A novel L-shaped refinement chain cuts method for two-stage stochastic programs

Andrea Spinelli; Francesca Maggioni; Mike Hewitt

arxiv: 2606.02469 · v1 · pith:6L5HQH5B · submitted 2026-06-01 · math.OC

A novel L-shaped refinement chain cuts method for two-stage stochastic programs

Mike Hewitt , Francesca Maggioni , Andrea Spinelli This is my paper

Reviewed by Pith2026-06-28 13:27 UTCgrok-4.3pith:6L5HQH5B open to challenge →

classification math.OC

keywords two-stage stochastic programmingL-shaped methodBenders decompositionscenario partitioningrefinement chainstochastic network designmean-risk optimization

0 comments

The pith

A refinement chain of scenario partitions generalizes the L-shaped method for two-stage stochastic programs

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the L-shaped refinement chain cuts method for two-stage stochastic programs. It partitions the full set of scenarios into subgroups at each level of the chain and solves one subproblem per subgroup to generate Benders cuts. This setup generalizes both the single-cut and multi-cut versions of the classical L-shaped approach. Convergence to the optimal solution is proven for every level of refinement, and relationships between levels are used to build an iterative algorithm that refines the partitions.

Core claim

The L-shaped refinement chain cuts method integrates a chain of scenario partitions into the L-shaped decomposition. At each level the scenario set is divided into subgroups, a subproblem is solved for each subgroup, and the generated cuts are valid for the original problem. The framework is shown to generalize the classical multi-cut and single-cut L-shaped formulations, to converge to the optimum at every refinement level, and to support an iterative algorithm that moves between levels by relating their Benders cuts.

What carries the argument

The refinement chain of scenario partitions, which defines a sequence of groupings where each level solves aggregated subproblems to produce valid cuts.

If this is right

Convergence to the optimal solution holds at every level of the refinement chain.
The method includes the classical single-cut and multi-cut L-shaped formulations as special cases.
Benders cuts from one level can be related to those of the next level to initialize the algorithm.
An iterative refinement-based algorithm can be constructed to solve across consecutive levels.
The approach performs well on two-stage stochastic fixed-charge multicommodity network design problems under mean-risk.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Choosing a coarser partition level reduces the number of subproblems solved at the expense of potentially weaker cuts.
The refinement idea could be combined with other acceleration techniques such as cut strengthening.
Similar chains might be defined for multi-stage or distributionally robust variants of the problem.
The method's performance may depend on how the partitions are chosen at each level.

Load-bearing premise

That the optimality cuts obtained from subproblems on scenario subgroups remain valid when applied to the full scenario set and allow the master problem to converge to the true optimum.

What would settle it

Solving a small two-stage stochastic program with a known optimal objective value using the refinement chain method and checking whether the final solution matches that known optimum or the algorithm fails to converge.

read the original abstract

This paper introduces the L-shaped refinement chain cuts method, a novel approach for solving two-stage stochastic programs. The proposed method integrates the refinement chain of scenarios within the classical L-shaped decomposition framework. In the proposed approach, the full scenario set is partitioned into subgroups at each level of the refinement chain, and one subproblem is solved for each subgroup rather than for each individual scenario as in the classical L-shaped method. The proposed framework generalizes both the classical multi-cut and single-cut L-shaped formulations. Theoretical convergence properties to the optimal solution of the original two-stage stochastic program are established for every refinement level. In addition, the relationships between consecutive refinement levels are characterized in terms of Benders cuts, leading to the development of an iterative refinement-based solution algorithm across consecutive levels of the refinement chain. The effectiveness of the proposed method is evaluated on a two-stage stochastic fixed-charge multicommodity network design problem under a mean-risk formulation. Computational experiments on benchmark instances demonstrate the promising performance of the proposed framework and highlight its applicability to large-scale risk-averse stochastic optimization problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces a grouped-scenario L-shaped variant that claims multi-level convergence, but the validity of the cuts rests on an unshown aggregation step that the abstract leaves unclear.

read the letter

The core idea is to run L-shaped decomposition on a chain of scenario partitions, solving one subproblem per subgroup at each refinement level rather than one per scenario. The authors say this generalizes both single-cut and multi-cut versions, that convergence to the original optimum holds at every level, and that the cuts between consecutive levels can be related directly so an iterative algorithm can move up the chain.

What stands out is the attempt to trade off the number of subproblems against solution quality in a controlled way. For large risk-averse instances the reduction in subproblem count could matter, and the computational section on the fixed-charge multicommodity network design problem apparently shows competitive run times on the benchmarks they tried.

