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Solving higher index DAE optimal control problems
An approximation scheme for stochastic programs with second order dominance constraints
1. | Department of Mathematics, Dalian Maritime University, Dalian 116026, China |
2. | School of Economics and Management, Nanjing University of Science and Techonolyogy, Nanjing, 210049, China |
3. | School of Mathematics, University of Southampton, SO17 1BJ, Southampton, United Kingdom |
References:
[1] |
E. Anderson, H. Xu and D. Zhang, CVaR approximations for minimax and robust convex optimization, manuscript, 2013. |
[2] |
R. J. Aumann, Integrals of set-valued functions, Journal of Mathematical Analysis and Applications, 12 (1965), 1-12. |
[3] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operational Research, Springer, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[4] |
G. Calafiore and M. C. Campi, Uncertain convex programs: randomized solutions and confidence levels, Mathematical Programming, 102 (2005), 25-46.
doi: 10.1007/s10107-003-0499-y. |
[5] |
X. Chen, Smoothing methods for nonsmooth, novonvex minimization, Mathematical Programming, 134 (2012), 71-99.
doi: 10.1007/s10107-012-0569-0. |
[6] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. |
[7] |
D. Dentcheva, R. Henrion and A. Ruszczyński, Stability and sensitivity of optimization problems with first order stochastic dominance constraints, SIAM Journal on Optimization, 18 (2007), 322-333.
doi: 10.1137/060650118. |
[8] |
D. Dentcheva and W. Römisch, Stability and sensitivity of stochastic dominance constrained optimization models, SIAM Journal on Optimization, 23 (2013), 1672-1688.
doi: 10.1137/120886790. |
[9] |
D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM Journal on Optimization, 14 (2003), 548-566.
doi: 10.1137/S1052623402420528. |
[10] |
D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization with nonlinear dominance constraints, Mathematical Programming, 99 (2004), 329-350.
doi: 10.1007/s10107-003-0453-z. |
[11] |
Y. Ermoliev and V. Norkin, Sample average approximation method for compound stochastic optimization problems, SIAM Journal on Optimization, 23 (2013), 2231-2263.
doi: 10.1137/120863277. |
[12] |
C. I. Fábián, G. Mitra and D.Roman, Processing second-order stochastic dominance models using cutting-plane representations, Mathematical Programming, 130 (2011), 33-57.
doi: 10.1007/s10107-009-0326-1. |
[13] |
H. Föllmer and T. Knispel, Entropic risk measures: coherence vs. convexity, model ambiguity, and robust large deviations, Stochastics and Dynamics, 11 (2011), 333-351.
doi: 10.1142/S0219493711003334. |
[14] |
M. Gugat, Error bounds for infinite systems of convex inequalities without Slater's condition, Mathematical Programming, 88 (2000), 255-275.
doi: 10.1007/s101070050016. |
[15] |
C. Hess, Set-valued integration and set-valued probability theory: an overview, Handbook of measure theory, North-Holland, Amsterdam, I-II (2002), 617-673.
doi: 10.1016/B978-044450263-6/50015-4. |
[16] |
T. Homem-de-Mello, A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints, SIAM Journal on Optimization, 20 (2010), 1250-1273.
doi: 10.1137/08074009X. |
[17] |
J. Hu, T. Homen-De-Mello and S. Mehrotra, Sample average approximation of stochastic dominance constrained programs, Mathematical Programming, 133 (2012), 171-201.
doi: 10.1007/s10107-010-0428-9. |
[18] |
P. Kall, Approximation to Optimization Problems: An Elementary Review, Mathematics of Operations Research, 11 (1986), 9-18.
doi: 10.1287/moor.11.1.9. |
[19] |
D. Klatte, On the stability of local and global optimal solutions in parametric problems of nonlinear programming, Part I: Basic results, Seminarbericht Nr. 75, Sektion Mathematik, Humboldt-Universität zu Berlin, Berlin, (1985) 1-21. |
[20] |
D. Klatte, A note on quantitative stability results in nonlinear optimization, Seminarbericht Nr. 90, Sektion Mathematik, Humboldt-Universität zu Berlin, Berlin, (1987), 77-86. |
[21] |
H. Levy, Stochastic dominance and expected utility: survey and analysis, Management Science, 38 (1992), 555-593. |
[22] |
G. H. Lin, M. W. Xu and J. J. Ye, On solving simple bilevel programs with a nonconvex lower level program, Mathematical Programming, 144 (2014), 277-305.
doi: 10.1007/s10107-013-0633-4. |
[23] |
Y. Liu and H. Xu, Stability analysis of stochastic programs with second order dominance constraints, Mathematical Programming, 142 (2013), 435-460.
