pith. sign in

arxiv: 2606.21483 · v1 · pith:QB7OZ7FJnew · submitted 2026-06-19 · 🧮 math.OC

A note on the convergence guarantees of RLT-based algorithms for polynomial optimization

Alejandro Barros-Gonz\'alez , Julio Gonz\'alez-D\'iaz , Brais Gonz\'alez-Rodr\'iguez , Iria Rodr\'iguez Acevedo This is my paper

Pith reviewed 2026-06-26 13:42 UTC · model grok-4.3

classification 🧮 math.OC
keywords Reformulation-Linearization TechniqueRLTpolynomial optimizationconvergence guaranteesglobal optimizationmathematical oversight
0
0 comments X

The pith

A mathematical oversight in foundational RLT proofs for polynomial optimization can be fixed by adding one minor natural assumption.

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

This note examines early convergence results for the Reformulation-Linearization Technique, or RLT, which turns polynomial optimization problems into sequences of linear programs. The authors locate a specific oversight in one of the key proofs that these methods reach the global optimum. They show that the original claims hold once a simple additional assumption is stated about the problem. Without awareness of this point, RLT-based algorithms risk operating without the convergence guarantees previously attributed to them. The observation matters for anyone relying on RLT to solve nonconvex polynomial problems with theoretical backing.

Core claim

The paper identifies a mathematical oversight in one of the foundational results on the Reformulation-Linearization Technique (RLT) for polynomial optimization. Although correctness of the original result can be easily recovered by adding a minor and natural assumption, not being aware of this nuance may lead to the loss of convergence guarantees in RLT-based algorithms.

What carries the argument

The convergence proofs for the Reformulation-Linearization Technique (RLT) applied to polynomial optimization problems.

If this is right

  • RLT-based algorithms require verification of the added assumption to retain their stated convergence properties.
  • Implementations that directly cite the original proofs without the extra assumption may produce results whose global optimality is no longer guaranteed.
  • Users of RLT solvers for polynomial problems should check whether their instances satisfy the repaired condition before claiming convergence.
  • Subsequent theoretical work that builds on the flawed proof inherits the same limitation unless the assumption is included.

Where Pith is reading between the lines

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

  • Similar subtle gaps could exist in other foundational linearization or relaxation proofs used in global optimization.
  • Empirical testing on problems that violate the minor assumption would quickly reveal whether practical convergence still occurs.
  • The note underscores the value of re-examining classic proofs when algorithms derived from them are deployed at scale.

Load-bearing premise

The foundational RLT convergence proofs contain a specific mathematical oversight whose existence and repair by one minor assumption the note takes as given.

What would settle it

An explicit polynomial optimization instance where an RLT algorithm fails to converge when the overlooked assumption is dropped, or where convergence is restored once the assumption is restored.

Figures

Figures reproduced from arXiv: 2606.21483 by Alejandro Barros-Gonz\'alez, Brais Gonz\'alez-Rodr\'iguez, Iria Rodr\'iguez Acevedo, Julio Gonz\'alez-D\'iaz.

Figure 1
Figure 1. Figure 1: Basic scheme of an RLT-based algorithm. 3 The theoretical oversight Below, we restate Lemma 2 in Sherali and Tuncbilek (1992) and provide a faithful transcription of its proof, adapted to our notation. Lemma 2 (Sherali and Tuncbilek (1992)). Consider δ ′ ∈ N, with 1 ≤ δ ′ < δ. Then, constraints [F δ ′ Ω (J1, J2)]L ≥ 0, where J1 + J2 ∈ Nδ ′ are all implied by those of degree δ, [F δ Ω (J1, J2)]L ≥ 0, where … view at source ↗
read the original abstract

This paper identifies and addresses a mathematical oversight in one of the foundational results on the Reformulation-Linearization Technique (RLT) for polynomial optimization. We then argue that, although correctness of the original result can be easily recovered by adding a minor and natural assumption, not being aware of this nuance may lead to the loss of convergence guarantees in RLT-based algorithms.

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.