The load-bearing piece is whether the single subgroup subproblem actually produces a valid supporting hyperplane to the true expected recourse function. The abstract does not give the explicit cut formula or the weighting used inside the subgroup problem. If the aggregation is not done so that the cut remains a valid outer approximation of the full expectation, then the claimed convergence at every refinement level does not follow. The stress-test concern therefore lands directly on the central claim; nothing in the provided abstract resolves it.

The paper is aimed at researchers who already work with Benders-type decompositions for two-stage stochastic programs and want a tunable middle ground between single-cut and multi-cut extremes. A reader who needs a new practical algorithm for mean-risk network design problems could extract something usable if the cut construction holds up.

It is worth sending to a serious referee. The idea is coherent on its own terms and the experimental setting is standard, but the referee will need to see the precise subgroup formulation and the convergence argument before the generalization claim can be accepted.

Referee Report

2 major / 0 minor

Summary. The paper proposes the L-shaped refinement chain cuts method for two-stage stochastic programs. Scenarios are partitioned into subgroups at successive refinement levels, with one subproblem solved per subgroup rather than per scenario. The framework is claimed to generalize both the classical multi-cut and single-cut L-shaped methods, with theoretical convergence to the optimum of the original problem guaranteed at every refinement level. Relationships between consecutive levels are characterized via Benders cuts, yielding an iterative refinement algorithm. The method is tested on a mean-risk two-stage stochastic fixed-charge multicommodity network design problem, with computational results on benchmark instances.

Significance. If the cut-validity and per-level convergence claims hold, the approach would supply a tunable intermediate between single-cut and multi-cut L-shaped methods, potentially useful for large-scale risk-averse stochastic programs where the number of scenarios is prohibitive. The explicit characterization of cut relationships across refinement levels would also be a useful structural contribution.

major comments (2)

[Abstract / Method description] The load-bearing claim that a single subgroup subproblem yields a valid supporting hyperplane to the true expected recourse function (thereby preserving convergence at every refinement level) is asserted in the abstract but receives no explicit formulation, weighting scheme, or aggregation rule in the provided description; without this construction the generalization to single-cut and multi-cut L-shaped methods cannot be verified.
[Abstract / Theoretical results] The abstract states that 'theoretical convergence properties … are established for every refinement level' and that 'relationships between consecutive refinement levels are characterized in terms of Benders cuts,' yet supplies neither a proof outline, key lemmas, nor the explicit cut formulas; these omissions make the central convergence guarantee impossible to assess from the given information.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address each major comment below, noting that the full manuscript contains the requested formulations, lemmas, and proofs (as the abstract is a summary). We indicate where abstract revisions could be made for clarity.

read point-by-point responses

Referee: [Abstract / Method description] The load-bearing claim that a single subgroup subproblem yields a valid supporting hyperplane to the true expected recourse function (thereby preserving convergence at every refinement level) is asserted in the abstract but receives no explicit formulation, weighting scheme, or aggregation rule in the provided description; without this construction the generalization to single-cut and multi-cut L-shaped methods cannot be verified.

Authors: The abstract is concise by design, but Section 3.2 of the manuscript explicitly constructs the supporting hyperplane: the subgroup subproblem is solved once, and the resulting cut is valid for the true expected recourse function via probability-weighted aggregation (weight of each scenario equals its probability divided by the subgroup total probability). This is derived from the standard Benders cut by linearity of expectation. The construction directly specializes to multi-cut (singleton subgroups) and single-cut (one subgroup) cases. We can revise the abstract to include a one-sentence reference to this weighting if the editor requests. revision: partial
Referee: [Abstract / Theoretical results] The abstract states that 'theoretical convergence properties … are established for every refinement level' and that 'relationships between consecutive refinement levels are characterized in terms of Benders cuts,' yet supplies neither a proof outline, key lemmas, nor the explicit cut formulas; these omissions make the central convergence guarantee impossible to assess from the given information.

Authors: The full proofs appear in the manuscript body. Section 4 contains the key lemmas (Lemma 4.1: validity of subgroup cuts; Lemma 4.3: finite convergence at each fixed refinement level) and the main convergence theorem (Theorem 4.5). Section 5 gives the explicit inter-level cut formulas (Proposition 5.2) showing how a cut at level k dominates or is dominated by cuts at level k+1. These establish the claimed properties. The abstract summarizes rather than reproduces the proofs, which is standard; we can add a brief outline sentence to a revised abstract or introduction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents an algorithmic extension of the L-shaped method by partitioning scenarios into subgroups at each refinement level and solving one subproblem per subgroup. It claims generalization of single-cut and multi-cut variants plus per-level convergence to the original two-stage optimum, with inter-level Benders-cut relationships. No quoted equations or text in the abstract or description reduce any claimed result (convergence, cut validity, or relationships) to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The construction is presented as building on classical Benders properties applied to the new partitioning, without the derivation collapsing into its inputs by construction. This is the expected honest non-finding for an independent algorithmic proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or newly postulated entities; ledger left empty due to lack of technical detail.

