{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T15:32:28Z","timestamp":1772724748654,"version":"3.50.1"},"reference-count":87,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T00:00:00Z","timestamp":1647907200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper presents a tutorial on the state-of-the-art software for the solution of two-stage (mixed-integer) linear stochastic programs and provides a list of software designed for this purpose. The methodologies are classified according to the decomposition alternatives and the types of the variables in the problem. We review the fundamentals of Benders decomposition, dual decomposition and progressive hedging, as well as possible improvements and variants. We also present extensive numerical results to underline the properties and performance of each algorithm using software implementations, including DECIS, FORTSP, PySP, and DSP. Finally, we discuss the strengths and weaknesses of each methodology and propose future research directions.<\/jats:p>","DOI":"10.3390\/a15040103","type":"journal-article","created":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T14:55:35Z","timestamp":1647960935000},"page":"103","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Software"],"prefix":"10.3390","volume":"15","author":[{"given":"Juan J.","family":"Torres","sequence":"first","affiliation":[{"name":"Laboratoire d\u2019Informatique de Paris Nord (LIPN), CNRS (UMR 7030), Universit\u00e9 Sorbonne Paris Nord, Sorbonne Paris Cit\u00e9, 93430 Villetaneuse, France"}]},{"given":"Can","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, USA"}]},{"given":"Robert M.","family":"Apap","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, USA"}]},{"given":"Ignacio E.","family":"Grossmann","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Birge, J.R., and Louveaux, F. (2011). Introduction to Stochastic Programming, Springer Science & Business Media.","DOI":"10.1007\/978-1-4614-0237-4"},{"key":"ref_2","unstructured":"Kall, P., and Wallace, S.W. (1996). Stochastic Programming, Wiley."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"17240","DOI":"10.1021\/ie5004174","article-title":"Design of Resilient Supply Chains with Risk of Facility Disruptions","volume":"53","author":"Wassick","year":"2014","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.compchemeng.2018.01.017","article-title":"An improved L-shaped method for two-stage convex 0\u20131 mixed integer nonlinear stochastic programs","volume":"112","author":"Li","year":"2018","journal-title":"Comput. Chem. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.compchemeng.2014.02.028","article-title":"Robust optimization and stochastic programming approaches for medium-term production scheduling of a large-scale steelmaking continuous casting process under demand uncertainty","volume":"66","author":"Ye","year":"2014","journal-title":"Comput. Chem. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1016\/j.compchemeng.2007.03.003","article-title":"A multistage stochastic programming approach with strategies for uncertainty reduction in the synthesis of process networks with uncertain yields","volume":"32","author":"Tarhan","year":"2008","journal-title":"Comput. Chem. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1137\/0117061","article-title":"L-Shaped Linear Programs with Applications to Optimal Control and Stochastic Programming","volume":"17","author":"Wets","year":"1969","journal-title":"SIAM J. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Geoffrion, A.M. (1974). Lagrangean relaxation for integer programming. Approaches to Integer Programming, Springer.","DOI":"10.1007\/BFb0120690"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1287\/moor.16.1.119","article-title":"Scenarios and Policy Aggregation in Optimization under Uncertainty","volume":"16","author":"Rockafellar","year":"1991","journal-title":"Math. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"K\u00fc\u00e7\u00fckyavuz, S., and Sen, S. (2017). An introduction to two-stage stochastic mixed-integer programming. Leading Developments from INFORMS Communities, INFORMS.","DOI":"10.1287\/educ.2017.0171"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1007\/s11081-019-09471-0","article-title":"Electric power infrastructure planning under uncertainty: Stochastic dual dynamic integer programming (SDDiP) and parallelization scheme","volume":"21","author":"Lara","year":"2020","journal-title":"Optim. