Optimal Information Blending with Measurements in the L² Sphere

References

• Agrawal S, Goyal N (2013) Thompson sampling for contextual bandits with linear payoffs. Proc. 30th Internat. Conf. Machine Learning, ICML ’13, Berlin, 337–344.Google Scholar
• Alizadeh F, Goldfarb D (2003) Second-order cone programming. Math. Programming 95(1):3–51.Crossref, Google Scholar
• Auer P, Cesa-Bianchi N, Fischer P (2002) Finite-time analysis of the multiarmed bandit problem. Machine Learning 47(2–3):235–256.Crossref, Google Scholar
• Barvinok A (2001) A remark on the rank of positive semidefinite matrices subject to affine constraints. Discrete Comput. Geometry 25(1):23–31.Crossref, Google Scholar
• Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust Optimization (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math. Programming 99(2):351–376.Crossref, Google Scholar
• Bertsekas DP, Shreve SE (1978) Stochastic Optimal Control: The Discrete-Time Case (Academic Press, New York).Google Scholar
• Bertsimas D, Gupta V, Kallus N (2013a) Data-driven robust optimization. Preprint. arXiv:1401.0212.Google Scholar
• Bertsimas D, Sim M (2004) The price of robustness. Oper. Res. 51(1):35–53.Link, Google Scholar
• Bertsimas D, O’Hair A, Relyea S, Silberholz J (2013b) An analytics approach to designing clinical trials for cancer. Working paper, MIT, http://josilber.scripts.mit.edu/CancerPaper_Revision1_names.pdf.Google Scholar
• Bickel PJ, Doksum KA (2007) Mathematical Statistics, Basic Ideas and Selected Topics, 2nd ed., Vol. 1 (Pearson Prentice-Hall, Upper Saddle River, NJ).Google Scholar
• Bubeck S, Cesa-Bianchi N, Kakade SM (2012) Towards minimax policies for online linear optimization with bandit feedback. Mannor S, Srebro N, Williamson RC, eds. Proc. 25th Conf. Learning Theory, COLT ’12, Edinburgh, Scotland, 1–14.Google Scholar
• Chapelle O, Li L (2011) An empirical evaluation of Thompson sampling. Shawe-Taylor J, Zemel RS, Bartlett PL, Pereira F, Weinberger KQ, eds. Adv. Neural Inform. Processing Systems 24, NIPS ’11, Granada, Spain: 2249–2257.Google Scholar
• Chick SE (2006) Subjective probability and Bayesian methodology. Henderson SG, Nelson BL, eds. Handbooks of Operations Research and Management Science, Simulation, Vol. 13 (North-Holland Publishing, Amsterdam), 225–258.Google Scholar
• Chick SE, Gans N (2009) Economic analysis of simulation selection problems. Management Sci. 55(3):421–437.Link, Google Scholar
• Cohn DA, Ghahramani Z, Jordan MI (1996) Active learning with statistical models. J. Artif. Intel. Res. 4(1):129–145.Google Scholar
• Dani V, Hayes TP, Kakade SM (2008) Stochastic linear optimization under bandit feedback. Servedio R, Zhang Tong, eds. Proc. 21st Conf. Learning Theory, COLT ’08, Helsinki, Finland, 355–366.Google Scholar
• Delage E, Mannor S (2007) Percentile optimization in uncertain Markov decision processes with application to efficient exploration. Ghahramani Z, ed. Proc. 24th Internat. Conf. Machine Learning, ICML ’07 (ACM, New York), 225–232.Crossref, Google Scholar
• Delage E, Mannor S (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(1):203–213.Link, Google Scholar
• Dellino G, Kleijnen JPC, Meloni C (2012) Robust optimization in simulation: Taguchi and Krige combined. Informs J. Comp. 24(3):471–484.Link, Google Scholar
