Risk aversion in two-stage stochastic integer programming. (English) Zbl 1246.90111
Infanger, Gerd (ed.), Stochastic programming. The state of the art. In honor of George B. Dantzig. New York, NY: Springer (ISBN 978-1-4419-1641-9/hbk; 978-1-4419-1642-6/ebook). International Series in Operations Research & Management Science 150, 165-187 (2011).
In this chapter the author considers the random optimization problem
\[
\min \{c^{T}x+q^{T}y+q^{\prime T}y^{\prime }:Tx+Wy+W^{\prime }y^{\prime }=z\left( \omega \right) ,x\in X,y\in \mathbb{Z}_{+}^{\bar{m}},y^{\prime }\in \mathbb{R}_{+}^{m^{\prime }}\},(1)
\]
where the ingredients of (1) have conformable dimensions, \(W,W^{\prime }\) are rational matrices, and \(X\subseteq \mathbb{R}^{m}\) is a nonempty polyhedron, possibly involving integer requirements to components of \(x.\) He survey structural and algorithmic results for optimization problems which result from (1) (the expectation model, riskaversion model and mean-risk model).
For the entire collection see [Zbl 1203.90008].
For the entire collection see [Zbl 1203.90008].
Reviewer: I. M. Stancu-Minasian (Bucureşti)
MSC:
| 90C15 | Stochastic programming |
Keywords:
stochastic programmingReferences:
| [1] | Ahmed, S., Convexity and decomposition of mean-risk stochastic programs, Math. Program., 106, 3, 433-446 (2006) · Zbl 1134.90025 · doi:10.1007/s10107-005-0638-8 |
| [2] | Alonso-Ayuso, A.; Escudero, L. F.; Garín, A.; Ortuño, M. T.; Pérez, G., An approach for strategic supply chain planning under uncertainty based on stochastic 0-1 programming, J. Global Optim., 26, 1, 97-124 (2003) · Zbl 1116.90383 · doi:10.1023/A:1023071216923 |
| [3] | Alonso-Ayuso, A.; Escudero, L. F.; Ortuño, M. T., BFC: A branch-and-fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0-1 programs, Eur. J. Oper. Res., 151, 3, 503-519 (2003) · Zbl 1053.90101 · doi:10.1016/S0377-2217(02)00628-8 |
| [4] | Artzner, P.; Delbaen, F.; Eber, J.-M.; Heath, D.; Ku, H., Coherent multiperiod risk adjusted values and bellman’s principle, Ann. Oper. Res., 152, 1, 5-22 (2007) · Zbl 1132.91484 · doi:10.1007/s10479-006-0132-6 |
| [5] | Bank, B.; Mandel, R., Parametric Integer Optimization (1988), Berlin: Akademie-Verlag, Berlin · Zbl 0643.90043 |
| [6] | Bank, B.; Guddat, J.; Klatte, D.; Kummer, B.; Tammer, K., Non-linear Parametric Optimization (1982), Berlin: Akademie-Verlag, Berlin |
| [7] | Bereanu, B., Programme de risque minimal en programmation linéaire stochastique, Comptes Rendus de l’ Académie des Sciences Paris, 259, 5, 981-983 (1964) · Zbl 0123.37301 |
| [8] | Bereanu, B., Minimum risk criterion in stochastic optimization, Econ. Comput. Econ. Cybern. Stud. Res., 2, 31-39 (1981) · Zbl 0457.90058 |
| [9] | Billingsley, P., Convergence of Probability Measures (1968), New York, NY: Wiley, New York, NY · Zbl 0172.21201 |
| [10] | Birge, J. R.; Louveaux, F., Introduction to Stochastic Programming (1997), New York, NY: Springer, New York, NY · Zbl 0892.90142 |
| [11] | Blair, C. E.; Jeroslow, R. G., The value function of a mixed integer program: I, Discr. Math., 19, 121-138 (1977) · Zbl 0364.90074 · doi:10.1016/0012-365X(77)90028-0 |
| [12] | Bonnans, J. F.; Shapiro, A., Perturbation Analysis of Optimization Problems (2000), New York, NY: Springer, New York, NY · Zbl 0966.49001 |
| [13] | Carøe, C. C.; Schultz, R., Dual decomposition in stochastic integer programming, Oper. Res. Lett., 24, 1-2, 37-45 (1999) · Zbl 1063.90037 · doi:10.1016/S0167-6377(98)00050-9 |