Referee Report

1 major / 1 minor

Summary. The paper identifies a mathematical oversight in one of the foundational results on the Reformulation-Linearization Technique (RLT) for polynomial optimization. It argues that the original convergence result can be recovered by adding a minor and natural assumption, but that failure to account for this nuance risks the loss of convergence guarantees when applying RLT-based algorithms.

Significance. If the claimed oversight exists and the proposed fix is valid, the note provides a useful clarification that could prevent misapplication of convergence results in polynomial optimization. The contribution is narrowly scoped and corrective rather than foundational; its value lies in highlighting a subtlety in existing proofs rather than introducing new theory or algorithms.

major comments (1)
  1. The manuscript asserts the existence of an oversight in prior RLT convergence arguments but supplies neither the specific theorem/equation from the referenced foundational papers nor a counter-example or proof sketch demonstrating where the gap occurs. Without this, the central claim that 'correctness of the original result can be easily recovered by adding a minor and natural assumption' cannot be verified.
minor comments (1)
  1. The abstract and title refer to 'one of the foundational results' without naming the specific reference or result; adding the citation in the opening paragraph would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address the major comment below.

read point-by-point responses

  1. Referee: The manuscript asserts the existence of an oversight in prior RLT convergence arguments but supplies neither the specific theorem/equation from the referenced foundational papers nor a counter-example or proof sketch demonstrating where the gap occurs. Without this, the central claim that 'correctness of the original result can be easily recovered by adding a minor and natural assumption' cannot be verified.

    Authors: We agree that the note would benefit from greater explicitness. The manuscript references the foundational RLT papers but does not quote the precise theorem or provide an illustrative gap. In revision we will add the exact theorem citation together with a short paragraph (or minimal counter-example sketch) showing where the original convergence argument is incomplete without the extra assumption. This addition is minor and preserves the note's corrective scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a short note that identifies an oversight in prior published RLT convergence proofs and shows that a minor natural assumption restores the original guarantees. No derivation, prediction, or first-principles result is constructed inside the paper from quantities that are themselves defined or fitted from the paper's own outputs. The argument rests on external literature rather than self-citation chains, self-definitional loops, or renaming of known results, so the claimed correction is independent of any internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities; the note concerns an existing proof gap rather than introducing new modeling constructs.

pith-pipeline@v0.9.1-grok · 5599 in / 1030 out tokens · 16720 ms · 2026-06-26T13:42:05.416306+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

112 extracted references · 16 canonical work pages

  1. [1]

    Google OR-Tools , year =

    Laurent Perron and Vincent Furnon , note =. Google OR-Tools , year =

  2. [2]

    COIN-OR Linear Programming (CLP) , year =

    John Forrest and Julian Hall , note =. COIN-OR Linear Programming (CLP) , year =

  3. [3]

    2014 , publisher=

    Misener, Ruth and Floudas, Christodoulos A , journal=. 2014 , publisher=

    2014

  4. [4]

    2024 , url =

    Suresh Bolusani and Mathieu Besan. 2024 , url =

    2024

  5. [5]

    The global solver in the

    Lin, Youdong and Schrage, Linus , journal=. The global solver in the. 2009 , publisher=

    2009

  6. [6]

    Power and Centrality: A Family of Measures , url =

    Phillip Bonacich , date-added =. Power and Centrality: A Family of Measures , url =. American Journal of Sociology , number =. 1987 , bdsk-url-1 =

    1987

  7. [7]

    Finding community structure in very large networks , volume =

    Aaron Clauset , date-added =. Finding community structure in very large networks , volume =. 2004 , bdsk-url-1 =. doi:10.1103/PhysRevE.70.066111 , journal =

    work page doi:10.1103/physreve.70.066111 2004

  8. [8]

    Social Network Analysis , year = 1994, bdsk-url-1 =

    Stanley Wasserman and Katherine Faust , date-added =. Social Network Analysis , year = 1994, bdsk-url-1 =

    1994

  9. [9]

    Graph Theory , year =

    Diestel, Reinhard , date-added =. Graph Theory , year =

  10. [10]