• D’Epenoux F (1960) Sur un problème de production et de stockage dans l’aléatoire. Revue Française de Recherche Opérationnelle 14:3–16 [English translation: Management Sci. 10(1):98-108.].Google Scholar
• Dontchev AL, Rockafellar RT (2009) Implicit Functions and Solution Mappings (Springer, New York).Crossref, Google Scholar
• Doob JL (1949) Application of the theory of martingales. Le calcul des probabilités et ses applications, Colloques Internationaux du Centre National de la Recherche Scientifique CNRS (Paris), 23–27.Google Scholar
• Fazel M, Hindi H, Boyd S (2001) A rank minimization heuristic with application to minimum order system approximation. Proc. 2001 Amer. Control Conf. (Arlington, VA), 4734–4739.Crossref, Google Scholar
• Gelman AB, Carlin JB, Stern HS, Rubin DB (2004) Bayesian Data Analysis, 2nd ed. (CRC Press, Boca Raton, FL).Google Scholar
• Ghaffari-Hadigheh A, Terlaky T (2006) Sensitivity analysis in linear optimization: Invariant support set intervals. Eur. J. Oper. Res. 169(3):1158–1175.Crossref, Google Scholar
• Gittins JC, Glazebrook KD, Weber R (2011) Multi-Armed Bandit Allocation Indices, 2nd ed. (John Wiley & Sons, Chichester, UK).Crossref, Google Scholar
• Graf S, Luschgy H (2000) Foundations of Quantization for Probability Distributions (Springer-Verlag, Berlin).Crossref, Google Scholar
• Grant M, Boyd S (2008) Graph implementations for nonsmooth convex programs. Blondel V, Boyd S, Kimura H, eds. Recent Advances in Learning and Control—Atribute to M. Vidyasagar, LNCIS (Springer, New York), 95–110.Crossref, Google Scholar
• Grant M, Boyd S (2011) CVX: MATLAB software for disciplined convex programming, version 1.21. http://cvxr.com/cvx.Google Scholar
• Gupta SS, Miescke KJ (1996) Bayesian look ahead one-stage sampling allocations for selection of the best population. J. Statist. Planning and Inference 54(2):229–244.Crossref, Google Scholar
• Iyengar GN (2005) Robust dynamic programming. Math. Oper. Res. 30(2):257–280.Link, Google Scholar
• Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4): 455–492.Crossref, Google Scholar
• Kim S-H, Nelson BL (2006) Selecting the best system. Henderson SG, Nelson BL, eds. Handbooks of Operations Research and Management Science, Simulation, Vol. 13 (North-Holland Publishing, Amsterdam), 501–534.Google Scholar
• Kim S-H, Nelson BL (2007) Recent advances in ranking and selection. Henderson SG, Biller B, Hsieh M-H, Shortle J, Tew JD, Barton RR, eds. Proc. 2007 Winter Simulation Conf. (IEEE, Piscataway, NJ), 162–172.Google Scholar
• Mangasarian OL, Shiau TH (1986) A variable-complexity norm maximization problem. SIAM J. Algebraic Discrete Methods 7(3): 455–461.Crossref, Google Scholar
• McMahan HB, Gordon GJ, Blum A (2003) Planning in the presence of cost functions controlled by an adversary. Proc. 20th Internat. Conf. Machine Learning, ICML ’03 (AAAI Press, Palo Alto, CA), 536–543.Google Scholar
• Minka TP (2000) Bayesian linear regression. Technical report, Microsoft Research, Redmond, WA.Google Scholar
• Negoescu DM, Frazier PI, Powell WB (2010) The knowledge-gradient algorithm for sequencing experiments in drug discovery. Informs J. Comp. 23(3):346–363.Link, Google Scholar
• Nesterov Y, Wolkowicz H, Ye Y (2000) Semidefinite programming relaxations of nonconvex quadratic optimization. Wolkowicz H, Saigal R, Vandenberghe L, eds. Handbook of Semidefinite Programming (Springer, New York), 361–419.Crossref, Google Scholar
