Stochastic polynomial optimization

References

• F. Bastin, C. Cirillo, and P.L. Toint, Convergence theory for nonconvex stochastic programming with an application to mixed logit, Math. Program. 108(23) (2006), pp. 207–234. doi: 10.1007/s10107-006-0708-6
   Google Scholar
• A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2001.
   Google Scholar
• G. Blekherman, P. Parrilo, and R. Thomas (eds.), Semidefinite Optimization and Convex Algebraic Geometry, MOS-SIAM Series on Optimization, SIAM, 2013.
   Google Scholar
• S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
   Google Scholar
• M. Branda, Sample approximation technique for mixed-integer stochastic programming problems with several chance constraints, Oper. Res. Lett. 40(3) (2012), pp. 207–211. doi: 10.1016/j.orl.2012.01.002
   Web of Science ®Google Scholar
• K. Chung, A Course in Probability Theory, Academic Press, 2001.
   Google Scholar
• R. Curto and L. Fialkow, Truncated K-moment problems in several variables, J Oper. Theory 54 (2005), pp. 189–226.
   Web of Science ®Google Scholar
• M. Fu, eds., Handbook of Simulation Optimization, Vol. 216, Springer, New York, 2015.
   Google Scholar
• S. Ghadimi and G. Lan, Stochastic first-and zeroth-order methods for nonconvex stochastic programming, SIAM J. Opt. 23(4) (2013), pp. 2341–2368. doi: 10.1137/120880811
   Web of Science ®Google Scholar
• S. Ghadimi, G. Lan, and H. Zhang, Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization, Math. Program. 155(1–2) (2016), pp. 267–305. doi: 10.1007/s10107-014-0846-1
   Web of Science ®Google Scholar
• J. Helton and J. Nie, Semidefinite representation of convex sets, Math. Program. Ser. A 122(1) (2010), pp. 21–64. doi: 10.1007/s10107-008-0240-y
   Web of Science ®Google Scholar
• J. Helton and J. Nie, A semidefinite approach for truncated K-moment problem, Found. Comput. Math. 12(6) (2012), pp. 851–881. doi: 10.1007/s10208-012-9132-x
   Web of Science ®Google Scholar
• D. Henrion and J. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly, in Positive Polynomials in Control, D. Henrion and A. Garulli, eds., Lecture Notes in Control and Inform. Sci., 312, Springer, Berlin, 2005, pp. 293–310.
   Google Scholar
• D. Henrion, J. Lasserre, and J. Löfberg, Gloptipoly 3: Moments, optimization and semidefinite programming, Optim. Methods Softw. 24(45) (2009), pp. 761–779. doi: 10.1080/10556780802699201
   Google Scholar
• K. Judd, The law of large numbers with a continuum of iid random variables, Theory J. Econ. Theory. 35(1) (1985), pp. 19–25. doi: 10.1016/0022-0531(85)90059-6
   Web of Science ®Google Scholar
• E. Klerk and M. Laurent, On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems, SIAM J. Optim. 21 (2011), pp. 824–832. doi: 10.1137/100814147
   Web of Science ®Google Scholar
• A. Kleywegt, A. Shapiro, and T. Mello, The sample average approximation method for stochastic discrete optimization, SIAM J. Optim. 12(2) (2002), pp. 479–502. doi: 10.1137/S1052623499363220
   Web of Science ®Google Scholar
• J. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11 (2001), pp. 796–817. doi: 10.1137/S1052623400366802
   Web of Science ®Google Scholar
• J. Lasserre, Convexity in semi-algebraic geometry and polynomial optimization, SIAM J. Optim. 19 (2009), pp. 1995–2014. doi: 10.1137/080728214
   Web of Science ®Google Scholar
• J. Lasserre, Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge University Press, Cambridge, 2015.
   Google Scholar
• J. Lasserre, M. Laurent, and P. Rostalski, Semidefinite characterization and computation of zero-dimensional real radical ideals, Found. Comput. Math. 8 (2008), pp. 607–647. doi: 10.1007/s10208-007-9004-y
   Web of Science ®Google Scholar
• M. Laurent, Semidefinite representations for finite varieties, Math. Program. 109 (2007), pp. 1–26. doi: 10.1007/s10107-004-0561-4
   Web of Science ®Google Scholar
• M. Laurent, Sums of squares, moment matrices and optimization over polynomials, in Emerging Applications of Algebraic Geometry of IMA Volumes in Mathematics and its Applications, M. Putinar and S. Sullivant, eds., Vol. 149, Springer, 2009, pp. 157–270.