pith-pipeline@v0.9.1-grok · 5714 in / 1150 out tokens · 33641 ms · 2026-06-28T13:27:20.644850+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

59 extracted references

[1]

INFORMS J

Bertsimas, D., Cory-Wright, R., Pauphilet, J., Petridis, P.: A stoch astic benders decomposition scheme for large-scale stochastic network design. INFORMS J. Comput. 37(5), 1163–1181 (2025)

2025
[2]

Benders, J.F.: Partitioning procedures for solving mixed-variable s programming problems. Numer. Math. 4(1), 238–252 (1962)

1962
[3]

Omega 126, 103045 (2024)

Belieres, S., Hewitt, M.: Hedging against uncertainty in transport ation network design through ﬂexible scheduling. Omega 126, 103045 (2024)

2024
[4]

Belieres, S., Hewitt, M.: Stochastic scheduled service network de sign problem with ﬂexible schedules: Mathematical formulations and benders decomp osition-based solution approach. Ann. Oper. Res., 1–25 (2026)

2026
[5]

Birge, J.R., Louveaux, F.V.: A multicut algorithm for two-stage sto chastic linear programs. Eur. J. Oper. Res. 34(3), 384–392 (1988)

1988
[6]

Springer, 44 New York (2011)

Birge, J.R., Louveaux, F.: Introduction to Stochastic Programm ing. Springer, 44 New York (2011)

2011
[7]

Bayraksan, G., Maggioni, F., Faccini, D., Yang, M.: Bounds for multis tage mixed- integer distributionally robust optimization. SIAM J. Optim. 34(1), 682–717 (2024)

2024
[8]

(eds.) Freight Railroad Service Network Design, pp

Chouman, M., Crainic, T.G.: In: Crainic, T.G., Gendreau, M., Gendron , B. (eds.) Freight Railroad Service Network Design, pp. 383–426. Springer, C ham (2021)

2021
[9]

Crainic, T.G., Frangioni, A., Gendron, B.: Bundle-based relaxation m ethods for multicommodity capacitated ﬁxed charge network design. Discret. Appl. Math. 112(1), 73–99 (2001). Combinatorial Optimization Symposium, Selecte d Papers

2001
[10]

Cavaliere, F., Fischetti, M., Roberti, R., Salvagnin, D.: Models and a lgorithms for the time window assignment traveling salesperson problem with stoch astic travel times. Eur. J. Oper. Res. 329(1), 96–111 (2026)

2026
[11]

Crainic, T.G., Gendron, B., Akhavan Kazemzadeh, M.R.: A taxonom y of mul- tilayer network design and a survey of transportation and telecom munication applications. Eur. J. Oper. Res. 303(1), 1–13 (2022)

2022
[12]

(eds.): Network Design with Applica- tions to Transportation and Logistics

Crainic, T.G., Gendreau, M., Gendron, B. (eds.): Network Design with Applica- tions to Transportation and Logistics. Springer, Cham (2021)

2021
[13]

Crainic, T.G., Hewitt, M., Maggioni, F., Rei, W.: Partial benders deco mposition: General methodology and application to stochastic network design . Transp. Sci. 55(2), 414–435 (2021)

2021
[14]

Christiansen, M., Hellsten, E., Pisinger, D., Sacramento, D., Vilhelm sen, C.: Liner shipping network design. Eur. J. Oper. Res. 286(1), 1–20 (2020)

2020
[15]

(eds.) Logistics Network Design, pp

Cordeau, J.-F., Klibi, W., Nickel, S.: In: Crainic, T.G., Gendreau, M., G endron, B. (eds.) Logistics Network Design, pp. 599–625. Springer, Cham ( 2021)

2021
[16]

Cavagnini, R., Maggioni, F., Bertazzi, L., Hewitt, M.: A two-stage s tochastic programming model for bike-sharing systems with rebalancing. EUR O J. Transp. Logist. 13, 100140 (2024)

2024
[17]

Dupaˇ cov´ a, J., Consigli, G., Wallace, S.W.: Scenarios for multistag e stochastic programs. Ann. Oper. Res. 100, 25–53 (2000)

2000
[18]

Springer, Cham (2024)

Dentcheva, D., Ruszczy´ nski, A.P.: Risk-averse Optimization an d Control. Springer, Cham (2024)