doi: 10.1007/s10107-012-0585-0. |
[24] |
Y. Liu, W. Römisch and H. Xu, Quantitative stability analysis of stochastic generalized equations, SIAM Journal on Optimization, 24 (2014), 467-497.
doi: 10.1137/120880434. |
[25] |
Y. Liu and H. Xu, Entropic approximation for mathematical programs with robust equilibrium constraints, SIAM Journal on Optimization, 24Z (2014), 933-958.
doi: 10.1137/130931011. |
[26] |
A. Müller and M. Scarsini, Stochastic Orders and Decision Under Risk, Institute of mathematical statistics, Hayward, CA, 1991. |
[27] |
D. Ralph and H. Xu, Asympototic analysis of stationary points of sample average two stage stochastic programs: A generalized equation approach, Mathematics of Operations Research, 36 (2011), 568-592.
doi: 10.1287/moor.1110.0506. |
[28] |
S. M. Robinson and R. J-B. Wets, Stability in two-stage stochastic programming, SIAM Journal on Control and Optimization, 25 (1987), 1409-1416.
doi: 10.1137/0325077. |
[29] |
S. M. Robinson, Analysis of sample-path optimization, Mathematics of Operations Research, 21 (1996), 513-528.
doi: 10.1287/moor.21.3.513. |
[30] |
R. T. Rockafellar and R. J-B. Wets, Variational Analysis, Springer, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[31] |
A. Ruszczyński and A. Shapiro, Stochastic Programming, Handbook in Operations Research and Management Science, Elsevier, 2003. |
[32] |
A. Shapiro and H. Xu, Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions, Journal of Mathematical Analysis and Applications, 325 (2007), 1390-1399.
doi: 10.1016/j.jmaa.2006.02.078. |
[33] |
H. Sun, H. Xu, R. Meskarian and Y. Wang, Exact penalization, level function method, and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints, SIAM Journal on Optimization, 23 (2013), 602-631.
doi: 10.1137/110850815. |
[34] |
H. Xu, Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming, Journal of Mathematical Analysis and Applications, 368 (2010), 692-710.
doi: 10.1016/j.jmaa.2010.03.021. |
[35] |
H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming, 119 (2009), 371-401.
doi: 10.1007/s10107-008-0214-0. |
[36] |
J. Zhang, H. Xu and L. W. Zhang, Quantitative stability analysis of stochastic quasi-variational inequality problems and applications, Manusicript, 2014. |
show all references
References:
[1] |
E. Anderson, H. Xu and D. Zhang, CVaR approximations for minimax and robust convex optimization, manuscript, 2013. |
[2] |
R. J. Aumann, Integrals of set-valued functions, Journal of Mathematical Analysis and Applications, 12 (1965), 1-12. |
[3] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operational Research, Springer, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[4] |
G. Calafiore and M. C. Campi, Uncertain convex programs: randomized solutions and confidence levels, Mathematical Programming, 102 (2005), 25-46.
doi: 10.1007/s10107-003-0499-y. |
[5] |
X. Chen, Smoothing methods for nonsmooth, novonvex minimization, Mathematical Programming, 134 (2012), 71-99.
doi: 10.1007/s10107-012-0569-0. |
[6] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. |
[7] |
D. Dentcheva, R. Henrion and A. Ruszczyński, Stability and sensitivity of optimization problems with first order stochastic dominance constraints, SIAM Journal on Optimization, 18 (2007), 322-333.
doi: 10.1137/060650118. |
[8] |
D. Dentcheva and W. Römisch, Stability and sensitivity of stochastic dominance constrained optimization models, SIAM Journal on Optimization, 23 (2013), 1672-1688.
doi: 10.1137/120886790. |
[9] |
D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM Journal on Optimization, 14 (2003), 548-566.
doi: 10.1137/S1052623402420528. |
[10] |
D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization with nonlinear dominance constraints, Mathematical Programming, 99 (2004), 329-350.
doi: 10.1007/s10107-003-0453-z. |
[11] |
Y. Ermoliev and V. Norkin, Sample average approximation method for compound stochastic optimization problems, SIAM Journal on Optimization, 23 (2013), 2231-2263.
doi: 10.1137/120863277. |
[12] |
C. I. Fábián, G. Mitra and D.Roman, Processing second-order stochastic dominance models using cutting-plane representations, Mathematical Programming, 130 (2011), 33-57.
doi: 10.1007/s10107-009-0326-1. |
[13] |
H. Föllmer and T. Knispel, Entropic risk measures: coherence vs. convexity, model ambiguity, and robust large deviations, Stochastics and Dynamics, 11 (2011), 333-351.
