{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T15:33:27Z","timestamp":1765812807295,"version":"3.48.0"},"reference-count":67,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,1,21]],"date-time":"2019-01-21T00:00:00Z","timestamp":1548028800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,1,21]],"date-time":"2019-01-21T00:00:00Z","timestamp":1548028800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Hungarian Government, co-financed by the European Social Fund","award":["EFOP-3.6.1-16-2016-00006"],"award-info":[{"award-number":["EFOP-3.6.1-16-2016-00006"]}]},{"name":"Hungarian Government and co-financed by the European Social Fund","award":["EFOP-3.6.1-16-2016-00006"],"award-info":[{"award-number":["EFOP-3.6.1-16-2016-00006"]}]},{"name":"Hungarian Government and co-financed by the European Social Fund","award":["EFOP-3.6.1-16-2016-00006"],"award-info":[{"award-number":["EFOP-3.6.1-16-2016-00006"]}]},{"name":"Hungarian Government and co-financed by the European Social Fund","award":["EFOP-3.6.1-16-2016-00006"],"award-info":[{"award-number":["EFOP-3.6.1-16-2016-00006"]}]},{"name":"Hungarian Government and co-financed by the European Social Fund","award":["EFOP-3.6.1-16-2016-00006"],"award-info":[{"award-number":["EFOP-3.6.1-16-2016-00006"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann Oper Res"],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We propose a randomized gradient method for handling a convex function whose gradient computation is demanding. The method bears a resemblance to the stochastic approximation family. But in contrast to stochastic approximation, the present method builds a model problem. The approach is adapted to probability maximization and probabilistic constrained problems. We discuss simulation procedures for gradient estimation.<\/jats:p>","DOI":"10.1007\/s10479-019-03143-z","type":"journal-article","created":{"date-parts":[[2019,1,21]],"date-time":"2019-01-21T12:32:17Z","timestamp":1548073937000},"page":"1073-1104","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A randomized method for handling a difficult function in a convex optimization problem, motivated by probabilistic programming"],"prefix":"10.1007","volume":"354","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9446-1566","authenticated-orcid":false,"given":"Csaba I.","family":"F\u00e1bi\u00e1n","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edit","family":"Csizm\u00e1s","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rajmund","family":"Drenyovszki","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tibor","family":"Vajnai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L\u00f3r\u00e1nt","family":"Kov\u00e1cs","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tam\u00e1s","family":"Sz\u00e1ntai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,1,21]]},"reference":[{"key":"3143_CR1","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0266-8920(98)00003-4","volume":"13","author":"R Ambartzumian","year":"1998","unstructured":"Ambartzumian, R., Der Kiureghian, A., Ohanian, V., & Sukiasian, H. (1998). Multinormal probability by sequential conditioned importance sampling: Theory and applications. Probabilistic Engineering Mechanics, 13, 299\u2013308.","journal-title":"Probabilistic Engineering Mechanics"},{"key":"3143_CR2","first-page":"5","volume":"10","author":"T Arnold","year":"2014","unstructured":"Arnold, T., Henrion, R., M\u00f6ller, A., & Vigerske, S. (2014). A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints. Pacific Journal of Optimization, 10, 5\u201320.","journal-title":"Pacific Journal of Optimization"},{"key":"3143_CR3","volume-title":"Adaptive algorithms and stochastic approximations","author":"A Benveniste","year":"1993","unstructured":"Benveniste, A., M\u00e9tivier, M., & Priouret, P. (1993). Adaptive algorithms and stochastic approximations. New York: Springer."},{"key":"3143_CR4","volume-title":"Introduction to stochastic programming","author":"J Birge","year":"1997","unstructured":"Birge, J., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer."},{"key":"3143_CR5","unstructured":"Boros, E., & Veneziani, P. (2002). Bounds of degree 3 for the probability of the union of events. Technical report, Rutgers Center for Operations Research, RUTCOR Research Report 3-2002."