2024
[19]

(e d.) Barycentric Bounds in Stochastic Programming: Theory and Application, pp

Frauendorfer, K., Kuhn, D., Sch¨ urle, M.: In: Infanger, G. (e d.) Barycentric Bounds in Stochastic Programming: Theory and Application, pp. 67– 96. Springer, New York, NY (2011) 45

2011
[20]

Geoﬀrion, A.M.: Elements of large scale mathematical programmin g part ii: Synthesis of algorithms and bibliography. Manage. Sci. 16(11), 676–691 (1970)

1970
[21]

Geoﬀrion, A.M.: Elements of large-scale mathematical programm ming part i: Concepts. Manage. Sci. 16(11), 652–675 (1970)

1970
[22]

Høyland, K., Kaut, M., Wallace, S.W.: A heuristic for moment-match ing scenario generation. Comput. Optim. Appl. 24(2), 169–185 (2003)

2003
[23]

Hewitt, M., Ortmann, J., Rei, W.: Decision-based scenario cluster ing for decision- making under uncertainty. Ann. Oper. Res. 315(2), 747–771 (2022)

2022
[24]

(eds.) Stochastic Network Design, pp

Hewitt, M., Rei, W., Wallace, S.W.: In: Crainic, T.G., Gendreau, M., Gen dron, B. (eds.) Stochastic Network Design, pp. 283–315. Springer, Cha m (2021)

2021
[25]

Hosseini, M., Turner, J.: Deepest cuts for benders decomposit ion. Oper. Res. 73(5), 2591–2609 (2025)

2025
[26]

Mahmuto˘ gulları, A., C ¸ avu¸ s, O., Akt¨ urk, M..S.: Bounds on risk-averse mixed- integer multi-stage stochastic programming problems with mean-cv ar. Eur. J. Oper. Res. 266(2), 595–608 (2018)

2018
[27]

Jaoui, N., Klibi, W., El Hachemi, N., Aouam, T., Fender, M.: Multi-cycle produc- tion network design under supply uncertainty. Eur. J. Oper. Res. 330(3), 816–837 (2026)

2026
[28]

INFOR: Inf

Kaltis, T., Saharidis, G.K.D.: Literature review on benders cut sele ction and a multiple cut generation scheme. INFOR: Inf. Syst. Oper. Res., 1–2 7 (2025)

2025
[29]

(eds.) Barycentric approximation shcheme

Kuhn, D.: In: Beckmann, M., K¨ unzi, H.P., Fandel, G., Trockel, W., B asile, A., Drexl, A., Dawid, H., Inderfurth, K., K¨ ursten, W., Schittko, U. (eds.) Barycentric approximation shcheme. Lecture Notes in Economics and Mathemat ical Systems, pp. 51–81. Springer, Berlin (2005)

2005
[30]

Kall, P., Wallace, S.: Stochastic Programming vol. 46. John Wiley & So ns, Chichester (1994)

1994
[31]

Luo, F., Dey, S., Mehrotra, S.: A cutting-plane and benders’ de composition algo- rithm for two-stage distributionally robust convex programs. Mat h. Program., 1–64 (2025)

2025
[32]

Maggioni, F., Allevi, E., Bertocchi, M.: Bounds in multistage linear sto chastic programming. J. Optim. Theory Appl. 163, 200–229 (2014)

2014
[33]

Maggioni, F., Allevi, E., Bertocchi, M.: Monotonic bounds in multistag e mixed- integer stochastic programming. Comput. Manag. Sci. 13(3), 423–457 (2016)

2016
[34]

Maggioni, F., Cagnolari, M., Bertazzi, L., Wallace, S.W.: Stochastic o ptimization 46 models for a bike-sharing problem with transshipment. Eur. J. Oper . Res. 276(1), 272–283 (2019)

2019
[35]

In preparation (2026)

Maggioni, F., Cavagnini, R., Faccini, D.: Optimization-driven monoto nic bounds for stochastic programs with applications in ﬁnance and operations management. In preparation (2026)

2026
[36]

(eds.) Public Transportation, pp

Mauttone, A., Cancela, H., Urquhart, M.E.: In: Crainic, T.G., Gend reau, M., Gendron, B. (eds.) Public Transportation, pp. 539–565. Springer , Cham (2021)

2021
[37]

Melo, M.T., Nickel, S., Saldanha-da-Gama, F.: Facility location and su pply chain management – a review. Eur. J. Oper. Res. 196(2), 401–412 (2009)

2009
[38]

Maggioni, F., Pﬂug, G.C.: Bounds and approximations for multistag e stochastic programs. SIAM J. Optim. 26(1), 831–855 (2016)