| [14] | Charnes, A.; Cooper, W. W., Deterministic equivalents for optimizing and satisficing under chance constraints, Oper. Res., 11, 1, 18-38 (1963) · Zbl 0117.15403 · doi:10.1287/opre.11.1.18 |
| [15] | Dentcheva, D.; Römisch, W., Duality gaps in nonconvex stochastic optimization, Math. Program., 101, 3, 515-535 (2004) · Zbl 1073.90025 · doi:10.1007/s10107-003-0496-1 |
| [16] | Dentcheva, D.; Ruszczyński, A., Stochastic optimization with dominance constraints, SIAM J. Optim., 14, 2, 548-566 (2003) · Zbl 1055.90055 · doi:10.1137/S1052623402420528 |
| [17] | Dentcheva, D.; Ruszczyński, A., Optimality and duality theory for stochastic optimization with nonlinear dominance constraints, Math. Program., 99, 2, 329-350 (2004) · Zbl 1098.90044 · doi:10.1007/s10107-003-0453-z |
| [18] | Eichhorn, A.; Römisch, W., Polyhedral risk measures in stochastic programming, SIAM J. Optim., 16, 1, 69-95 (2005) · Zbl 1114.90077 · doi:10.1137/040605217 |
| [19] | Helmberg, C.; Kiwiel, K. C., A spectral bundle method with bounds, Math. Program., 93, 7, 173-194 (2002) · Zbl 1065.90059 · doi:10.1007/s101070100270 |
| [20] | Kall, P.; Wallace, S. W., Stochastic Programming (1994), Chichester: Wiley, Chichester · Zbl 0812.90122 |
| [21] | Kibzun, A. I.; Kan, Y. S., Stochastic Programming Problems with Probability and Quantile Functions (1996), Chichester: Wiley, Chichester · Zbl 0885.90088 |
| [22] | Kiwiel, K. C., Proximity control in bundle methods for convex nondifferentiable optimization, Math. Program., 46, 1-2, 105-122 (1990) · Zbl 0697.90060 · doi:10.1007/BF01585731 |
| [23] | Kristoffersen, T., Deviation measures in linear two-stage stochastic programming, Math. Methods Oper. Res., 62, 2, 255-274 (2005) · Zbl 1109.90065 · doi:10.1007/s00186-005-0006-8 |
| [24] | Levy, H., Stochastic dominance and expected utility: survey and analysis, Manage. Sci., 38, 4, 555-593 (1992) · Zbl 0764.90004 · doi:10.1287/mnsc.38.4.555 |
| [25] | Louveaux, F. V.; Schultz, R.; Ruszczyński, A.; Shapiro, A., Stochastic integer programming., Stochastic Programming, Handbooks in Operations Research and Management Science, 213-266 (2003), Amsterdam: Elsevier, Amsterdam · Zbl 1115.90001 · doi:10.1016/S0927-0507(03)10004-7 |
| [26] | Märkert, A.; Schultz, R., On deviation measures in stochastic integer programming, Oper. Res. Lett., 33, 5, 441-449 (2005) · Zbl 1099.90037 · doi:10.1016/j.orl.2004.09.003 |
| [27] | Markowitz, H. M., Portfolio selection, J. Finance, 7, 1, 77-91 (1952) |
| [28] | Müller, A.; Stoyan, D., Comparison Methods for Stochastic Models and Risks (2002), Chichester: Wiley, Chichester · Zbl 0999.60002 |
| [29] | Mulvey, J. M.; Vanderbei, R. J.; Zenios, S. A., Robust optimization of large-scale systems, Oper. Res., 43, 2, 264-281 (1995) · Zbl 0832.90084 · doi:10.1287/opre.43.2.264 |
| [30] | Ogryczak, W.; Ruszczyński, A., From stochastic dominance to mean-risk models: Semideviations as risk measures, Eur. J. Oper. Res., 116, 1, 33-50 (1999) · Zbl 1007.91513 · doi:10.1016/S0377-2217(98)00167-2 |
| [31] | Ogryczak, W.; Ruszczyński, A., Dual stochastic dominance and related mean-risk models, SIAM J. Optim., 13, 1, 60-78 (2002) · Zbl 1022.91017 · doi:10.1137/S1052623400375075 |
| [32] | Pflug, G. C.; Uryasev, S., Some remarks on the value-at-risk and the conditional value-at-risk., Probabilistic Constrained Optimization: Methodology and Applications, 272-281 (2000), Dordrecht: Kluwer, Dordrecht · Zbl 0994.91031 |
| [33] | Pflug, G. C., A value-of-information approach to measuring risk in multiperiod economic activity, J. Bank. Finance., 30, 2, 695-715 (2006) · doi:10.1016/j.jbankfin.2005.04.006 |
