r ωt) r j } (59) j:r R q ωt,γr ωt >0,j J1 In the sufficient condition given by Proposition 3, we try to find disjunction t such that f t is cose to zero, i.e., a the (vωt r /γr ωt ) j, j J 1, γωt r > 0 satisfy the integraity constraints. Here, we sighty abuse notation and denote the disjunction t with the east f t as t, t = arg min f t (60) t Tω q If f t is cose to zero, we can prove (x q ω ) conv(sq ω) by Proposition 3. Otherwise, we appy a basic step between t and the disjunction that specifies g t. The idea is to make the most fractiona (vωt r /γr ωt ) j in disjunction t satisfy the integraity constraint in the next iteration. Remark 3 For the probem with a arge number of binary variabes in the second stage, the sequentia convexification scheme coud potentiay reduce the computationa time by avoiding appying unnecessary basic steps. Whie the sequentia convexification scheme is ess expensive than using the DNF representation directy, soving (P CBABq) in the first iteration with the CNF representation can aso be expensive. Therefore, one may start with the NLP reaxation of Sω q0 and start using the CNF representation when the ower bound cannot be improved by the NLP reaxation anymore. In fact, this strategy is used when we sove the panning probem using GBDBAB with sequentia convexification scheme in section 6. 6 Computationa resuts In this section, computationa resuts of three different probems are presented to demonstrate the effectiveness of GBDBAB. The toerance for the reative

17 Tite Suppressed Due to Excessive Length 17 optimaity gap for GBDBAB is set to 0.1%. The GBDBAB agorithm to sove the three probems with increasing number of scenarios is impemented in JuMP/Juia [11]. A the probems are soved on the 12 processors of an Inte Xeon (2.67GHz) machine with 64 GB RAM. Depending on the computationa efficiency on different probems, different NLP sovers incuding Knitro [7], and IPOPT [35] are used to sove the NLP subprobems in parae. The choice of NLP sovers for each probem are discussed in detai in the subsequent subsections. CPLEX [10] is used to sove the MILP master probems. The corresponding deterministic equivaent formuations (DEFs) are impemented in GAMS and soved using convex MINLP sovers incuding DICOPT, AphaECP, and SBB. 6.1 Computationa resuts of the iustrative exampe with quadratic constraints In section 4, an iustrative exampe with 2 scenarios is soved using GBDBAB. The convex noninear constraints in the mode are a quadratic as shown in (40)-(44). d ω is the ony uncertain parameter in the probem. In this section, we generate stochastic convex MINLP probems with increasing number of scenarios by uniformy samping d ω in the interva [0, 3]. We assume that a the scenarios are of equa probabiity. The time imit for a DEFs of this sma probem is set to 10,000 secs. The sizes of the DEFs, the tota running time and the reative optimaity gap of each sover are shown in Tabe 2. Athough AphaECP and DICOPT are abe to sove the 20-scenario probem to optimaity within 10 secs, the computationa time increases dramaticay with the increase in the number of scenarios. For instance, for the DEF with 300 scenarios, neither SBB nor DICOPT is abe to obtain the optima soution within the time imit. AphaECP soves the probem in 917 secs. Tabe 2 Computationa statistics of the DEFs of the iustrative probem Scenarios Linear Noninear Binary Continuous SBB AphaECP DICOPT Constr Constr Var Var sec(gap) sec(gap) sec(gap) Timed out(7%) Timed out(234%) Timed out(245%) Timed out(247%) 917 Timed out(9%) GBDBAB is aso impemented to sove this iustrative probem. Since there are ony two binary variabes in the second stage, conv(s q ω) can be constructed easiy with the DNF representation. The NLP subprobems are soved using IPOPT in parae. The upper bound at each node is obtained by appying Heuristic 1 where the upper bound subprobems are soved using Pajarito. The tota wa time, the wa time corresponding to soving the master probems, subprobems and upper bound subprobems, and the number of nodes