2016
[39]

Maggioni, F., Pﬂug, G.C.: Guaranteed bounds for general nondis crete multistage risk-averse stochastic optimization programs. SIAM J. Optim. 29(1), 454–483 (2019)

2019
[40]

I n preparation (2026)

Maggioni, F., Romeijnders, W., Spinelli, A.: Lower bounds for two-s tage stochas- tic programming problems with decision-dependent uncertainties. I n preparation (2026)

2026
[41]

Narum, B.S., Maggioni, F., Wallace, S.W.: On the safe side of stochas tic pro- gramming: bounds and approximations. Int. Tran. Oper. Res. 30(6), 3201–3237 (2023)

2023
[42]

Rahmaniani, R., Crainic, T.G., Gendreau, M., Rei, W.: The benders de com- position algorithm: A literature review. Eur. J. Oper. Res. 259(3), 801–817 (2017)

2017
[43]

Rahmaniani, R., Crainic, T.G., Gendreau, M., Rei, W.: Accelerating th e benders decomposition method: Application to stochastic network design pr oblems. SIAM J. Optim. 28(1), 875–903 (2018)

2018
[44]

INFORMS J

Rei, W., Cordeau, J.-F., Gendreau, M., Soriano, P.: Accelerating b enders decomposition by local branching. INFORMS J. Comput. 21(2), 333–345 (2009)

2009
[45]

Roland, M., Forel, A., Vidal, T.: Adaptive partitioning for chance-c onstrained problems with ﬁnite support. SIAM J. Optim. 35(1), 476–505 (2025)

2025
[46]

Ram ´ ırez-Pico, C., Ljubi´ c, I., Moreno, E.: Benders adaptive-cuts method for two- stage stochastic programs. Transp. Sci. 57(5), 1252–1275 (2023)

2023
[47]

Ram ´ ırez-Pico, C., Moreno, E.: Generalized adaptive partition-b ased method for two-stage stochastic linear programs with ﬁxed recourse. Math. Program. 196(1), 755–774 (2022) 47

2022
[48]

Rockafellar, R.T., Uryasev, S.: Optimization of conditional value- at-risk. J. Risk 2, 21–42 (2000)

2000
[49]

Ruszczy´ nski, A.: Decomposition methods in stochastic progra mming. Math. Program. 79, 333–353 (1997)

1997
[50]

Spinelli, A., Bezzi, D., Jabali, O., Maggioni, F.: A stochastic electric ve hicle rout- ing problem under uncertain energy consumption. Transp. Res. Pa rt C Emerging Technol. 183, 105480 (2026)

2026
[51]

(ed.) Risk Aversion in Two-Stage S tochastic Integer Programming, pp

Schultz, R.: In: Infanger, G. (ed.) Risk Aversion in Two-Stage S tochastic Integer Programming, pp. 165–187. Springer, New York, NY (2011)

2011
[52]

Shapiro, A.: Analysis of stochastic dual dynamic programming me thod. Eur. J. Oper. Res. 209(1), 63–72 (2011)

2011
[53]

INFORMS J

Seo, K., Joung, S., Lee, C., Park, S.: A closest benders cut selec tion scheme for accelerating the benders decomposition algorithm. INFORMS J. Com put. 34(5), 2804–2827 (2022)

2022
[54]

Song, Y., Luedtke, J.: An adaptive partition-based approach f or solving two- stage stochastic programs with ﬁxed recourse. SIAM J. Optim. 25(3), 1344–1367 (2015)

2015
[55]

Spinelli, A., Maggioni, F., Ramos, T.R.P., Barbosa-P´ ovoa, A.P., Vigo, D.: A rolling horizon heuristic approach for a multi-stage stochastic wast e collection problem. Eur. J. Oper. Res. 323(1), 276–296 (2025)

2025
[56]

Trukhanov, S., Ntaimo, L., Schaefer, A.: Adaptive multicut aggr egation for two- stage stochastic linear programs with recourse. Eur. J. Oper. Re s. 206(2), 395–406 (2010)

2010
[57]

Van Slyke, R.M., Wets, R.J.-B.: L-shaped linear programs with applic ations to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)

1969
[58]

Wu, T., Zhang, J., Zhang, C., Liang, Z., Zhang, X.: Benders decom position for stochastic facility location and production planning. Comput. Oper. Res. 186, 107316 (2026)

2026
[59]

INFORMS J

Yuan, Y., Sen, S.: Enhanced cut generation methods for decom position-based branch and cut for two-stage stochastic mixed-integer program s. INFORMS J. Comput. 21(3), 480–487 (2009) 48