    Learning

    Martina Fischetti and Andrea Lodi and Giulia Zarpellon , booktitle =. Learning. doi:10.1007/978-3-030-19212-9_18 , pages =

    work page doi:10.1007/978-3-030-19212-9_18

  11. [11]

    Learning a classification of mixed-integer quadratic programming problems , year =

    Bonami, Pierre and Lodi, Andrea and Zarpellon, Giulia , booktitle =. Learning a classification of mixed-integer quadratic programming problems , year =

  12. [12]

    A classifier to decide on the linearization of mixed-integer quadratic problems in

    Bonami, Pierre and Lodi, Andrea and Zarpellon, Giulia , journal=. A classifier to decide on the linearization of mixed-integer quadratic problems in. 2022 , volume=

    2022

  13. [13]

    Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning , volume =

    Morrison, David R and Jacobson, Sheldon H and Sauppe, Jason J and Sewell, Edward C , journal =. Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning , volume =

  14. [14]

    Computational advances in polynomial optimization:

    Gonz. Computational advances in polynomial optimization:. Journal of Global Optimization , volume =

  15. [15]

    Sherali and Evrim Dalkiran and Jitamitra Desai , journal =

    Hanif D. Sherali and Evrim Dalkiran and Jitamitra Desai , journal =. Enhancing. 2012 , number =

    2012

  16. [16]

    Sherali and Evrim Dalkiran and Leo Liberti , journal =

    Hanif D. Sherali and Evrim Dalkiran and Leo Liberti , journal =. Reduced. 2012 , issn =

    2012

  17. [17]

    Sherali , journal =

    Evrim Dalkiran and Hanif D. Sherali , journal =. Theoretical filtering of. 2013 , number =

    2013

  18. [18]

    Incidence matrices and interval graphs , volume =

    Delbert Fulkerson and Oliver Gross , date-added =. Incidence matrices and interval graphs , volume =. doi:10.2140/pjm.1965.15.835 , journal =

    work page doi:10.2140/pjm.1965.15.835 1965

  19. [19]

    2018 , bdsk-url-1 =

    Bienstock, Daniel and Mu. 2018 , bdsk-url-1 =. doi:10.1137/15M1054079 , journal =

    work page doi:10.1137/15m1054079 2018

  20. [20]

    Sherali and Cihan H

    Hanif D. Sherali and Cihan H. Tuncbilek , date-added =. A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique , volume =. doi:10.1007/bf00121304 , journal =

    work page doi:10.1007/bf00121304

  21. [21]

    On learning and branching: a survey , volume =

    Lodi, Andrea and Zarpellon, Giulia , journal =. On learning and branching: a survey , volume =

  22. [22]

    Machine learning for combinatorial optimization: a methodological tour d'horizon , volume =

    Bengio, Yoshua and Lodi, Andrea and Prouvost, Antoine , journal =. Machine learning for combinatorial optimization: a methodological tour d'horizon , volume =

  23. [23]

    Machine-learning-based column selection for column generation , volume =

    Morabit, Mouad and Desaulniers, Guy and Lodi, Andrea , journal =. Machine-learning-based column selection for column generation , volume =

  24. [24]

    Parameterizing Branch-and-Bound Search Trees to Learn Branching Policies , url =

    Giulia Zarpellon and Jason Jo and Andrea Lodi and Yoshua Bengio , bibsource =. Parameterizing Branch-and-Bound Search Trees to Learn Branching Policies , url =. Thirty-Fifth. 2021 , bdsk-url-1 =

    2021

  25. [25]

    arXiv preprint arXiv:2112.02195 , year=

    Learning to search in local branching , author=. arXiv preprint arXiv:2112.02195 , year=

    arXiv

  26. [26]

    Mathematical Programming , pages=

    Machine learning augmented branch and bound for mixed integer linear programming , author=. Mathematical Programming , pages=. 2024 , publisher=

    2024

  27. [27]

    arXiv preprint arXiv:2102.09544 , year=

    Combinatorial optimization and reasoning with graph neural networks , author=. arXiv preprint arXiv:2102.09544 , year=

    arXiv

  28. [28]

    arXiv preprint arXiv:2103.10294 , year=

    Learning to Schedule Heuristics in Branch-and-Bound , author=. arXiv preprint arXiv:2103.10294 , year=

    arXiv

  29. [29]