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106659","DOI":"10.1016\/j.ijepes.2020.106659","article-title":"Two-stage and multi-stage decompositions for the medium-term hydrothermal scheduling problem: A computational comparison of solution techniques","volume":"127","author":"Finardi","year":"2021","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Colonetti, B., Finardi, E.C., and de Oliveira, W. (2020). A Mixed-Integer and Asynchronous Level Decomposition with Application to the Stochastic Hydrothermal Unit-Commitment Problem. Algorithms, 13.","DOI":"10.3390\/a13090235"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Li, C., and Grossmann, I.E. (2021). A review of stochastic programming methods for optimization of process systems under uncertainty. Front. Chem. Eng., 34.","DOI":"10.3389\/fceng.2020.622241"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"e16933","DOI":"10.1002\/aic.16933","article-title":"Shale gas pad development planning under price uncertainty","volume":"66","author":"Li","year":"2020","journal-title":"AIChE J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/BF02614323","article-title":"Decomposition methods in stochastic programming","volume":"79","year":"1997","journal-title":"Math. Program."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/S0167-6377(98)00050-9","article-title":"Dual decomposition in stochastic integer programming","volume":"24","author":"Schultz","year":"1999","journal-title":"Oper. Res. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lim, C., Cochran, J.J., Cox, L.A., Keskinocak, P., Kharoufeh, J.P., and Smith, J.C. (2010). Relationship among Benders, Dantzig\u2013Wolfe, and Lagrangian Optimization. Wiley Encyclopedia of Operations Research and Management Science, John Wiley & Sons, Inc.","DOI":"10.1002\/9780470400531"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1007\/BF01386316","article-title":"Partitioning procedures for solving mixed-variables programming problems","volume":"4","author":"Benders","year":"1962","journal-title":"Numer. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/BF00934810","article-title":"Generalized Benders decomposition","volume":"10","author":"Geoffrion","year":"1972","journal-title":"J. Optim. Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10107-003-0375-9","article-title":"Logic-based Benders decomposition","volume":"96","author":"Hooker","year":"2003","journal-title":"Math. Program."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1023\/A:1013827731218","article-title":"A modification of Benders\u2019 decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse","volume":"22","author":"Sherali","year":"2002","journal-title":"J. Glob. Optim."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s10107-012-0615-y","article-title":"Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs","volume":"144","author":"Gade","year":"2014","journal-title":"Math. Program."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1933","DOI":"10.1137\/13092678X","article-title":"Finitely convergent decomposition algorithms for two-stage stochastic pure integer programs","volume":"24","author":"Zhang","year":"2014","journal-title":"SIAM J. Optim."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s10107-003-0475-6","article-title":"A finite branch-and-bound algorithm for two-stage stochastic integer programs","volume":"100","author":"Ahmed","year":"2004","journal-title":"Math. Program."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.ejor.2016.12.005","article-title":"The Benders decomposition algorithm: A literature review","volume":"259","author":"Rahmaniani","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1287\/opre.29.3.464","article-title":"Accelerating Benders decomposition: Algorithmic enhancement and model selection criteria","volume":"29","author":"Magnanti","year":"1981","journal-title":"Oper. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1287\/mnsc.20.5.822","article-title":"Multicommodity distribution system design by Benders decomposition","volume":"20","author":"Geoffrion","year":"1974","journal-title":"Manag. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10479-010-0806-y","article-title":"An interior-point Benders based branch-and-cut algorithm for mixed integer programs","volume":"210","author":"Elhedhli","year":"2013","journal-title":"Ann. Oper. Res."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Vanderbeck, F., and Wolsey, L.A. (2009). Reformulation and Decomposition of Integer Programs, Springer.","DOI":"10.1007\/978-3-540-68279-0_13"},{"key":"ref_31","unstructured":"Zaourar, S., and Malick, J. (2022, January 15). Quadratic Stabilization of Benders Decomposition. Available online: https:\/\/hal.archives-ouvertes.fr\/hal-01181273\/document."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1287\/mnsc.24.3.312","article-title":"A modified Benders\u2019 partitioning algorithm for mixed integer programming","volume":"24","author":"McDaniel","year":"1977","journal-title":"Manag. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1590\/S0101-74382012005000005","article-title":"Accelerating Benders decomposition with heuristic master problem solutions","volume":"32","author":"Costa","year":"2012","journal-title":"Pesqui. Oper."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Raidl, G.R., Baumhauer, T., and Hu, B. (2014). Speeding up logic-based Benders\u2019 decomposition by a metaheuristic for a bi-level capacitated vehicle routing problem. International Workshop on Hybrid Metaheuristics, Springer.","DOI":"10.1007\/978-3-319-07644-7_14"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0377-2217(94)90032-9","article-title":"On using approximations of the Benders master problem","volume":"77","author":"Holmberg","year":"1994","journal-title":"Eur. J. Oper. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1137\/S1052623497318700","article-title":"Inexact cuts in Benders decomposition","volume":"10","author":"Zakeri","year":"2000","journal-title":"SIAM J. Optim."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/0377-2217(88)90159-2","article-title":"A multicut algorithm for two-stage stochastic linear programs","volume":"34","author":"Birge","year":"1988","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1111\/j.1475-3995.2009.00706.x","article-title":"Accelerating Benders method using covering cut bundle generation","volume":"17","author":"Saharidis","year":"2010","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/BF02591718","article-title":"Cross decomposition for mixed integer programming","volume":"25","year":"1983","journal-title":"Math. Program."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1016\/S0377-2217(96)00401-8","article-title":"Accelerating the regularized decomposition method for two stage stochastic linear problems","volume":"101","year":"1997","journal-title":"Eur. J. Oper. Res."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1023\/A:1021858008222","article-title":"Decomposition Algorithms for Stochastic Programming on a Computational Grid","volume":"24","author":"Linderoth","year":"2003","journal-title":"Comput. Optim. Appl."},{"key":"ref_42","unstructured":"Zehtabian, S., and Bastin, F. (2022, January 15). Penalty Parameter Update Strategies in Progressive Hedging Algorithm. Available online: https:\/\/www.cirrelt.ca\/documentstravail\/cirrelt-2016-12.pdf."},{"key":"ref_43","unstructured":"Hiriart-Urruty, J.B., and Lemar\u00e9chal, C. (2013). Convex Analysis and Minimization Algorithms II, Springer Science & Business Media."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Kallrath, J. (2004). General Algebraic Modeling System (GAMS). Modeling Languages in Mathematical Optimization, Springer. Volume 88, Applied Optimization.","DOI":"10.1007\/978-1-4613-0215-5"},{"key":"ref_45","unstructured":"Infanger, G. (2022, January 15). GAMS\/DECIS User\u2019S Guide. Available online: https:\/\/www.gams.com\/docs\/DECIS-Users_Guide.pdf."},{"key":"ref_46","unstructured":"Ellison, F., Mitra, G., and Zverovich, V. (2022, January 15). FortSP: A Stochastic Programming Solver. Available online: https:\/\/citeseerx.ist.psu.edu\/viewdoc\/download?doi=10.1.1.178.5858&rep=rep1&type=pdf."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/BF02579036","article-title":"Lagrangean relaxation","volume":"11","author":"Guignard","year":"2003","journal-title":"Top"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1137\/17M1148189","article-title":"An asynchronous bundle-trust-region method for dual decomposition of stochastic mixed-integer programming","volume":"29","author":"Kim","year":"2019","journal-title":"SIAM J. Optim."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s12532-017-0128-z","article-title":"Algorithmic innovations and software for the dual decomposition method applied to stochastic mixed-integer programs","volume":"10","author":"Kim","year":"2017","journal-title":"Math. Program. Comput."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s10107-016-1001-y","article-title":"A cross-decomposition scheme with integrated primal\u2013dual multi-cuts for two-stage stochastic programming investment planning problems","volume":"157","author":"Mitra","year":"2016","journal-title":"Math. Program."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12532-021-00212-y","article-title":"Scalable branching on dual decomposition of stochastic mixed-integer programming problems","volume":"14","author":"Kim","year":"2022","journal-title":"Math. Program. Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s10287-010-0125-4","article-title":"Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems","volume":"8","author":"Watson","year":"2011","journal-title":"Comput. Manag. Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.orl.2013.02.003","article-title":"On parallelizing dual decomposition in stochastic integer programming","volume":"41","author":"Lubin","year":"2013","journal-title":"Oper. Res. Lett."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Gabay, D., and Mercier, B. (1975). A Dual Algorithm for the Solution of Non Linear Variational Problems via Finite Element Approximation.","DOI":"10.1016\/0898-1221(76)90003-1"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF01581204","article-title":"On the Douglas\u2014Rachford splitting method and the proximal point algorithm for maximal monotone operators","volume":"55","author":"Eckstein","year":"1992","journal-title":"Math. Program."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s10107-016-1000-z","article-title":"Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs","volume":"157","author":"Gade","year":"2016","journal-title":"Math. Program."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.epsr.2011.09.006","article-title":"Applying different decomposition schemes using the progressive hedging algorithm to the operation planning problem of a hydrothermal system","volume":"83","author":"Finardi","year":"2012","journal-title":"Electr. Power Syst. Res."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1137\/16M1076290","article-title":"Combining Progressive Hedging with a Frank\u2013Wolfe Method to Compute Lagrangian Dual Bounds in Stochastic Mixed-Integer Programming","volume":"28","author":"Boland","year":"2018","journal-title":"SIAM J. Optim."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/j.orl.2015.03.008","article-title":"Integration of progressive hedging and dual decomposition in stochastic integer programs","volume":"43","author":"Guo","year":"2015","journal-title":"Oper. Res. Lett."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s12532-012-0036-1","article-title":"PySP: Modeling and solving stochastic programs in Python","volume":"4","author":"Watson","year":"2012","journal-title":"Math. Program. Comput."},{"key":"ref_61","unstructured":"Ahmed, S., Garcia, R., Kong, N., Ntaimo, L., Parija, G., Qiu, F., and Sen, S. (2022, March 10). SIPLIB: A Stochastic Integer Programming Test Problem Library. Available online: https:\/\/www2.isye.gatech.edu\/~sahmed\/siplib\/."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/s10898-004-5910-6","article-title":"The million-variable \u201cmarch\u201d for stochastic combinatorial optimization","volume":"32","author":"Ntaimo","year":"2005","journal-title":"J. Glob. Optim."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1287\/opre.1090.0693","article-title":"Disjunctive decomposition for two-stage stochastic mixed-binary programs with random recourse","volume":"58","author":"Ntaimo","year":"2010","journal-title":"Oper. Res."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1023\/B:ANOR.0000004773.66339.df","article-title":"Dynamic capacity acquisition and assignment under uncertainty","volume":"124","author":"Ahmed","year":"2003","journal-title":"Ann. Oper. Res."},{"key":"ref_65","unstructured":"Hart, W.E., Laird, C.D., Watson, J.P., Woodruff, D.L., Hackebeil, G.A., Nicholson, B.L., and Siirola, J.D. (2012). Pyomo-Optimization Modeling in Python, Springer."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.compchemeng.2013.11.011","article-title":"A new decomposition algorithm for multistage stochastic programs with endogenous uncertainties","volume":"62","author":"Gupta","year":"2014","journal-title":"Comput. Chem. Eng."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Ryan, S.M., Wets, R.J.B., Woodruff, D.L., Silva-Monroy, C., and Watson, J.P. (2013, January 21\u201325). Toward scalable, parallel progressive hedging for stochastic unit commitment. Proceedings of the 2013 IEEE Power & Energy Society General Meeting, Vancouver, BC, Canada.","DOI":"10.1109\/PESMG.2013.6673013"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.cor.2017.04.007","article-title":"Scenario cluster Lagrangean decomposition for risk averse in multistage stochastic optimization","volume":"85","author":"Escudero","year":"2017","journal-title":"Comput. Oper. Res."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1212","DOI":"10.1137\/S1052623497324242","article-title":"Computational experience with an interior point cutting plane algorithm","volume":"10","author":"Mitchell","year":"2000","journal-title":"SIAM J. Optim."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1287\/ijoc.2020.0987","article-title":"SDDP. jl: A Julia package for stochastic dual dynamic programming","volume":"33","author":"Dowson","year":"2021","journal-title":"INFORMS J. Comput."