},{"key":"3143_CR6","unstructured":"Buksz\u00e1r, J., Pr\u00e9kopa, A. (2000). Probability bounds with cherry-trees. Technical report, Rutgers Center for Operations Research, RUTCOR Research Report 44-2000."},{"key":"3143_CR7","first-page":"69","volume":"2","author":"J Buksz\u00e1r","year":"1999","unstructured":"Buksz\u00e1r, J., & Sz\u00e1ntai, T. (1999). Probability bounds given by hyper-cherry-trees. Alkalmazott Matematikai Lapok, 2, 69\u201385. (in Hungarian) .","journal-title":"Alkalmazott Matematikai Lapok"},{"key":"3143_CR8","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1080\/10556788.2013.871282","volume":"29","author":"W de Oliveira","year":"2014","unstructured":"de Oliveira, W., & Sagastiz\u00e1bal, C. (2014). Level bundle methods for oracles with on-demand accuracy. Optimization Methods and Software, 29, 1180\u20131209.","journal-title":"Optimization Methods and Software"},{"key":"3143_CR9","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1137\/100808289","volume":"21","author":"W de Oliveira","year":"2011","unstructured":"de Oliveira, W., Sagastiz\u00e1bal, C., & Scheimberg, S. (2011). Inexact bundle methods for two-stage stochastic programming. SIAM Journal on Optimization, 21, 517\u2013544.","journal-title":"SIAM Journal on Optimization"},{"key":"3143_CR10","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/BF01399006","volume":"35","author":"I De\u00e1k","year":"1980","unstructured":"De\u00e1k, I. (1980). Three digit accurate multiple normal probabilities. Numerische Mathematik, 35, 369\u2013380.","journal-title":"Numerische Mathematik"},{"key":"3143_CR11","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1080\/00949658608810951","volume":"26","author":"I De\u00e1k","year":"1986","unstructured":"De\u00e1k, I. (1986). Computing probabilities of rectangles in case of multinormal distributions. Journal of Statistical Computation and Simulation, 26, 101\u2013114.","journal-title":"Journal of Statistical Computation and Simulation"},{"key":"3143_CR12","first-page":"920","volume":"11","author":"I De\u00e1k","year":"2002","unstructured":"De\u00e1k, I., Gassmann, H., & Sz\u00e1ntai, T. (2002). Computing multivariate normal probabilities: A new look. Journal of Statistical Computation and Simulation, 11, 920\u2013949.","journal-title":"Journal of Statistical Computation and Simulation"},{"key":"3143_CR13","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s001860400371","volume":"60","author":"D Dentcheva","year":"2004","unstructured":"Dentcheva, D., Lai, B., & Ruszczy\u0144ski, A. (2004). Dual methods for probabilistic optimization problems. Mathematical Methods of Operations Research, 60, 331\u2013346.","journal-title":"Mathematical Methods of Operations Research"},{"key":"3143_CR14","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s10107-012-0539-6","volume":"138","author":"D Dentcheva","year":"2013","unstructured":"Dentcheva, D., & Martinez, G. (2013). Regularization methods for optimization problems with probabilistic constraints. Mathematical Programming, 138, 223\u2013251.","journal-title":"Mathematical Programming"},{"key":"3143_CR15","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/PL00011393","volume":"89","author":"D Dentcheva","year":"2000","unstructured":"Dentcheva, D., Pr\u00e9kopa, A., & Ruszczy\u0144ski, A. (2000). Concavity and efficient points of discrete distributions in probabilistic programming. Mathematical Programming, 89, 55\u201377.","journal-title":"Mathematical Programming"},{"key":"3143_CR16","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1007\/BF01071091","volume":"5","author":"Y Ermoliev","year":"1969","unstructured":"Ermoliev, Y. (1969). On the stochastic quasigradient method and stochastic quasi-Feyer sequences. Cybernetics, 5, 208\u2013220.","journal-title":"Cybernetics"},{"key":"3143_CR17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/17442508308833246","volume":"9","author":"Y Ermoliev","year":"1983","unstructured":"Ermoliev, Y. (1983). Stochastic quasigradient methods and their application to system optimization. Stochastics, 9, 1\u201336.","journal-title":"Stochastics"},{"key":"3143_CR18","unstructured":"F\u00e1bi\u00e1n, C., & Sz\u00e1ntai, T. (2017). A randomized method for smooth convexminimization, motivated by probability maximization. Technical report, OptimizationOnline, March 2017."