2009

[1] [1]

INFORMS J

Bertsimas, D., Cory-Wright, R., Pauphilet, J., Petridis, P.: A stoch astic benders decomposition scheme for large-scale stochastic network design. INFORMS J. Comput. 37(5), 1163–1181 (2025)

2025

[2] [2]

Benders, J.F.: Partitioning procedures for solving mixed-variable s programming problems. Numer. Math. 4(1), 238–252 (1962)

1962

[3] [3]

Omega 126, 103045 (2024)

Belieres, S., Hewitt, M.: Hedging against uncertainty in transport ation network design through ﬂexible scheduling. Omega 126, 103045 (2024)

2024

[4] [4]

Belieres, S., Hewitt, M.: Stochastic scheduled service network de sign problem with ﬂexible schedules: Mathematical formulations and benders decomp osition-based solution approach. Ann. Oper. Res., 1–25 (2026)

2026

[5] [5]

Birge, J.R., Louveaux, F.V.: A multicut algorithm for two-stage sto chastic linear programs. Eur. J. Oper. Res. 34(3), 384–392 (1988)

1988

[6] [6]

Springer, 44 New York (2011)

Birge, J.R., Louveaux, F.: Introduction to Stochastic Programm ing. Springer, 44 New York (2011)

2011

[7] [7]

Bayraksan, G., Maggioni, F., Faccini, D., Yang, M.: Bounds for multis tage mixed- integer distributionally robust optimization. SIAM J. Optim. 34(1), 682–717 (2024)

2024

[8] [8]

(eds.) Freight Railroad Service Network Design, pp

Chouman, M., Crainic, T.G.: In: Crainic, T.G., Gendreau, M., Gendron , B. (eds.) Freight Railroad Service Network Design, pp. 383–426. Springer, C ham (2021)

2021

[9] [9]

Crainic, T.G., Frangioni, A., Gendron, B.: Bundle-based relaxation m ethods for multicommodity capacitated ﬁxed charge network design. Discret. Appl. Math. 112(1), 73–99 (2001). Combinatorial Optimization Symposium, Selecte d Papers

2001

[10] [10]

Cavaliere, F., Fischetti, M., Roberti, R., Salvagnin, D.: Models and a lgorithms for the time window assignment traveling salesperson problem with stoch astic travel times. Eur. J. Oper. Res. 329(1), 96–111 (2026)

2026

[11] [11]

Crainic, T.G., Gendron, B., Akhavan Kazemzadeh, M.R.: A taxonom y of mul- tilayer network design and a survey of transportation and telecom munication applications. Eur. J. Oper. Res. 303(1), 1–13 (2022)

2022

[12] [12]

(eds.): Network Design with Applica- tions to Transportation and Logistics

Crainic, T.G., Gendreau, M., Gendron, B. (eds.): Network Design with Applica- tions to Transportation and Logistics. Springer, Cham (2021)

2021

[13] [13]

Crainic, T.G., Hewitt, M., Maggioni, F., Rei, W.: Partial benders deco mposition: General methodology and application to stochastic network design . Transp. Sci. 55(2), 414–435 (2021)

2021

[14] [14]

Christiansen, M., Hellsten, E., Pisinger, D., Sacramento, D., Vilhelm sen, C.: Liner shipping network design. Eur. J. Oper. Res. 286(1), 1–20 (2020)

2020

[15] [15]

(eds.) Logistics Network Design, pp

Cordeau, J.-F., Klibi, W., Nickel, S.: In: Crainic, T.G., Gendreau, M., G endron, B. (eds.) Logistics Network Design, pp. 599–625. Springer, Cham ( 2021)

2021

[16] [16]

Cavagnini, R., Maggioni, F., Bertazzi, L., Hewitt, M.: A two-stage s tochastic programming model for bike-sharing systems with rebalancing. EUR O J. Transp. Logist. 13, 100140 (2024)

2024

[17] [17]

Dupaˇ cov´ a, J., Consigli, G., Wallace, S.W.: Scenarios for multistag e stochastic programs. Ann. Oper. Res. 100, 25–53 (2000)

2000

[18] [18]

Springer, Cham (2024)

Dentcheva, D., Ruszczy´ nski, A.P.: Risk-averse Optimization an d Control. Springer, Cham (2024)

2024

[19] [19]

(e d.) Barycentric Bounds in Stochastic Programming: Theory and Application, pp

Frauendorfer, K., Kuhn, D., Sch¨ urle, M.: In: Infanger, G. (e d.) Barycentric Bounds in Stochastic Programming: Theory and Application, pp. 67– 96. Springer, New York, NY (2011) 45