| [34] | Pflug, G. C.; Ruszczyński, A.; Szegö, G., Risk measures for income streams., Risk Measures for the 21st. Century, 249-269 (2004), Chichester: Wiley, Chichester |
| [35] | Prékopa, A., Stochastic Programming (1995), Dordrecht: Kluwer, Dordrecht · Zbl 0863.90116 |
| [36] | Raik, E., Qualitative research into the stochastic nonlinear programming problems, Eesti NSV Teaduste Akademia Toimetised / Füüsika, Matemaatica (News of the Estonian Academy of Sciences / Physics, Mathematics), 20, 8-14 (1971) · Zbl 0219.90038 |
| [37] | Raik, E., On the stochastic programming problem with the probability and quantile functionals, Eesti NSV Teaduste Akademia Toimetised / Füüsika, Matemaatica (News of the Estonian Academy of Sciences / Physics, Mathematics), 21, 142-148 (1972) · Zbl 0243.90030 |
| [38] | Riis, M.; Schultz, R., Applying the minimum risk criterion in stochastic recourse programs, Comput. Optim. Appl., 24, 2-3, 267-287 (2003) · Zbl 1094.90028 · doi:10.1023/A:1021862109131 |
| [39] | Robinson, S. M., Local epi-continuity and local optimization, Math. Program., 37, 2, 208-222 (1987) · Zbl 0623.90078 · doi:10.1007/BF02591695 |
| [40] | Rockafellar, R. T.; Uryasev, S., Conditional value-at-risk for general loss distributions, J. Bank. Finance., 26, 7, 1443-1471 (2002) · doi:10.1016/S0378-4266(02)00271-6 |
| [41] | Römisch, W.; Ruszczyński, A. A.; Shapiro, A., Stability of stochastic programming problems., Stochastic Programming, Handbooks in Operations Research and Management Science, 483-554 (2003), Amsterdam: Elsevier, Amsterdam · Zbl 1115.90001 · doi:10.1016/S0927-0507(03)10008-4 |
| [42] | Ruszczyński, A.; Shapiro, A., Stochastic Programming. Handbooks in Operations Research and Management Science (2003), Amsterdam: Elsevier, Amsterdam · Zbl 1115.90001 |
| [43] | Ruszczyński, A.; Vanderbei, R. J., Frontiers of stochastically nondominated portfolios, Econometrica, 71, 4, 1287-1297 (2003) · Zbl 1154.91475 · doi:10.1111/1468-0262.t01-1-00448 |
| [44] | Schultz, R., On structure and stability in stochastic programs with random technology matrix and complete integer recourse, Math. Program., 70, 1-3, 73-89 (1995) · Zbl 0841.90101 |
| [45] | Schultz, R., Some aspects of stability in stochastic programming, Ann. Oper. Res., 100, 1-4, 55-84 (2000) · Zbl 1017.90073 · doi:10.1023/A:1019258932012 |
| [46] | Schultz, R.; Jünger, M.; Reinelt, G.; Rinaldi, G., Mixed-integer value functions in stochastic programming., Combinatorial Optimization - Eureka, You Shrink!, 171-184 (2003), Berlin: Springer, Berlin · Zbl 1024.90051 · doi:10.1007/3-540-36478-1_16 |
| [47] | Schultz, R.; Tiedemann, S., Risk aversion via excess probabilities in stochastic programs with mixed-integer recourse, SIAM J. Optim., 14, 1, 115-138 (2003) · Zbl 1043.90059 · doi:10.1137/S1052623402410855 |
| [48] | Schultz, R.; Tiedemann, S., Conditional value-at-risk in stochastic programs with mixed-integer recourse, Math. Program., 105, 2-3, 365-386 (2006) · Zbl 1085.90042 · doi:10.1007/s10107-005-0658-4 |
| [49] | Takriti, S.; Ahmed, S., On robust optimization of two-stage systems, Math. Program., 99, 1, 109-126 (2004) · Zbl 1111.90080 · doi:10.1007/s10107-003-0373-y |
| [50] | Tiedemann, S.: Risk measures with preselected tolerance levels in two-stage stochastic mixed-integer programming. PhD thesis, University of Duisburg-Essen, Cuvillier Verlag, Göttingen (2005) |
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