18 18 Can Li, Ignacio E. Grossmann visited are shown in Tabe 3. In a the cases, ony three nodes are visited in order to sove the probems to optimaity due to the tight reaxations given by the DNF representation. It is easy to observe that the wa time for soving the subprobems is the most significant part of the tota wa time. For the probems with 150 and 300 scenarios, GBDBAB performs better than using the sovers to sove the DEFs. Tabe 3 Computationa statistics of soving the iustrative probem using GBDBAB Scenarios Time (s) Master (s) Subprobem (s) UB subprobem (s) Nodes Computationa resuts of the constrained ayout probem The constrained ayout probem under price uncertainty described in Appendix 2 is tested with 3 rectanges and 2 areas. Since the mode is formuated as a GDP, the DEFs of the mode are converted to MINLP with the big-m and the hu reformuation respectivey. The time imit is set to 10,000 secs. The sizes of the DEFs with the big-m and the hu reformuation are shown in Tabe 4 and Tabe 5 respectivey. Athough a the sovers can sove the DEFs with 3 and 9 scenarios within the time imit, they a fai to obtain the optima soution for the probems with 36 and 100 scenarios. Tabe 4 Computationa statistics of the big-m reformuation of the constrained ayout probem Scenarios Linear Noninear Binary Continuous SBB AphaECP DICOPT Constr Constr Var Var sec(gap) sec(gap) sec(gap) Timed out(94%) Timed out Timed out(84%) Timed out(98%) Timed out Timed out(97%) GBDBAB is aso impemented to sove the constrained ayout probems with increasing number of scenarios. In order to obtain conv(s q ω), constraint (64) and the constraints setting the upper and ower bound of a the variabes are added to each disjunct shown in (65). We further appy basic steps to convert the CNF to DNF and reformuate the DNF with hu reaxation, which gives us conv(s q ω). The NLP subprobems are soved using IPOPT in parae. No feasibe soution found in 10,000 secs time imit.

19 Tite Suppressed Due to Excessive Length 19 Tabe 5 Computationa statistics of the hu reformuation of the constrained ayout probem Scenarios Linear Noninear Binary Continuous SBB AphaECP DICOPT Constr Constr Var Var sec(gap) sec(gap) sec(gap) Timed out(52%) Timed out (46%) Timed out Timed out(82%) Timed out (67%) Timed out The statistics of GBDBAB to sove these probems are shown in Tabe 6. For the probems with 3, 36, and 100 scenarios, optimaity is proved at the root node. For the probem with 9 scenarios, 3 nodes are visited in order to prove optimaity. Most of the computationa time is sti in soving the NLP subprobems. Tabe 6 Computationa statistics of soving the constrained ayout probem with GBDBAB Scenarios Time (s) Master (s) Subprobem (s) UB subprobem (s) Nodes Computationa resuts of the panning probem The panning probem reported in Li and Grossmann [21] is tested with 2 suppiers, 2 pants, 2 customers, and the number of scenarios from 3 to 81. The time imit for soving the DEFs is set to 50,000 secs. For the 81-scenario probem, none of the sovers is abe to obtain the optima soution within the time imit. Tabe 7 Computationa statistics of the DEFs of the panning probem Scenarios Linear Noninear Binary Continuous AphaECP SBB DICOPT Constr Constr Var Var sec(gap) sec(gap) sec(gap) Timed out(2%) , , Timed out(19%) , ,918 Timed out(2%) Timed out(40%) Timed out(0.2%) GBDBAB is aso appied to sove the panning probem. The NLP subprobems are soved with Knitro in this case since it behaves best for this probem among the NLP sovers avaiabe in JuMP. GBDBAB is abe to sove a the panning probems at the root node. Sequentia convexification scheme