},{"key":"3143_CR19","first-page":"105","volume":"15","author":"C F\u00e1bi\u00e1n","year":"2018","unstructured":"F\u00e1bi\u00e1n, C., Csizm\u00e1s, E., Drenyovszki, R., van Ackooij, W., Vajnai, T., Kov\u00e1cs, L., et al. (2018). Probability maximization by inner approximation. Acta Polytechnica Hungarica, 15, 105\u2013125.","journal-title":"Acta Polytechnica Hungarica"},{"key":"3143_CR20","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1007\/s10100-014-0344-9","volume":"23","author":"C F\u00e1bi\u00e1n","year":"2015","unstructured":"F\u00e1bi\u00e1n, C., Eretnek, K., & Papp, O. (2015). A regularized simplex method. Central European Journal of Operations Research, 23, 877\u2013898.","journal-title":"Central European Journal of Operations Research"},{"key":"3143_CR21","unstructured":"Frangioni, A. (2018). Standard bundle methods: Untrusted models and duality. Technical reports, Department of Informatics, University of Pisa, Italy. http:\/\/eprints.adm.unipi.it\/2378\/1\/StandardBundle.pdf. Accessed August 26, 2018"},{"key":"3143_CR22","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1137\/S1052623498342186","volume":"13","author":"A Frangioni","year":"2002","unstructured":"Frangioni, A. (2002). Generalized bundle methods. SIAM Journal on Optimization, 13, 117\u2013156.","journal-title":"SIAM Journal on Optimization"},{"key":"3143_CR23","first-page":"89","volume":"4","author":"A Gaivoronski","year":"1978","unstructured":"Gaivoronski, A. (1978). Nonstationary stochastic programming problems. Kybernetika, 4, 89\u201392.","journal-title":"Kybernetika"},{"key":"3143_CR24","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/978-3-642-61370-8_11","volume-title":"Numerical techniques for stochastic optimization","author":"H Gassmann","year":"1988","unstructured":"Gassmann, H. (1988). Conditional probability and conditional expectation of a random vector. In Y. Ermoliev & R. B. Wets (Eds.), Numerical techniques for stochastic optimization (pp. 237\u2013254). Berlin: Springer."},{"key":"3143_CR25","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1080\/10618600.1992.10477010","volume":"1","author":"A Genz","year":"1992","unstructured":"Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1, 141\u2013150.","journal-title":"Journal of Computational and Graphical Statistics"},{"key":"3143_CR26","doi-asserted-by":"publisher","unstructured":"Hantoute, A., Henrion, R., P\u00e9rez-Aros, P. (2018). Subdifferential characterization of probability functions under Gaussian distribution. Mathematical Programming. https:\/\/doi.org\/10.1007\/s10107-018-1237-9","DOI":"10.1007\/s10107-018-1237-9"},{"key":"3143_CR27","unstructured":"Henrion, R. (2004). Introduction to chance constraint programming. Technical report, Weierstrass-Institut f\u00fcr Angewandte Analysis und Stochastik. www.wias-berlin.de\/people\/henrion\/ccp.ps"},{"key":"3143_CR28","doi-asserted-by":"crossref","unstructured":"Higle, J., Sen, S. (1996). Stochastic decomposition: A statistical method for large scale stochastic linear programming. In: Nonconvex optimization and its applications vol. 8. Springer.","DOI":"10.1007\/978-1-4615-4115-8"},{"key":"3143_CR29","doi-asserted-by":"crossref","first-page":"597","DOI":"10.2307\/3212481","volume":"13","author":"D Hunter","year":"1976","unstructured":"Hunter, D. (1976). Bounds for the probability of a union. Journal of Applied Probbility, 13, 597\u2013603.","journal-title":"Journal of Applied Probbility"},{"key":"3143_CR30","doi-asserted-by":"crossref","unstructured":"Impagliazzo, R., & Kabanets, V. (2010). Constructive proofs of concentration bounds. In: M. Serna, R. Shaltiel, K. Jansen, J. Rolim (Eds) Approximation, randomization, and combinatorial optimization. Algorithms and techniques, RANDOM 2010, APPROX 2010. Lecture Notes in Computer Science vol. 6302 (pp. 617\u2013631). Berlin: Springer.","DOI":"10.1007\/978-3-642-15369-3_46"},{"key":"3143_CR31","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s10107-010-0434-y","volume":"133","author":"G Lan","year":"2012","unstructured":"Lan, G. (2012). An optimal method for stochastic composite optimization. Mathematical Programming, 133, 365\u2013397.","journal-title":"Mathematical Programming"},{"key":"3143_CR32","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF01585555","volume":"69","author":"C