    An abstract model for branching and its application to mixed integer programming , volume =

    Le Bodic, Pierre and Nemhauser, George , journal =. An abstract model for branching and its application to mixed integer programming , volume =

  30. [30]

    Further results on an abstract model for branching and its application to mixed integer programming , url =

    Daniel Anderson and Pierre Le Bodic and Kerri Morgan , bibsource =. Further results on an abstract model for branching and its application to mixed integer programming , url =. 2021 , bdsk-url-1 =. doi:10.1007/s10107-020-01556-4 , journal =

    work page doi:10.1007/s10107-020-01556-4 2021

  31. [31]

    Further results on an abstract model for branching and its application to mixed integer programming , year =

    Anderson, Daniel and Le Bodic, Pierre and Morgan, Kerri , journal =. Further results on an abstract model for branching and its application to mixed integer programming , year =

  32. [32]

    Advances in neural information processing systems , volume=

    Exact combinatorial optimization with graph convolutional neural networks , author=. Advances in neural information processing systems , volume=

  33. [33]

    Sherali , doi =

    Evrim Dalkiran and Hanif D. Sherali , doi =. Mathematical Programming Computation , pages =. 2016 , bdsk-url-1 =

    2016

  34. [34]

    Bussieck and Arne Stolbjerg Drud and Alexander Meeraus , doi =

    Michael R. Bussieck and Arne Stolbjerg Drud and Alexander Meeraus , doi =. INFORMS Journal on Computing , pages =. 2003 , bdsk-url-1 =

    2003

  35. [35]

    Mathematical Programming Computation , pages =

    Fabio Furini and Emiliano Traversi and Pietro Belotti and Antonio Frangioni and Ambros Gleixner and Nick Gould and Leo Liberti and Andrea Lodi and Ruth Misener and Hans Mittelmann and Nikolaos Sahinidis and Stefan Vigerske and Angelika Wiegele , date-modified =. Mathematical Programming Computation , pages =. 2018 , bdsk-url-1 =. doi:10.1007/s12532-018-01...

    work page doi:10.1007/s12532-018-0147-4 2018

  36. [36]

    Intersection Graphs: An Introduction , volume =

    Madhumangal Pal , journal =. Intersection Graphs: An Introduction , volume =

  37. [37]

    Integer Programming Techniques for Polynomial Optimization , year =

    Gonzalo Mu\. Integer Programming Techniques for Polynomial Optimization , year =

  38. [38]

    A Machine Learning-Based Approximation of Strong Branching , url =

    Alejandro Marcos Alvarez and Quentin Louveaux and Louis Wehenkel , bibsource =. A Machine Learning-Based Approximation of Strong Branching , url =. 2017 , bdsk-url-1 =. doi:10.1287/ijoc.2016.0723 , journal =

    work page doi:10.1287/ijoc.2016.0723 2017

  39. [39]

    Learning to Search in Branch and Bound Algorithms , url =

    He He and Hal Daum. Learning to Search in Branch and Bound Algorithms , url =. Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, December 8-13 2014, Montreal, Quebec, Canada , editor =. 2014 , bdsk-url-1 =

    2014

  40. [40]

    European Journal of Operational Research , number =

    Giovanni. European Journal of Operational Research , number =. 2016 , bdsk-url-1 =. doi:https://doi.org/10.1016/j.ejor.2015.08.018 , issn =

    work page doi:10.1016/j.ejor.2015.08.018 2016

  41. [41]

    Learning to Branch , volume =

    Balcan, Maria-Florina and Dick, Travis and Sandholm, Tuomas and Vitercik, Ellen , booktitle =. Learning to Branch , volume =

  42. [42]

    Gomes and Bart Selman , doi =

    Carla P. Gomes and Bart Selman , doi =. Algorithm portfolios , url =. Artificial Intelligence , keywords =. 2001 , bdsk-url-1 =