},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Biel, M., and Johansson, M. (2022). Efficient stochastic programming in Julia. INFORMS J. Comput.","DOI":"10.1287\/ijoc.2022.1158"},{"key":"ref_72","unstructured":"Ding, L., Ahmed, S., and Shapiro, A. (2022, March 15). A python Package for Multi-Stage Stochastic Programming. Available online: http:\/\/www.optimization-online.org\/DB_FILE\/2019\/05\/7199.pdf."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1007\/s10898-019-00820-y","article-title":"A finite \u03f5-convergence algorithm for two-stage stochastic convex nonlinear programs with mixed-binary first and second-stage variables","volume":"75","author":"Li","year":"2019","journal-title":"J. Glob. Optim."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s10898-019-00816-8","article-title":"A generalized Benders decomposition-based branch and cut algorithm for two-stage stochastic programs with nonconvex constraints and mixed-binary first and second stage variables","volume":"75","author":"Li","year":"2019","journal-title":"J. Glob. Optim."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"1245","DOI":"10.1007\/s11081-020-09563-2","article-title":"Sample average approximation for stochastic nonconvex mixed integer nonlinear programming via outer-approximation","volume":"22","author":"Li","year":"2021","journal-title":"Optim. Eng."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s10898-019-00769-y","article-title":"A scalable global optimization algorithm for stochastic nonlinear programs","volume":"75","author":"Cao","year":"2019","journal-title":"J. Glob. Optim."},{"key":"ref_77","doi-asserted-by":"crossref","unstructured":"F\u00fcllner, C., and Rebennack, S. (2022). Non-convex nested Benders decomposition. Math. Program., 1\u201338.","DOI":"10.1007\/s10107-021-01740-0"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1007\/s11750-016-0413-4","article-title":"Regularized optimization methods for convex MINLP problems","volume":"24","year":"2016","journal-title":"Top"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1007\/s11228-017-0437-4","article-title":"Solving stochastic programming problems with risk measures by progressive hedging","volume":"26","author":"Rockafellar","year":"2018","journal-title":"Set-Valued Var. Anal."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Kall, P., and Mayer, J. (1998). On testing SLP codes with SLP-IOR. New Trends in Mathematical Programming, Springer.","DOI":"10.1007\/978-1-4757-2878-1_10"},{"key":"ref_81","first-page":"83","article-title":"Optimierung des Gasbezugs im liberalisierten Gasmarkt unter Berucksichtigung von Rohren-und Untertagespeichern","volume":"2018","author":"Konig","year":"2007","journal-title":"VDI BERICHTE"},{"key":"ref_82","unstructured":"Keller, E. (1984). GENSLP: A Program for Generating Input for Stochastic Linear Programs with Complete Fixed Recourse, IOR, University of Zurich. Manuscript."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s12532-012-0038-z","article-title":"A computational study of a solver system for processing two-stage stochastic LPs with enhanced Benders decomposition","volume":"4","author":"Zverovich","year":"2012","journal-title":"Math. Program. Comput."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"47","DOI":"10.21314\/JEM.2011.061","article-title":"Modeling and optimizing risk in the strategic gas-purchase planning problem of local distribution companies","volume":"4","author":"Koberstein","year":"2011","journal-title":"J. Energy Mark."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/BF01580883","article-title":"A regularized decomposition method for minimizing a sum of polyhedral functions","volume":"35","year":"1986","journal-title":"Math. Program."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF01585555","article-title":"New variants of bundle methods","volume":"69","author":"Nemirovskii","year":"1995","journal-title":"Math. Program."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1023\/A:1011336210885","article-title":"Variable Neighborhood Decomposition Search","volume":"7","author":"Hansen","year":"2001","journal-title":"J. Heuristics"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/4\/103\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:41:13Z","timestamp":1760136073000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/4\/103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,22]]},"references-count":87,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["a15040103"],"URL":"https:\/\/doi.org\/10.3390\/a15040103","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,22]]}}}