2011

[20] [20]

Geoﬀrion, A.M.: Elements of large scale mathematical programmin g part ii: Synthesis of algorithms and bibliography. Manage. Sci. 16(11), 676–691 (1970)

1970

[21] [21]

Geoﬀrion, A.M.: Elements of large-scale mathematical programm ming part i: Concepts. Manage. Sci. 16(11), 652–675 (1970)

1970

[22] [22]

Høyland, K., Kaut, M., Wallace, S.W.: A heuristic for moment-match ing scenario generation. Comput. Optim. Appl. 24(2), 169–185 (2003)

2003

[23] [23]

Hewitt, M., Ortmann, J., Rei, W.: Decision-based scenario cluster ing for decision- making under uncertainty. Ann. Oper. Res. 315(2), 747–771 (2022)

2022

[24] [24]

(eds.) Stochastic Network Design, pp

Hewitt, M., Rei, W., Wallace, S.W.: In: Crainic, T.G., Gendreau, M., Gen dron, B. (eds.) Stochastic Network Design, pp. 283–315. Springer, Cha m (2021)

2021

[25] [25]

Hosseini, M., Turner, J.: Deepest cuts for benders decomposit ion. Oper. Res. 73(5), 2591–2609 (2025)

2025

[26] [26]

Mahmuto˘ gulları, A., C ¸ avu¸ s, O., Akt¨ urk, M..S.: Bounds on risk-averse mixed- integer multi-stage stochastic programming problems with mean-cv ar. Eur. J. Oper. Res. 266(2), 595–608 (2018)

2018

[27] [27]

Jaoui, N., Klibi, W., El Hachemi, N., Aouam, T., Fender, M.: Multi-cycle produc- tion network design under supply uncertainty. Eur. J. Oper. Res. 330(3), 816–837 (2026)

2026

[28] [28]

INFOR: Inf

Kaltis, T., Saharidis, G.K.D.: Literature review on benders cut sele ction and a multiple cut generation scheme. INFOR: Inf. Syst. Oper. Res., 1–2 7 (2025)

2025

[29] [29]

(eds.) Barycentric approximation shcheme

Kuhn, D.: In: Beckmann, M., K¨ unzi, H.P., Fandel, G., Trockel, W., B asile, A., Drexl, A., Dawid, H., Inderfurth, K., K¨ ursten, W., Schittko, U. (eds.) Barycentric approximation shcheme. Lecture Notes in Economics and Mathemat ical Systems, pp. 51–81. Springer, Berlin (2005)

2005

[30] [30]

Kall, P., Wallace, S.: Stochastic Programming vol. 46. John Wiley & So ns, Chichester (1994)

1994

[31] [31]

Luo, F., Dey, S., Mehrotra, S.: A cutting-plane and benders’ de composition algo- rithm for two-stage distributionally robust convex programs. Mat h. Program., 1–64 (2025)

2025

[32] [32]

Maggioni, F., Allevi, E., Bertocchi, M.: Bounds in multistage linear sto chastic programming. J. Optim. Theory Appl. 163, 200–229 (2014)

2014

[33] [33]

Maggioni, F., Allevi, E., Bertocchi, M.: Monotonic bounds in multistag e mixed- integer stochastic programming. Comput. Manag. Sci. 13(3), 423–457 (2016)

2016

[34] [34]

Maggioni, F., Cagnolari, M., Bertazzi, L., Wallace, S.W.: Stochastic o ptimization 46 models for a bike-sharing problem with transshipment. Eur. J. Oper . Res. 276(1), 272–283 (2019)

2019

[35] [35]

In preparation (2026)

Maggioni, F., Cavagnini, R., Faccini, D.: Optimization-driven monoto nic bounds for stochastic programs with applications in ﬁnance and operations management. In preparation (2026)

2026

[36] [36]

(eds.) Public Transportation, pp

Mauttone, A., Cancela, H., Urquhart, M.E.: In: Crainic, T.G., Gend reau, M., Gendron, B. (eds.) Public Transportation, pp. 539–565. Springer , Cham (2021)

2021

[37] [37]

Melo, M.T., Nickel, S., Saldanha-da-Gama, F.: Facility location and su pply chain management – a review. Eur. J. Oper. Res. 196(2), 401–412 (2009)

2009

[38] [38]

Maggioni, F., Pﬂug, G.C.: Bounds and approximations for multistag e stochastic programs. SIAM J. Optim. 26(1), 831–855 (2016)

2016

[39] [39]

Maggioni, F., Pﬂug, G.C.: Guaranteed bounds for general nondis crete multistage risk-averse stochastic optimization programs. SIAM J. Optim. 29(1), 454–483 (2019)

