Abstract
In 1987, J. Dulá considered the problem of finding an upper bound for the expectation of a so-called “simplicial” function of a random vector and used for this purpose first and total second moments. Under the same moment conditions we consider some different cases of “recourse” functions and demonstrate how the related moment problems can be solved by solving nonsmooth (unconstrained) optimization problems and thereafter satisfying simple linear constraint systems.
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References
J.R. Birge and R.J.-B. Wets, Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse, Math. Progr.Study 27 (1986) 54–102.
J.R. Birge and R.J.-B. Wets, Computing bounds for stochastic programming problems by means of a generalized moment problem, Math. Oper. Res. 12 (1987) 149–162.
J.H. Dulá, An upper bound on the expectation of sublinear functions of multivariate random variables, Math. Progr., to appear.
J. Dupačová, Minimax stochastic programs with nonconvex nonseparable penalty functions, in:Progress in Operations Research, ed. A. Prékopa (North-Holland, Amsterdam, 1976) pp. 303–316.
J. Dupačová, Minimax stochastic programs with nonseparable penalties in:Optimization Techniques, Part I, Lecture Notes in Control and Information Science 22, eds. K. Iracki, K. Malanowski and S. Walukiewicz (Springer, Berlin, 1980) pp. 157–163.
J. Dupačová, The minimax approach to stochastic programming and an illustrative application, Stochastics 20 (1987) 73–88.
K. Frauendorfer, Solving SLP recourse problems with arbitrary multivariate distributions — the dependent case, Math. Oper. Res. 13 (1988) 377–394.
K. Frauendorfer and P. Kall, A solution method for SLP recourse problems with arbitrary multivariate distributions — the independent case, Probl. Control Inf. Theory 17 (1988) 177–205.
H. Gassmann and W.T. Ziemba, A right upper bound for the expectation of a convex function of a multivariate random variable, Math. Progr. Study 27 (1986) 39–53.
K. Glashoff and S.A. Gustafson,Einführung in die Lineare Optimierung, (Wissenschaftliche Buchgesellschaft, Darmstadt, 1978).
D.B. Hausch and W.T. Ziemba, Bounds on the value of information in uncertain decision problems, II, Stochastics 10 (1983) 181–217.
P. Kall, Stochastic programs with recourse: an upper bound and the related moment problem, ZOR 31 (1987) A119-A141.
P. Kall, Stochastic programming with recourse: upper bounds and moment problems — a review, in:Advances in Mathematical Optimization, eds. J. Guddat et al. (Akademie-Verlag, Berlin, 1988) pp. 86–103.
S. Karlin and W.J. Studden,Tschebycheff Systems: With Applications in Analysis and Statistics, (Interscience, New York, 1966).
J.H.B. Kemperman, The general moment problem, a geometric approach, Ann. Math. Statist. 39 (1968) 93–122.
M.G. Krein and A.A. Nudel'man, The Markov moment problem and extremal problems, Trans. Math. Mon. 50 (AMS, Providence, RI, 1977).
H. Richter, Parameterfreie Abschätzung und Realisierung von Erwartungswerten, Blätter Dt. Ges. Versicherungsmath. 3 (1957) 147–161.
W.W. Rogosinski, Moment of nonnegative mass, Proc. Roy. Soc. London A 245 (1958) 1–27.
H. Schramm, Eine Kombination von Bundle- und Trust-Region-Verfahren zur Lösung nicht-differenzierbarer Optimierungsprobleme, Dissertation, Univ. Bayreuth, Bayreuther Mathematische Schriften 30 (1989).
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Kall, P. An upper bound for SLP using first and total second moments. Ann Oper Res 30, 267–276 (1991). https://doi.org/10.1007/BF02204820
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DOI: https://doi.org/10.1007/BF02204820