Lemar\u00e9chal","year":"1995","unstructured":"Lemar\u00e9chal, C., Nemirovski, A., & Nesterov, Y. (1995). New variants of bundle methods. Mathematical Programming, 69, 111\u2013147.","journal-title":"Mathematical Programming"},{"key":"3143_CR33","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s10107-008-0247-4","volume":"122","author":"J Luedtke","year":"2010","unstructured":"Luedtke, J., Ahmed, S., & Nemhauser, G. (2010). An integer programming approach for linear programs with probabilistic constraints. Mathematical Programming, 122, 247\u2013272.","journal-title":"Mathematical Programming"},{"key":"3143_CR34","doi-asserted-by":"crossref","unstructured":"Luenberger, D., Ye, Y. (2008). Linear and nonlinear programming. In International series in operations research and management science. Springer.","DOI":"10.1007\/978-0-387-74503-9"},{"key":"3143_CR35","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1287\/moor.1030.0064","volume":"29","author":"G M\u00e1di-Nagy","year":"2004","unstructured":"M\u00e1di-Nagy, G., & Pr\u00e9kopa, A. (2004). On multivariate discrete moment problems and their applications to bounding expectations and probabilities. Mathematics of Operations Research, 29, 229\u2013258.","journal-title":"Mathematics of Operations Research"},{"key":"3143_CR36","volume-title":"Stochastic linear programming algorithms: A comparison based on a model management system","author":"J Mayer","year":"1998","unstructured":"Mayer, J. (1998). Stochastic linear programming algorithms: A comparison based on a model management system. Philadelphia: Gordon and Breach Science Publishers."},{"key":"3143_CR37","unstructured":"Nemirovski, A., Yudin, D. (1978). On Cezari\u2019s convergence of the steepest descent method for approximating saddle point of convex-concave functions. Soviet Mathematics Doklady, 19."},{"key":"3143_CR38","doi-asserted-by":"crossref","first-page":"1574","DOI":"10.1137\/070704277","volume":"19","author":"A Nemirovski","year":"2009","unstructured":"Nemirovski, A., Juditsky, A., Lan, G., & Shapiro, A. (2009). Robust stochastic approximation approach to stochastic programming. SIAM Journal on Optimization, 19, 1574\u20131609.","journal-title":"SIAM Journal on Optimization"},{"key":"3143_CR39","volume-title":"Problem complexity and method efficiency in optimization, Wiley-interscience series in discrete mathematics","author":"A Nemirovski","year":"1983","unstructured":"Nemirovski, A., & Yudin, D. (1983). Problem complexity and method efficiency in optimization, Wiley-interscience series in discrete mathematics (Vol. 15). New York: Wiley."},{"key":"3143_CR40","first-page":"543","volume":"269","author":"Y Nesterov","year":"1983","unstructured":"Nesterov, Y. (1983). A method for unconstrained convex minimization with the rate of convergence of $$o(1\/k^2)$$. Doklady AN SSSR, 269, 543\u2013547.","journal-title":"Doklady AN SSSR"},{"key":"3143_CR41","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s10107-007-0149-x","volume":"120","author":"Y Nesterov","year":"2009","unstructured":"Nesterov, Y. (2009). Primal-dual subgradient methods for convex problems. Mathematical Programming, 120, 221\u2013259.","journal-title":"Mathematical Programming"},{"key":"3143_CR42","doi-asserted-by":"crossref","first-page":"1559","DOI":"10.1016\/j.automatica.2008.01.017","volume":"44","author":"Y Nesterov","year":"2008","unstructured":"Nesterov, Y., & Vial, J. P. (2008). Confidence level solutions for stochastic programming. Automatica, 44, 1559\u20131568.","journal-title":"Automatica"},{"key":"3143_CR43","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1137\/S0097539793250767","volume":"26","author":"A Panconesi","year":"1997","unstructured":"Panconesi, A., & Srinivasan, A. (1997). Randomized distributed edge coloring via an extension of the Chernoff\u2013Hoeffding bounds. SIAM Journal on Computing, 26, 350\u2013368.","journal-title":"SIAM Journal on Computing"},{"key":"3143_CR44","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/978-3-642-61370-8_17","volume-title":"Numerical techniques for stochastic optimization","author":"G Pflug","year":"1988","unstructured":"Pflug, G. (1988). Stepsize rules, stopping times and their implementation in stochastic quasigradient algorithms. In Y. Ermoliev & R. Wets (Eds.), Numerical techniques for stochastic optimization (pp. 353\u2013372). Berlin: Springer."