参考文献:Anderson EJ, Ferris MC (2001) A direct search algorithm for optimization with noisy function evaluations. SIAM J Optim 11(3):837–857CrossRef MathSciNet MATH Audet C, Dennis JE (2003) Analysis of generalized pattern searches. SIAM J Optim 13:889–903CrossRef MathSciNet MATH Audet C, Dennis JE (2006) Mesh adaptive direct search algorithms for constrained optimization. SIAM J Optim 17:188–217CrossRef MathSciNet MATH Audet C, Dennis JE (2009) A progressive barrier for derivative-free nonlinear programming. SIAM J Optim 20:445–472CrossRef MathSciNet MATH Audet C, Booker AJ, Dennis JE, Moore DW, Frank PD (2000) A surrogate-model-based method for constrained optimization, AIAA-2000-4891. In: Proceedings of the symposium on multidisciplinary analysis and optimization Booker AJ (1994) DOE for computer output. Technical report BCSTECH-94-052. Boeing Computer Services, Seattle Booker AJ (2000) Well–conditioned kriging models for optimization of computer models. Technical report M&CT-TECH-002. Boeing Phantom Works, Mathematics and Computing Technology Booker AJ, Dennis JE, Frank PD, Serafini DB, Torczon V, Trosset MW (1999) A rigorous framework for optimization of expensive functions by surrogates. Struct Optim 17:1–13CrossRef Broyden CG (1969) A new double-rank minimization algorithm. AMS Not 16:670 Carter RG, Gablonsky JM, Patrick A, Kelley CT, Eslinger OJ (2001) Algorithms for noisy problems in gas transmission pipeline optimization. Optim Eng 2:139–157CrossRef MathSciNet MATH Characklis GW, Kirsch BR, Ramsey J, Dillard KEM, Kelley CT (2006) Developing portfolios of water supply transfers. Water Resour Res 42:W05403–1–W05403–14 Choi TD, Eslinger OJ, Kelley CT, David JW, Etheridge M (2000) Optimization of automotive valve train components with implicit filtering. Optim Eng 1:9–27CrossRef MATH Clarke FH (1990) Number 5. Classics in applied mathematics. Optimization and nonsmooth analysis. SIAM, PhiladelphiaCrossRef Conn AR, Scheinberg K, Vicente LN (2009) Introduction to derivative-free optimization., MPS-SIAM series on optimizationSIAM, PhiladelphiaCrossRef MATH David JW, Kelley CT, Cheng CY (1996) Use of an implicit filtering algorithm for mechanical system parameter identification. In: SAE paper 960358, 1996 SAE international congress and exposition conference proceedings, modeling of CI and SI engines, society of automotive engineers, Washington, DC, pp 189–194 Deng G, Ferris MC (2006) Adaptation of the UOBYQA algorithm for noisy functions. In: Perrone LP, Lawson B, Liu J, Wieland F (ed) 2007 Winter simulation conference. pp 312–219 Deng G, Ferris MC (2007) Extension of the DIRECT optimization algorithm for noisy functions. In: Biller B, Henderson S, Hsieh M, Shortle J (ed) 2007 Winter simulation conference. IEEE, pp 497–504 Deng G, Ferris MC (2009) Variable-number sample-path optimization. Math Program 117:81–109CrossRef MathSciNet MATH Dennis JE, Torczon V (1991) Direct search methods on parallel machines. SIAM J Optim 1:448–474CrossRef MathSciNet MATH Dillard KEM (2007) An application of implicit filtering to water resources management. PhD thesis. North Carolina State University, Raleigh, North Carolina Finkel DE, Kelley CT (2009) Convergence analysis of sampling methods for perturbed Lipschitz functions. Pac J Optim 5:339–350MathSciNet MATH Fletcher R (1970) A new approach to variable metric methods. Comput J 13:317–322CrossRef MATH Gilmore P, Kelley CT (1995) An implicit filtering algorithm for optimization of functions with many local minima. SIAM J Optim 5:269–285CrossRef MathSciNet MATH Gilmore PA, Berger SS, Burr RF, Burns JA (1997) Automated optimization techniques for phase change piezoelectric ink jet performance enhancement. In: 1997 International conference on digital printing technologies. Society for imaging science and technology, IS&T’s NIP 13, pp 716–721 Goldfarb D (1970) A family of variable metric methods derived by variational means. Math Comput 24:23–26CrossRef MathSciNet MATH Hooke R, Jeeves TA (1961) ‘Direct search’ solution of numerical and statistical problems. JACM 8:212–229CrossRef MATH Jahn J (1996) Introduction to the theory of nonlinear optimization. Springer, BerlinCrossRef MATH Jones DR, Perttunen CD, Stuckman BE (1993) Lipschitzian