• Nilim A, Ghaoui LE (2005) Robust control of Markov decision processes with uncertain transition matrices. Oper. Res. 53(5):780–798.Link, Google Scholar
• Pages G, Printems J (2003) Optimal quadratic quantization for numerics: The Gaussian case. Monte Carlo Methods Appl. 9(2):135–166.Crossref, Google Scholar
• Pataki G (1998) On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23(2):339–358.Link, Google Scholar
• Powell WR (2011) Approximate Dynamic Programming: Solving the Curses of Dimensionality, 2nd ed. (Wiley, Hoboken, NJ).Crossref, Google Scholar
• Powell WB, Ryzhov IO (2012) Optimal Learning (Wiley, Hoboken, NJ).Crossref, Google Scholar
• Puterman ML (1994) Markov Decision Processes: Discrete Stochastic Dynamic Programming (Wiley, Hoboken, NJ).Crossref, Google Scholar
• Regan K, Boutilier C (2009) Regret-based reward elicitation for Markov decision processes. Proc. 25th Conf. Uncertainty in Artificial Intelligence (The AAAI Press, Menlo Park, CA), 444–451.Google Scholar
• Regan K, Boutilier C (2010) Robust policy computation in reward-uncertain MDPs using nondominated policies. Proc. 24th AAAI Conf. Artificial Intelligence (AAAI-10) (The AAAI Press, Menlo Park, CA), 1127–1133.Google Scholar
• Rockafellar RT (1970) Convex Analysis (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• Rockafellar RT, Wets RJ-B (1998) Variational Analysis, 3rd ed. (Springer, New York).Crossref, Google Scholar
• Russo D, Van Roy B (2013) Learning to optimize via posterior sampling. arXiv preprint arXiv:1301.2609.Google Scholar
• Ruszczyński A (2010) Risk-averse dynamic programming for Markov decision processes. Math. Programming 125(2):235–261.Crossref, Google Scholar
• Ryzhov IO, Powell WB (2011) Information collection on a graph. Oper. Res. 59(1):188–201.Link, Google Scholar
• Ryzhov IO, Powell WB (2012) Information collection for linear programs with uncertain objective coefficients. SIAM J. Optim. 22(4):1344–1368.Crossref, Google Scholar
• Ryzhov IO, Defourny B, Powell WB (2012) Ranking and selection meets robust optimization. Proc. 2012 Winter Simulation Conf. (ACM, New York), 532–542.Crossref, Google Scholar
• Scott WR, Frazier PI, Powell WB (2011) The correlated knowledge gradient for simulation optimization of continuous parameters using Gaussian process regression. SIAM J. Optim. 21(3):996–1026.Crossref, Google Scholar
• Shor NZ (1987) Quadratic optimization problems. Soviet J. Circuits and Systems Sci. 25(6):1–11.Google Scholar
• Thompson WR (1933) On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika 25(3–4):285–294.Crossref, Google Scholar
• Tütüncü RH, Toh KC, Todd MJ (2003) Solving semidefinite-quadratic-linear programs using SDPT3. Math. Programming 95(2):189–217.Crossref, Google Scholar
• Waeber R, Frazier PI, Henderson SG (2010) Performance measures for ranking and selection procedures. Johansson B, Jain S, Montoya-Torres J, Hugan J, Yücesan E, eds. Proc. 2010 Winter Simulation Conf. (IEEE, Piscataway, NJ), 1235–1245.Crossref, Google Scholar
• Xu H, Mannor S (2009) Parametric regret in uncertain Markov decision processes. Proc. 48th IEEE Conf. Decision and Control (IEEE, Piscataway, NJ), 3606–3613.Crossref, Google Scholar
• Zheng XJ, Sun XL, Li D (2011) Convex relaxations for nonconvex quadratically constrained quadratic programming: Matrix cone decomposition and polyhedral approximation. Math. Programming, Ser. B 129(2):301–329.Crossref, Google Scholar