   Google Scholar
• J. Linderoth, A. Shapiro, and S. Wright, The empirical behavior of sampling methods for stochastic programming, Ann. Oper. Res. 142(1) (2006), pp. 215–241. doi: 10.1007/s10479-006-6169-8
   Web of Science ®Google Scholar
• J. Luedtke and S. Ahmed, A sample approximation approach for optimization with probabilistic constraints, SIAM J. Optim. 19(2) (2008), pp. 674–699. doi: 10.1137/070702928
   Web of Science ®Google Scholar
• J. Nie, Certifying convergence of Lasserre's hierarchy via flat truncation, Math. Program. Ser. A 142(1–2) (2013), pp. 485–510. doi: 10.1007/s10107-012-0589-9
   Web of Science ®Google Scholar
• J. Nie, Polynomial optimization with real varieties, SIAM J. Optim. 23(3) (2013), pp. 1634–1646. doi: 10.1137/120898772
   Web of Science ®Google Scholar
• J. Nie, Optimality conditions and finite convergence of Lasserre's hierarchy, Math. Program. Ser. A 146(12) (2014), pp. 97–121. doi: 10.1007/s10107-013-0680-x
   Google Scholar
• J. Nie, The A-truncated K-moment problem, Found. Comput. Math. 14(6) (2014), pp. 1243–1276. doi: 10.1007/s10208-014-9225-9
   Web of Science ®Google Scholar
• J. Nie, Linear optimization with cones of moments and nonnegative polynomials, Math. Program. Ser. B 153(1) (2015), pp. 247–274. doi: 10.1007/s10107-014-0797-6
   Web of Science ®Google Scholar
• P. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program. Ser. B 96(2) (2003), pp. 293–320. doi: 10.1007/s10107-003-0387-5
   Web of Science ®Google Scholar
• P. Parrilo and B. Sturmfels, Minimizing polynomial functions, in Algorithmic and Quantitative Real Algebraic Geometry (Piscataway, NJ, 2001), S. Basu and L. Gonzalez-Vega, eds., DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 60, American Mathematical Society, Providence, RI, 2003, pp. 83–99.
   Google Scholar
• E. Plambeck, B. Fu, S. Robinson, and R. Suri, Sample-path optimization of convex stochastic performance functions, Math. Program. 75(2) (1996), pp. 137–176. doi: 10.1007/BF02592150
   Web of Science ®Google Scholar
• M. Putinar, Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42 (1993), pp. 969–984. doi: 10.1512/iumj.1993.42.42045
   Web of Science ®Google Scholar
• S. Robinson, Analysis of sample-path optimization, Math. Oper. Res. 21(3) (1996), pp. 513–528. doi: 10.1287/moor.21.3.513
   Web of Science ®Google Scholar
• T. Santoso, S. Ahmed, M. Goetschalckx, and A. Shapiro, A stochastic programming approach for supply chain network design under uncertainty, Eur. J. Oper. Res. 167(1) (2005), pp. 96–115. doi: 10.1016/j.ejor.2004.01.046
   Web of Science ®Google Scholar
• C. Scheiderer, Positivity and sums of squares: A guide to recent results, in Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant, eds., IMA Volumes Math. Appl. Vol. 149, Springer, 2009, pp. 271–324.
   Google Scholar
• S. Shalev-Shwartz, O. Shamir, N. Srebro, and K. Sridharan, Stochastic convex optimization, in 22nd Annual Conference on Learning Theory (COLT 2009), 2009.
   Google Scholar
• A. Shapiro, D. Dentcheva, and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, 2009.
   Google Scholar
• J. Sturm, SeDuMi 1.02: A MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw. 11&12 (1999), pp. 625–653. Available at http://sedumi.mcmaster.ca/. doi: 10.1080/10556789908805766
   Web of Science ®Google Scholar
• B. Verweij, S. Ahmed, A. Kleywegt, G. Nemhauser, and A. Shapiro, The sample average approximation method applied to stochastic routing problems: a computational study, Comput. Optim. Appl. 24(2–3) (2003), pp. 289–333. doi: 10.1023/A:1021814225969
   Web of Science ®Google Scholar