20 20 Can Li, Ignacio E. Grossmann with Heuristic A is appied and the maximum and the minimum number of basic steps that are appied in a the scenarios are shown in the ast coumn of Tabe 8. In each of the four cases, the number of basic steps needed varies for different scenarios. For some of the scenarios, we are abe to prove the soution (x, y ω ) is in conv(sω q0 ) by Proposition 3 and therefore no more basic steps need to be appied. The computationa statistics of GBDBAB are shown in Tabe 8. The probems with increasing number of scenarios can a be soved to optimaity by GBDBAB within 10,000 secs. Tabe 8 Computationa statistics of soving the panning probem using GBDBAB Scenarios Time (s) Master (s) Subprobem (s) UB subprobem (s) Basic step (max, min) (7,2) 9 1, (5,1) 27 1, , (0,0) 81 7,994 1,534 5, (1,0) Remark 4 The computationa resuts in this section demonstrate that GB- DBAB can outperform the convex MINLP sovers in soving the DEFs with a arge number of scenarios. However, for the first sma probem and the panning probem, some of the sovers can obtain soutions to the DEFs with sma optimaity gaps. The effect of using GBDBAB is most significant for the constrained ayout probem. First of a, it is reativey easy to construct the convex hu of each subprobem as there are ony 3 disjunctions in each subprobem. Second, since this probem is highy noninear, it woud take sovers that use inearization strategies ike DICOPT and AphaECP many iterations to converge. Third, the constrained ayout probem aso has poor NLP reaxation. Sovers that are sensitive to the quaity of NLP reaxation of the origina probem ike SBB woud need more iterations to converge. Therefore, GBDBAB is most effective in soving highy noninear probems with poor reaxations but with sma number of binary variabes in the second stage decisions. 7 Concusion In this paper, a generaized Benders decomposition-based branch-and-bound agorithm, GBDBAB, is proposed to sove two-stage convex mixed integer noninear stochastic programs with mixed-integer variabes in both first and second stage decisions. In order to obtain the convex hu of the mixed-integer noninear subprobems in cosed-form, each subprobem is first represented in conjunctive norma form (CNF), and we appy basic steps to transform CNF to disjunctive norma form (DNF). The convex hu of each subprobem can be represented by the hu reaxation of the DNF. For the probems with pure binary first stage variabes, we are abe to prove that GBD has finite

21 Tite Suppressed Due to Excessive Length 21 ɛ-convergence if the convex hu of each subprobem is constructed in cosedform. For the probems with mixed-integer first stage variabes, we aso prove that in order to have finite ɛ-convergence, we may need to branch on the first stage continuous variabes. A sequentia convexification scheme is proposed to adaptivey appy basic steps for the subprobems with a arge number of binary variabes. Three probems with increasing number of scenarios are soved with GB- DBAB to within 0.1% optimaity gap. Convex MINLP sovers incuding SBB, AphaECP, and DICOPT are used to sove the corresponding DEFs to benchmark the proposed agorithm. It is shown that the proposed agorithm outperforms the sovers for the probems with a arge number of scenarios. Especiay, the agorithm is preferabe for highy noninear probems with poor reaxations but with sma number of binary variabes in the second stage decisions ike the constrained ayout probem. Acknowedgements The authors gratefuy acknowedge financia support from the Center of Advanced Process Decion-making at Carnegie Meon University and from the Department of Energy as part of the IDAES Project. 8 Appendix 1: Benders cuts for the iustrative exampe The Benders cuts that are generated in each node of the branch-and-bound tree for the iustrative exampe in section 4 are shown in Tabes Appendix 2: Constrained ayout probem under price uncertainty The constrained ayout probem is adapted from the PhD thesis of Sawaya [28]. In this probem, we need to decide the ayout of some units represented by rectanges with known engths and widths. The units have to be manufactured before we insta them at certain ocations. In the manufacturing process, we need to provide the reative position of the units in order to produce the pipes that connect those units. The first stage decisions incude the designed reative position of the units. After the units are manufactured, they can be instaed at severa ocations. In order to insta some unit(s) at a given ocation, we need to purchase a circed area around the center of that ocation. The rectange(s) instaed at that ocation has to be constrained within the circe that is purchased. The price of each area is uncertain and the designed units are instaed according to the prices. The sets, parameters, and variabes that are used in the mode is defined as foows: Sets i N = rectanges (units) q Q = areas ω Ω = scenarios