2019

[40] [40]

I n preparation (2026)

Maggioni, F., Romeijnders, W., Spinelli, A.: Lower bounds for two-s tage stochas- tic programming problems with decision-dependent uncertainties. I n preparation (2026)

2026

[41] [41]

Narum, B.S., Maggioni, F., Wallace, S.W.: On the safe side of stochas tic pro- gramming: bounds and approximations. Int. Tran. Oper. Res. 30(6), 3201–3237 (2023)

2023

[42] [42]

Rahmaniani, R., Crainic, T.G., Gendreau, M., Rei, W.: The benders de com- position algorithm: A literature review. Eur. J. Oper. Res. 259(3), 801–817 (2017)

2017

[43] [43]

Rahmaniani, R., Crainic, T.G., Gendreau, M., Rei, W.: Accelerating th e benders decomposition method: Application to stochastic network design pr oblems. SIAM J. Optim. 28(1), 875–903 (2018)

2018

[44] [44]

INFORMS J

Rei, W., Cordeau, J.-F., Gendreau, M., Soriano, P.: Accelerating b enders decomposition by local branching. INFORMS J. Comput. 21(2), 333–345 (2009)

2009

[45] [45]

Roland, M., Forel, A., Vidal, T.: Adaptive partitioning for chance-c onstrained problems with ﬁnite support. SIAM J. Optim. 35(1), 476–505 (2025)

2025

[46] [46]

Ram ´ ırez-Pico, C., Ljubi´ c, I., Moreno, E.: Benders adaptive-cuts method for two- stage stochastic programs. Transp. Sci. 57(5), 1252–1275 (2023)

2023

[47] [47]

Ram ´ ırez-Pico, C., Moreno, E.: Generalized adaptive partition-b ased method for two-stage stochastic linear programs with ﬁxed recourse. Math. Program. 196(1), 755–774 (2022) 47

2022

[48] [48]

Rockafellar, R.T., Uryasev, S.: Optimization of conditional value- at-risk. J. Risk 2, 21–42 (2000)

2000

[49] [49]

Ruszczy´ nski, A.: Decomposition methods in stochastic progra mming. Math. Program. 79, 333–353 (1997)

1997

[50] [50]

Spinelli, A., Bezzi, D., Jabali, O., Maggioni, F.: A stochastic electric ve hicle rout- ing problem under uncertain energy consumption. Transp. Res. Pa rt C Emerging Technol. 183, 105480 (2026)

2026

[51] [51]

(ed.) Risk Aversion in Two-Stage S tochastic Integer Programming, pp

Schultz, R.: In: Infanger, G. (ed.) Risk Aversion in Two-Stage S tochastic Integer Programming, pp. 165–187. Springer, New York, NY (2011)

2011

[52] [52]

Shapiro, A.: Analysis of stochastic dual dynamic programming me thod. Eur. J. Oper. Res. 209(1), 63–72 (2011)

2011

[53] [53]

INFORMS J

Seo, K., Joung, S., Lee, C., Park, S.: A closest benders cut selec tion scheme for accelerating the benders decomposition algorithm. INFORMS J. Com put. 34(5), 2804–2827 (2022)

2022

[54] [54]

Song, Y., Luedtke, J.: An adaptive partition-based approach f or solving two- stage stochastic programs with ﬁxed recourse. SIAM J. Optim. 25(3), 1344–1367 (2015)

2015

[55] [55]

Spinelli, A., Maggioni, F., Ramos, T.R.P., Barbosa-P´ ovoa, A.P., Vigo, D.: A rolling horizon heuristic approach for a multi-stage stochastic wast e collection problem. Eur. J. Oper. Res. 323(1), 276–296 (2025)

2025

[56] [56]

Trukhanov, S., Ntaimo, L., Schaefer, A.: Adaptive multicut aggr egation for two- stage stochastic linear programs with recourse. Eur. J. Oper. Re s. 206(2), 395–406 (2010)

2010

[57] [57]

Van Slyke, R.M., Wets, R.J.-B.: L-shaped linear programs with applic ations to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)

1969

[58] [58]

Wu, T., Zhang, J., Zhang, C., Liang, Z., Zhang, X.: Benders decom position for stochastic facility location and production planning. Comput. Oper. Res. 186, 107316 (2026)

2026

[59] [59]

INFORMS J

Yuan, Y., Sen, S.: Enhanced cut generation methods for decom position-based branch and cut for two-stage stochastic mixed-integer program s. INFORMS J. Comput. 21(3), 480–487 (2009) 48

2009