    2001

  43. [43]

    A Portfolio Approach to Algorithm Selection , year =

    Leyton-Brown, Kevin and Nudelman, Eugene and Andrew, Galen and McFadden, Jim and Shoham, Yoav , booktitle =. A Portfolio Approach to Algorithm Selection , year =

  44. [44]

    Learning combinatorial optimization algorithms over graphs , year =

    Khalil, Elias and Dai, Hanjun and Zhang, Yuyu and Dilkina, Bistra and Song, Le , booktitle =. Learning combinatorial optimization algorithms over graphs , year =

  45. [45]

    and Le Bodic, Pierre and Song, Le and Nemhauser, George and Dilkina, Bistra , booktitle =

    Khalil, Elias B. and Le Bodic, Pierre and Song, Le and Nemhauser, George and Dilkina, Bistra , booktitle =. Learning to Branch in Mixed Integer Programming , url =. 2016 , bdsk-url-1 =

    2016

  46. [46]

    Machine Learning , issn =

    Random Forests , author =. Machine Learning , issn =

  47. [47]

    Journal of Machine Learning Research , year =

    Nicolai Meinshausen , title =. Journal of Machine Learning Research , year =

  48. [48]

    doi:10.1201/9781315120256 , year=

    Handbook of Quantile Regression , publisher=. doi:10.1201/9781315120256 , year=

    work page doi:10.1201/9781315120256

  49. [49]

    Quantile Regression , url =

    Koenker, Roger , biburl =. Quantile Regression , url =

  50. [50]

    Wood and Margaux Zaffran and Raphaël Nedellec and Yannig Goude , title =

    Matteo Fasiolo and Simon N. Wood and Margaux Zaffran and Raphaël Nedellec and Yannig Goude , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =

    2021

  51. [51]

    Proceedings of the 9th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems , pages =

    Sabharwal, Ashish and Samulowitz, Horst and Reddy, Chandra , title =. Proceedings of the 9th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems , pages =. 2012 , isbn =. doi:10.1007/978-3-642-29828-8_23 , abstract =

    work page doi:10.1007/978-3-642-29828-8_23 2012

  52. [52]

    Branching rules revisited , journal =

    Tobias Achterberg and Thorsten Koch and Alexander Martin , keywords =. Branching rules revisited , journal =. 2005 , issn =. doi:https://doi.org/10.1016/j.orl.2004.04.002 , url =

    work page doi:10.1016/j.orl.2004.04.002 2005

  53. [53]

    Linderoth, J. T. and Savelsbergh, M. W. P. , title =. INFORMS Journal on Computing , volume =. 1999 , doi =

    1999

  54. [54]

    2016 , isbn =

    Gupta, Rishi and Roughgarden, Tim , title =. 2016 , isbn =. doi:10.1145/2840728.2840766 , booktitle =

    work page doi:10.1145/2840728.2840766 2016

  55. [55]

    Branching and bounds tightening techniques for non-convex

    Pietro Belotti and Jon Lee and Leo Liberti and Fran. Branching and bounds tightening techniques for non-convex

  56. [56]

    Optimization Methods and Software , volume =

    Pietro Belotti and Jon Lee and Leo Liberti and François Margot and Andreas Wächter , title =. Optimization Methods and Software , volume =. 2009 , publisher =

    2009

  57. [57]

    Scoring positive semidefinite cutting planes for quadratic optimization via trained neural networks , year =

    Radu Baltean-Lugojan and Pierre Bonami and Ruth Misener and Andrea Tramontani , institution =. Scoring positive semidefinite cutting planes for quadratic optimization via trained neural networks , year =

  58. [58]

    Friedman

    Stochastic gradient boosting , journal =. 2002 , note =. doi:https://doi.org/10.1016/S0167-9473(01)00065-2 , url =

    work page doi:10.1016/s0167-9473(01)00065-2 2002

  59. [59]

    2009.The Elements of Statistical Learning: Data Mining, Inference, and Prediction(2nd ed.)

