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An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes

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Abstract

This paper develops an algorithm for uniform random generation over a constrained simplex, which is the intersection of a standard simplex and a given set. Uniform sampling from constrained simplexes has numerous applications in different fields, such as portfolio optimization, stochastic multi-criteria decision analysis, experimental design with mixtures and decision problems involving discrete joint distributions with imprecise probabilities. The proposed algorithm is developed by combining the acceptance–rejection and conditional methods along with the use of optimization tools. The acceptance rate of the algorithm is analytically compared to that of a crude acceptance–rejection algorithm, which generates points over the simplex and then rejects any points falling outside the intersecting set. Finally, using convex optimization, the setup phase of the algorithm is detailed for the special cases where the intersecting set is a general convex set, a convex set defined by a finite number of convex constraints or a polyhedron.

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References

  • Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Application. Wiley, New York (2006)

    Book  MATH  Google Scholar 

  • Butler, J., Dia, J., Dyer, J.: Simulation techniques for the sensitivity analysis of multi-criteria decision models. Eur. J. Oper. Res. 103, 531–545 (1997)

    Article  MATH  Google Scholar 

  • Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  • Burns, P.: Random portfolios for performance measurement. In: Kontoghiorghes, E., Gatu, C. (eds.) Optimisation, Econometric and Financial Analysis. Springer, Berlin (2007)

    Google Scholar 

  • Dawson, R., Young, R.: Near-uniformly distributed, stochastically generated portfolios. In: Satchell, S.E., Scowcroft, A.E. (eds.) Advances in Portfolio Construction and Implementation. Butterworth and Heinemann, Oxford (2003)

    Google Scholar 

  • Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986)

    Book  MATH  Google Scholar 

  • Fang, K.T., Yang, Z.H.: On uniform design of experiments with restricted mixtures and generation of uniform distribution on some domains. Stat. Probab. Lett. 46, 113–120 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Jia, J., Fischer, G.W., Dyer, J.S.: Attribute weighting methods and decision quality in the presence of response error: a simulation study. J. Behav. Decis. Mak. 11, 85–105 (1998)

    Article  Google Scholar 

  • Hormann, W., Leydold, J., Derflinger, G.: Automatic Nonuniform Random Variate Generation. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  • Jeleva, M., Villeneuve, B.: Insurance contracts with imprecise probabilities and adverse selection. Econ. Theory 23, 777–794 (2004)

  • Johnson, M.E.: Multivariate Statistical Simulation. Wiley, New York (1987)

    Book  MATH  Google Scholar 

  • Kroese, D.P., Taimre, T., Botev, Z.I.: Handbook of Monte Carlo Methods. Wiley, New York (2011)

    Book  MATH  Google Scholar 

  • Lahdelma, R., Hokkanen, J., Salminen, P.: Stochastic multiobjective acceptability analysis. Eur. J. Oper. Res. 106, 137–143 (1998)

  • Lahdelma, R., Salminen, P.: Prospect theory and stochastic multicriteria acceptability analysis (SMAA). Omega 37, 961–971 (2009)

    Article  Google Scholar 

  • Leydold, J., Hormann, W.: A sweep-plane algorithm for generating random tuples in simple polytopes. Math. Comput. 67, 1617–1635 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Moeini, A., Abbasi, B., Mahlooji, H.: Conditional distribution inverse method in generating uniform random vectors over a simplex. Commun. Stat. Simul. Comput. 40, 685–693 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Montiel, L.V., Bickel, J.E.: Generating a random collection of discrete joint probability distributions subject to partial information. Methodol. Comput. Appl. Probab. 15, 951–967 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Park, K.S., Kim, S.H.: Tools for interactive multiattribute decision making with incompletely identified information. Eur. J. Oper. Res. 98, 111–123 (1997)

    Article  MATH  Google Scholar 

  • Pouchkarev, I.: Performance Evaluation of Constrained Portfolios (ERIM Ph.D. Series Research in Management). Erasmus School of Economics, Erasmus University, Rotterdam (2005)

  • Rubin, P.A.: Generating random points on a polytope. Commun. Stat. Simul. Comput. J. 13, 375–396 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Rubinstein, R.Y.: Generating random vectors uniformly distributed inside and on the surface of different regions. Eur. J. Oper. Res. 10, 205–209 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Rubinstein, R.Y., Kroese, D.P.: Simulation and the Monte Carlo Method. Wiley-Interscience, New York (2007)

    Book  MATH  Google Scholar 

  • Tervonen, T., van Valkenhoef, G., Baştürk, N., Postmus, D.: Hit-and-run enables efficient weight generation for simulation-based multiple criteria decision analysis. Eur. J. Oper. Res. 224, 552–559 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Tian, G.L., Fang, K.T.: Uniform designs for mixture-amount experiments and for mixture experiments under order restrictions. Sci. China. Ser. A 42, 456–470 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Tian, G.L., Fang, H.B., Tan, M., Qin, H., Tang, M.L.: Uniform distributions in a class of convex polyhedrons with applications to drug combination studies. J. Multivar. Anal. 100, 1854–1865 (2009)

  • Vempala, S.: Geometric random walks: a survey. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, pp. 577–616. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  • Vetschera, R., Chen, Y., Hipel, K.W., Kilgour, D.M.: Robustness and information levels in case-based multiple criteria sorting. Eur. J. Oper. Res. 202, 841–852 (2010)

    Article  Google Scholar 

  • Utkin, L., Coolen, F.: Imprecise reliability: an introductory overview. Comput. Intell. Reliab. Eng. 40, 261–306 (2007)

  • Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)

    Book  MATH  Google Scholar 

  • Wang, Y., Fang, K.: Uniform design of experiments with mixtures. Sci. China Ser. A 39, 264–275 (1996)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Amir Ahmadi-Javid.

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Ahmadi-Javid, A., Moeini, A. An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes. Stat Comput 26, 703–713 (2016). https://doi.org/10.1007/s11222-015-9553-x

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