doi: 10.1142/S0219493711003334. |
[14] |
M. Gugat, Error bounds for infinite systems of convex inequalities without Slater's condition, Mathematical Programming, 88 (2000), 255-275.
doi: 10.1007/s101070050016. |
[15] |
C. Hess, Set-valued integration and set-valued probability theory: an overview, Handbook of measure theory, North-Holland, Amsterdam, I-II (2002), 617-673.
doi: 10.1016/B978-044450263-6/50015-4. |
[16] |
T. Homem-de-Mello, A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints, SIAM Journal on Optimization, 20 (2010), 1250-1273.
doi: 10.1137/08074009X. |
[17] |
J. Hu, T. Homen-De-Mello and S. Mehrotra, Sample average approximation of stochastic dominance constrained programs, Mathematical Programming, 133 (2012), 171-201.
doi: 10.1007/s10107-010-0428-9. |
[18] |
P. Kall, Approximation to Optimization Problems: An Elementary Review, Mathematics of Operations Research, 11 (1986), 9-18.
doi: 10.1287/moor.11.1.9. |
[19] |
D. Klatte, On the stability of local and global optimal solutions in parametric problems of nonlinear programming, Part I: Basic results, Seminarbericht Nr. 75, Sektion Mathematik, Humboldt-Universität zu Berlin, Berlin, (1985) 1-21. |
[20] |
D. Klatte, A note on quantitative stability results in nonlinear optimization, Seminarbericht Nr. 90, Sektion Mathematik, Humboldt-Universität zu Berlin, Berlin, (1987), 77-86. |
[21] |
H. Levy, Stochastic dominance and expected utility: survey and analysis, Management Science, 38 (1992), 555-593. |
[22] |
G. H. Lin, M. W. Xu and J. J. Ye, On solving simple bilevel programs with a nonconvex lower level program, Mathematical Programming, 144 (2014), 277-305.
doi: 10.1007/s10107-013-0633-4. |
[23] |
Y. Liu and H. Xu, Stability analysis of stochastic programs with second order dominance constraints, Mathematical Programming, 142 (2013), 435-460.
doi: 10.1007/s10107-012-0585-0. |
[24] |
Y. Liu, W. Römisch and H. Xu, Quantitative stability analysis of stochastic generalized equations, SIAM Journal on Optimization, 24 (2014), 467-497.
doi: 10.1137/120880434. |
[25] |
Y. Liu and H. Xu, Entropic approximation for mathematical programs with robust equilibrium constraints, SIAM Journal on Optimization, 24Z (2014), 933-958.
doi: 10.1137/130931011. |
[26] |
A. Müller and M. Scarsini, Stochastic Orders and Decision Under Risk, Institute of mathematical statistics, Hayward, CA, 1991. |
[27] |
D. Ralph and H. Xu, Asympototic analysis of stationary points of sample average two stage stochastic programs: A generalized equation approach, Mathematics of Operations Research, 36 (2011), 568-592.
doi: 10.1287/moor.1110.0506. |
[28] |
S. M. Robinson and R. J-B. Wets, Stability in two-stage stochastic programming, SIAM Journal on Control and Optimization, 25 (1987), 1409-1416.
doi: 10.1137/0325077. |
[29] |
S. M. Robinson, Analysis of sample-path optimization, Mathematics of Operations Research, 21 (1996), 513-528.
doi: 10.1287/moor.21.3.513. |
[30] |
R. T. Rockafellar and R. J-B. Wets, Variational Analysis, Springer, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[31] |
A. Ruszczyński and A. Shapiro, Stochastic Programming, Handbook in Operations Research and Management Science, Elsevier, 2003. |
[32] |
A. Shapiro and H. Xu, Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions, Journal of Mathematical Analysis and Applications, 325 (2007), 1390-1399.
doi: 10.1016/j.jmaa.2006.02.078. |
[33] |
H. Sun, H. Xu, R. Meskarian and Y. Wang, Exact penalization, level function method, and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints, SIAM Journal on Optimization, 23 (2013), 602-631.
doi: 10.1137/110850815. |
[34] |
H. Xu, Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming, Journal of Mathematical Analysis and Applications, 368 (2010), 692-710.
doi: 10.1016/j.jmaa.2010.03.021. |
[35] |
H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming, 119 (2009), 371-401.
doi: 10.1007/s10107-008-0214-0. |
[36] |
J. Zhang, H. Xu and L. W. Zhang, Quantitative stability analysis of stochastic quasi-variational inequality problems and applications, Manusicript, 2014. |
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