A Novel Sampling Approach to Combinatorial Optimization Under Uncertainty

The generalized approach to stochastic optimization involves two computationally intensive recursive loops: (1) the outer optimization loop, (2) the inner sampling loop. Furthermore, inclusion of discrete decision variables adds to the complexity. The focus of the current endeavor is to reduce the computational intensity of the two recursive loops. The study achieves the goals through an improved understanding and description of the sampling phenomena based on the concepts of fractal geometry and incorporating the knowledge of the accuracy of the sampling (fractal model) in the stochastic optimization framework thereby, automating and improving the combinatorial optimization algorithm. The efficiency of the algorithm is presented in the context of a large scale real world problem, related to the nuclear waste at Hanford, involving discrete and continuous decision variables, and uncertainties. These new developments reduced the computational intensity for solving this problem from an estimated 20 days of CPU time on a dedicated Alpha workstation to 18 hours of CPU time on the same machine.

[1]  Samy Bengio,et al.  Variance Reduction Techniques in . . . , 2003 .

[2]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[3]  W. J. Whiten,et al.  Computational investigations of low-discrepancy sequences , 1997, TOMS.

[4]  Peter Kall,et al.  Stochastic Programming , 1995 .

[5]  R. Wets,et al.  Stochastic programming , 1989 .

[6]  Urmila M. Diwekar,et al.  Process synthesis under uncertainty: A penalty function approach , 1996 .

[7]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[8]  Urmila M. Diwekar,et al.  Synthesizing optimal design configurations for a brayton cycle power plant , 1994 .

[9]  U. Diwekar,et al.  Stochastic annealing for synthesis under uncertainty , 1995 .

[10]  S. Andradóttir,et al.  A Simulated Annealing Algorithm with Constant Temperature for Discrete Stochastic Optimization , 1999 .

[11]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[12]  Max Henrion,et al.  Uncertainty: A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis , 1990 .

[13]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[14]  D. Rogers,et al.  Variance-Reduction Techniques , 1988 .

[15]  Ronald L. Iman,et al.  A FORTRAN-77 PROGRAM AND USER'S GUIDE FOR THE GENERATION OF LATIN HYPERCUBE AND RANDOM SAMPLES FOR USE WITH COMPUTER MODELS , 1984 .

[16]  Urmila M. Diwekar,et al.  An efficient sampling technique for off-line quality control , 1997 .

[17]  Shi-Chune Yao,et al.  Correlation of thermal conductivities of unidirectional fibrous composites using local fractal techniques , 1991 .

[18]  Urmila M. Diwekar,et al.  Synthesizing optimal waste blends , 1996 .

[19]  John P. Lehoczky,et al.  Path Generation for Quasi-Monte Carlo Simulation of Mortgage-Backed Securities , 2000 .

[20]  J. C. Helton,et al.  An Investigation of Uncertainty and Sensitivity Analysis Techniques for Computer Models , 1988 .

[21]  Marek J. Druzdzel,et al.  Computational Investigation of Low-Discrepancy Sequences in . . . , 2000 .

[22]  Stochastic Programming,et al.  Logarithmic Concave Measures and Related Topics , 1980 .

[23]  Hartmut Jürgens,et al.  Fractals for the Classroom: Part One Introduction to Fractals and Chaos , 1991 .

[24]  H. Wozniakowski.  Average case complexity of multivariate integration , 1991 .

[25]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[26]  George B. Dantzig,et al.  Parallel processors for planning under uncertainty , 1990 .

[27]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[28]  Grzegorz W. Wasilkowski,et al.  On the average complexity of multivariate problems , 1990, J. Complex..

[29]  R. Toral,et al.  Simulated Annealing Using Hybrid Monte Carlo , 1997 .

[30]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[31]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1987, Math. Program..

[32]  R. E. Gephart,et al.  Hanford Tank Cleanup: A Guide to Understanding the Technical Issues , 1998 .

[33]  J Figueira,et al.  Stochastic Programming , 1998, J. Oper. Res. Soc..

[34]  Roger J.-B. Wets.  Challenges in stochastic programming , 1996, Math. Program..

[35]  U. Diwekar,et al.  Efficient sampling technique for optimization under uncertainty , 1997 .

[36]  Emile H. L. Aarts,et al.  Simulated Annealing: Theory and Applications , 1987, Mathematics and Its Applications.

[37]  Eduardo Saliby,et al.  Descriptive Sampling: A Better Approach to Monte Carlo Simulation , 1990 .

[38]  H. Szu.  Fast simulated annealing , 1987 .

[39]  Ronald L. Iman,et al.  Risk methodology for geologic disposal of radioactive waste: small sample sensitivity analysis techniques for computer models, with an application to risk assessment , 1980 .

[40]  John R. Birge,et al.  Stochastic Programming Computation and Applications , 1997, INFORMS J. Comput..

[41]  Julia L. Higle,et al.  Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse , 1991, Math. Oper. Res..