optimization without the Lipschitz constant. J Optim Theory Appl 79:157–181CrossRef MathSciNet MATH Kelley CT (1999) Number 18 in frontiers in applied mathematics. In: Kelley CT (ed) Iterative methods for optimization. SIAM, PhiladelphiaCrossRef Kelley CT (2011) Number 23 in software environments and tools. In: Kelley CT (ed) Implicit filtering. SIAM, PhiladelphiaCrossRef Kim S, Zhang D (2010) Convergence properties of direct search methods for stochastic optimization. In: Proceedings of the 2009 winter simulation conference, pp 1003–1011. http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5679089 Kirsch BR, Characklis GW, Dillard KEM, Kelley CT (2009) More efficient optimization of long-term water supply portfolios. Water Resour Res 45:W03414–1–W03414–12. doi:10.1029/2008WR007018 Kolda TG, Lewis RM, Torczon V (2003) Optimization by direct search: New perspectives on some classical and modern methods. SIAM Rev 45:385–482CrossRef MathSciNet MATH Kolda TG, Lewis RM, Torczon V (2006) Stationarity results for generating set search for linearly constrained optimization. SIAM J Optim 17:943–968CrossRef MathSciNet MATH Lewis RM, Torczon V (1996) Rank ordering and positive bases in pattern search algorithms. Technical report 96-71. Institute for Computer Applications in Science and Engineering Lewis RM, Torczon V (2000) Pattern search algorithms for linearly constrained minimization. SIAM J Optim 10:917–941CrossRef MathSciNet MATH Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313CrossRef MATH Powell MJD (2002) UOBYQA: unconstrained optimization by quadratic approximation. Math Program Ser B 92:555–582CrossRef MATH Rockafellar RT, Wets JB (1998) Variational analysis. Springer, BerlinCrossRef MATH Ross SM (2007) Introduction to probability models, 9th edn. Academic Press, New York Shanno DF (1970) Conditioning of quasi-Newton methods for function minimization. Math Comput 24:647–657CrossRef MathSciNet Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming., MPS-SIAM series on optimizationSIAM, PhiladelphiaCrossRef MATH Stoneking DE, Bilbro GL, Gilmore P, Trew RJ, Kelley CT (1992) Yield optimization using a GaAs process simulator coupled to a physical device model. IEEE Trans Microw Theory Tech 40:1353–1363CrossRef Trosset MW (2000) On the use of direct search methods for stochastic optimization. Technical report. Rice University, Department of Computational and Applied Mathematics Vicente LN, Custódio AL (2012) Analysis of direct searches for discontinuous functions. Math Program 133(1–2):299–325CrossRef MathSciNet MATH Willert J, Kelley CT, Knoll DA, Park HK (2013a) Hybrid deterministic/Monte Carlo neutronics. SIAM J Sci Comput 35:S62–S83. doi:10.1137/120880021 CrossRef MathSciNet MATH Willert J, Kelley CT, Knoll DA, Park HK (2013b) A hybrid approach to the neutron transport k-eigenvalue problem using NDA-based algorithms. In: Proceedings of international conference on mathematics and computational methods applied to nuclear science & engineering, pp 1934–1941 Willert Jeffrey, Chen Xiaojun, Kelley CT (2015) Newton’s method for Monte Carlo-based residuals. SIAM J Numer Anal 53:1738–1757. doi:10.1137/130905691 CrossRef MathSciNet Williams D (1991) Probability with martingales. Cambridge University Press, CambridgeCrossRef MATH Yu W (1979) Positive basis and a class of direct search techniques. Sci Sin Spec Issue Math 1:53–67
作者单位:Xiaojun Chen (1) C. T. Kelley (2)
1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 2. Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC, 27695-8205, USA
刊物类别:Mathematics and Statistics
刊物主题:Mathematics Optimization Engineering, general Systems Theory and Control Environmental Management Agriculture Operation Research and Decision Theory
出版者:Springer Netherlands
ISSN:1573-2924
文摘
In this paper we explore the convergence properties of deterministic direct search methods when the objective function contains a stochastic or Monte Carlo simulation. We present new results for the case where the objective is only defined on a set with certain minimal regularity properties. We present two numerical examples to illustrate the ideas. Keywords Sampling methods Monte Carlo simulation Water resource policy Hidden constraints