Non-parametric stochastic subset optimization utilizing multivariate boundary kernels and adaptive stochastic sampling

Abstract The implementation of NP-SSO (non-parametric stochastic subset optimization) to general design under uncertainty problems and its enhancement through various soft computing techniques is discussed. NP-SSO relies on iterative simulation of samples of the design variables from an auxiliary probability density, and approximates the objective function through kernel density estimation (KDE) using these samples. To deal with boundary correction in complex domains, a multivariate boundary KDE based on local linear estimation is adopted in this work. Also, a non-parametric characterization of the search space at each iteration using a framework based on support vector machine is formulated. To further improve computational efficiency, an adaptive kernel sampling density formulation is integrated and an adaptive, iterative selection of the number of samples needed for the KDE implementation is established.

[1]  David M. Boore,et al.  Simulation of Ground Motion Using the Stochastic Method , 2003 .

[2]  J. Bray,et al.  Characterization of forward-directivity ground motions in the near-fault region , 2004 .

[3]  Gerhart I. Schuëller,et al.  A survey on approaches for reliability-based optimization , 2010 .

[4]  J. Beck,et al.  A new adaptive importance sampling scheme for reliability calculations , 1999 .

[5]  Robert Tibshirani,et al.  Estimating the number of clusters in a data set via the gap statistic , 2000 .

[6]  Alexandros A. Taflanidis,et al.  Non-parametric stochastic subset optimization for optimal-reliability design problems , 2013 .

[7]  J. A. Hartigan,et al.  A k-means clustering algorithm , 1979 .

[8]  Alaa Chateauneuf,et al.  Reliability-based optimization of structural systems by adaptive target safety – Application to RC frames , 2008 .

[9]  R. Baierlein.  Probability Theory: The Logic of Science , 2004 .

[10]  Alexandros A. Taflanidis,et al.  Adaptive importance sampling for optimization under uncertainty problems , 2014 .

[11]  E. Polak,et al.  Reliability-based optimal design using sample average approximations , 2004 .

[12]  Jens Perch Nielsen.  Multivariate Boundary Kernels from Local Linear Estimation , 1999 .

[13]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[14]  James L. Beck,et al.  Life-cycle cost optimal design of passive dissipative devices , 2009 .

[15]  Bernhard Sendhoff,et al.  Robust Optimization - A Comprehensive Survey , 2007 .

[16]  J. Beck,et al.  A New Adaptive Rejection Sampling Method Using Kernel Density Approximations and Its Application to Subset Simulation , 2017 .

[17]  James L. Beck,et al.  An efficient framework for optimal robust stochastic system design using stochastic simulation , 2008 .

[18]  James C. Spall,et al.  Introduction to stochastic search and optimization - estimation, simulation, and control , 2003, Wiley-Interscience series in discrete mathematics and optimization.

[19]  A. Atiya,et al.  Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond , 2005, IEEE Transactions on Neural Networks.

[20]  Alexandros A. Taflanidis,et al.  A simulation‐based framework for risk assessment and probabilistic sensitivity analysis of base‐isolated structures , 2011 .

[21]  André T. Beck,et al.  A comparison of deterministic‚ reliability-based and risk-based structural optimization under uncertainty , 2012 .

[22]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[23]  Gerhart I. Schuëller,et al.  Computational methods in optimization considering uncertainties – An overview , 2008 .

[24]  George P. Mavroeidis,et al.  A Mathematical Representation of Near-Fault Ground Motions , 2003 .

[25]  Christian P. Robert,et al.  Monte Carlo Statistical Methods (Springer Texts in Statistics) , 2005 .

[26]  A. Taflanidis,et al.  Sample-based evaluation of global probabilistic sensitivity measures , 2014 .

[27]  Armen Der Kiureghian,et al.  Optimal design with probabilistic objective and constraints , 2006 .

[28]  Johannes O. Royset,et al.  Efficient sample sizes in stochastic nonlinear programming , 2008 .

[29]  Ian Abramson.  On Bandwidth Variation in Kernel Estimates-A Square Root Law , 1982 .

[30]  J. Beck,et al.  Stochastic Subset Optimization for optimal reliability problems , 2008 .

[31]  I. Doltsinis,et al.  ROBUST DESIGN OF STRUCTURES USING OPTIMIZATION METHODS , 2004 .