    Trevor Hastie and Robert Tibshirani and Jerome Friedman , year =. The Elements of Statistical Learning , subtitle =. doi:https://doi.org/10.1007/978-0-387-84858-7 , edition =

    work page doi:10.1007/978-0-387-84858-7

  60. [60]

    Mathematical Programming , volume=

    New limits of treewidth-based tractability in optimization , author=. Mathematical Programming , volume=. 2022 , publisher=

    2022

  61. [61]

    Friedman , title =

    Jerome H. Friedman , title =. The Annals of Statistics , number =. 2001 , doi =

    2001

  62. [62]

    The Annals of Applied Statistics , number =

    Brian Kriegler and Richard Berk , title =. The Annals of Applied Statistics , number =. 2010 , doi =

    2010

  63. [63]

    Dolan and Jorge J

    Elizabeth D. Dolan and Jorge J. Mor\'e , journal =. Benchmarking optimization software with performance profiles , year =

  64. [64]

    2017 , publisher=

    Complex networks: principles, methods and applications , author=. 2017 , publisher=

    2017

  65. [65]

    Graph minors

    Robertson, Neil and Seymour, Paul D , journal=. Graph minors. 1984 , publisher=

    1984

  66. [66]

    Wright and Andreas Ziegler , journal =

    Marvin N. Wright and Andreas Ziegler , journal =. 2017 , volume =

    2017

  67. [67]

    Wood and Margaux Zaffran and Rapha\"el Nedellec and Yannig Goude , journal =

    Matteo Fasiolo and Simon N. Wood and Margaux Zaffran and Rapha\"el Nedellec and Yannig Goude , journal =. 2021 , volume =

    2021

  68. [68]

    2020 , note =

    gbm: Generalized Boosted Regression Models , author =. 2020 , note =

    2020

  69. [69]

    Journal of the American Statistical Association , author =

    Huixia Judy Wang and Deyuan Li , title =. Journal of the American Statistical Association , volume =. 2013 , publisher =. doi:10.1080/01621459.2013.820134 , URL =

    work page doi:10.1080/01621459.2013.820134 2013

  70. [70]

    Advances in Polynomial Optimization

    Gonz \'a lez-Rodr \' guez, Brais. Advances in Polynomial Optimization

  71. [71]

    N. V. Sahinidis , title =. 2017 , OPTnote =

    2017

  72. [72]

    Mathematical Programming Computation , volume=

    A hybrid LP/NLP paradigm for global optimization relaxations , author=. Mathematical Programming Computation , volume=. 2018 , publisher=

    2018

  73. [73]

    1996 , publisher=

    Sahinidis, Nikolaos V , journal=. 1996 , publisher=

    1996

  74. [74]

    Proceedings of the 27th International Joint Conference on Artificial Intelligence , pages=

    Neural networks for predicting algorithm runtime distributions , author=. Proceedings of the 27th International Joint Conference on Artificial Intelligence , pages=

  75. [75]

    Artificial Intelligence , volume=

    Algorithm runtime prediction: Methods & evaluation , author=. Artificial Intelligence , volume=. 2014 , publisher=

    2014

  76. [76]

    Tornede, Alexander and Wever, Marcel and Werner, Stefan and Mohr, Felix and H. Run2. Asian Conference on Machine Learning , pages=. 2020 , organization=

    2020

  77. [77]

    arXiv preprint arXiv:2205.13028 , year=

    Formalizing Preferences Over Runtime Distributions , author=. arXiv preprint arXiv:2205.13028 , year=

    arXiv

  78. [78]

    Optimization Letters , volume=

    Some applications of polynomial optimization in operations research and real-time decision making , author=. Optimization Letters , volume=. 2016 , publisher=

    2016

  79. [79]

    IEEE Transactions on Power Systems , volume=

    Optimal power flow as a polynomial optimization problem , author=. IEEE Transactions on Power Systems , volume=. 2015 , publisher=

    2015

  80. [80]

    European Journal of Operational Research , volume=

    Polynomial optimization for water networks: Global solutions for the valve setting problem , author=. European Journal of Operational Research , volume=. 2017 , publisher=

    2017

Showing first 80 references.