},{"key":"3143_CR45","volume-title":"Optimization of stochastic models. The interface between simulation and optimization","author":"G Pflug","year":"1996","unstructured":"Pflug, G. (1996). Optimization of stochastic models. The interface between simulation and optimization. Boston: Kluwer."},{"key":"3143_CR46","first-page":"98","volume":"7","author":"B Polyak","year":"1990","unstructured":"Polyak, B. (1990). New stochastic approximation type procedures. Automat i Telemekh, 7, 98\u2013107.","journal-title":"Automat i Telemekh"},{"key":"3143_CR47","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1137\/0330046","volume":"30","author":"B Polyak","year":"1992","unstructured":"Polyak, B., & Juditsky, A. (1992). Acceleration of stochastic approximation by averaging. SIAM Journal on Control and Optimization, 30, 838\u2013855.","journal-title":"SIAM Journal on Control and Optimization"},{"key":"3143_CR48","unstructured":"Pr\u00e9kopa, A., Vizv\u00e1ri, B., Reg\u0151s, G. (1995). Lower and upper bounds on probabilities of Boolean functions of events. Technical report, Rutgers Center for Operations Research, RUTCOR Research Report 36-95."},{"key":"3143_CR49","first-page":"441","volume":"34","author":"A Pr\u00e9kopa","year":"1990","unstructured":"Pr\u00e9kopa, A. (1990). Dual method for a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution. ZOR: Methods and Models of Operations Research, 34, 441\u2013461.","journal-title":"ZOR: Methods and Models of Operations Research"},{"key":"3143_CR50","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-017-3087-7","volume-title":"Stochastic programming","author":"A Pr\u00e9kopa","year":"1995","unstructured":"Pr\u00e9kopa, A. (1995). Stochastic programming. Dordrecht: Kluwer Academic Publishers."},{"key":"3143_CR51","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/978-1-4757-2878-1_18","volume-title":"New trends in mathematical programming","author":"A Pr\u00e9kopa","year":"1998","unstructured":"Pr\u00e9kopa, A., Vizv\u00e1ri, B., & Badics, T. (1998). Programming under probabilistic constraint with discrete random variable. In F. Giannesi, T. Rapcs\u00e1k, & S. Koml\u00f3si (Eds.), New trends in mathematical programming (pp. 235\u2013255). Dordrecht: Kluwer."},{"key":"3143_CR52","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1214\/aoms\/1177729586","volume":"22","author":"H Robbins","year":"1951","unstructured":"Robbins, H., & Monro, S. (1951). A stochastic approximation method. Annals of Mathematical Statistics, 22, 400\u2013407.","journal-title":"Annals of Mathematical Statistics"},{"key":"3143_CR53","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex analysis","author":"R Rockafellar","year":"1970","unstructured":"Rockafellar, R. (1970). Convex analysis. Princeton: Princeton University Press."},{"key":"3143_CR54","doi-asserted-by":"crossref","unstructured":"Ruszczy\u0144ski, A., Syski, W. (1986). A method of aggregate stochastic subgradients with on-line stepsize rules for convex stochastic programming problems. In: Pr\u00e9kopa A, Wets R (eds) Stochastic programming 84 Part II, Mathematical Programming Studies (vol. 28, pp. 113\u2013131) Berlin: Springer.","DOI":"10.1007\/BFb0121128"},{"key":"3143_CR55","doi-asserted-by":"crossref","DOI":"10.1515\/9781400841059","volume-title":"Nonlinear optmization","author":"A Ruszczy\u0144ski","year":"2006","unstructured":"Ruszczy\u0144ski, A. (2006). Nonlinear optmization. Princeton: Princeton University Press."},{"key":"3143_CR56","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1214\/aoms\/1177731088","volume":"16","author":"C Stein","year":"1945","unstructured":"Stein, C. (1945). A two-sample test for a linear hypothesis whose power is indpendent of the variance. Annals of Mathematical Statistics, 16, 243\u2013258.","journal-title":"Annals of Mathematical Statistics"},{"key":"3143_CR57","unstructured":"Sz\u00e1ntai, T. (1985). Numerical evaluation of probabilities concerning multidimensional probability distributions. Thesis, Hungarian Academy of Sciences, Budapest."},{"key":"3143_CR58","first-page":"27","volume":"2","author":"T Sz\u00e1ntai","year":"1976","unstructured":"Sz\u00e1ntai, T. (1976). A procedure for determination of the multivariate normal probability distribution function and its gradient values. Alkalmazott Matematikai Lapok, 2, 27\u201339. (in Hungarian) .","journal-title":"Alkalmazott Matematikai Lapok"},{"key":"3143_CR59","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/978-3-642-61370-8_10","volume-title":"Numerical techniques for stochastic optimization","author":"T Sz\u00e1ntai","year":"1988","unstructured":"Sz\u00e1ntai, T. (1988). A computer code for solution of probabilistic-constrained stochastic programming problems. In Y. Ermoliev & R. B. Wets (Eds.), Numerical techniques for stochastic optimization (pp. 229\u2013235). Berlin: Springer."},{"key":"3143_CR60","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1023\/A:1019211000153","volume":"100","author":"T Sz\u00e1ntai","year":"2000","unstructured":"Sz\u00e1ntai, T. (2000). Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function. Annals of Operations Research, 100, 85\u2013101.","journal-title":"Annals of Operations Research"},{"key":"3143_CR61","unstructured":"Sz\u00e1sz, P. (1951). Elements of differential and integral calculus. Budapest: K\u00f6zoktat\u00e1s\u00fcgyi Kiad\u00f3v\u00e1llalat (in Hungarian)."},{"key":"3143_CR62","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1239\/aap\/1444308883","volume":"47","author":"HW Teng","year":"2015","unstructured":"Teng, H. W., Kang, M. H., & Fuh, C. D. (2015). On spherical Monte Carlo simulations for multivariate normal probabilities. Advances in Applied Probability, 47, 817\u2013836.","journal-title":"Advances in Applied Probability"},{"key":"3143_CR63","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/0095-8956(86)90044-4","volume":"41","author":"I Tomescu","year":"1986","unstructured":"Tomescu, I. (1986). Hypertrees and Bonferroni inequalities. Journal of Combinatorial Theory, Series B, 41, 209\u2013217.","journal-title":"Journal of Combinatorial Theory, Series B"},{"key":"3143_CR64","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/978-3-642-61370-8_18","volume-title":"Numerical techniques for stochastic optimization","author":"S Uryasev","year":"1988","unstructured":"Uryasev, S. (1988). Adaptive stochastic quasigradient methods. In Y. Ermoliev & R. Wets (Eds.), Numerical techniques for stochastic optimization (pp. 373\u2013384). Berlin: Springer."},{"key":"3143_CR65","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.cor.2016.08.002","volume":"77","author":"W van Ackooij","year":"2017","unstructured":"van Ackooij, W., Berge, V., de Oliveira, W., & Sagastiz\u00e1bal, C. (2017). Probabilistic optimization via approximate p-efficient points and bundle methods. Computers & Operations Research, 77, 177\u2013193.","journal-title":"Computers & Operations Research"},{"key":"3143_CR66","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1137\/120903099","volume":"24","author":"W van Ackooij","year":"2014","unstructured":"van Ackooij, W., & Sagastiz\u00e1bal, C. (2014). Constrained bundle methods for upper inexact oracles with application to joint chance constrained energy problems. SIAM Journal on Optimization, 24, 733\u2013765.","journal-title":"SIAM Journal on Optimization"},{"key":"3143_CR67","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1093\/biomet\/69.2.297","volume":"69","author":"K Worsley","year":"1982","unstructured":"Worsley, K. (1982). An improved Bonferroni inequality and applications. Biometrika, 69, 297\u2013302.","journal-title":"Biometrika"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-019-03143-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10479-019-03143-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-019-03143-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T15:29:28Z","timestamp":1765812568000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10479-019-03143-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,21]]},"references-count":67,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["3143"],"URL":"https:\/\/doi.org\/10.1007\/s10479-019-03143-z","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"type":"print","value":"0254-5330"},{"type":"electronic","value":"1572-9338"}],"subject":[],"published":{"date-parts":[[2019,1,21]]},"assertion":[{"value":"21 January 2019","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}