Optimisation in space of measures and optimal design

Ilya Molchanov; Sergei Zuyev

ESAIM: Probability and Statistics (2010)

  • Volume: 8, page 12-24
  • ISSN: 1292-8100

Abstract

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The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes it possible to use the whole arsenal of descent algorithms from the general optimisation literature for finding optimal designs. The corresponding steepest descent involves adding a signed measure at every step and converges faster than the conventional sequential algorithms used to construct optimal designs. We study a new class of design problems when the observation points are distributed according to a Poisson point process arising in the situation when the total control on the placement of measurements is impossible.

How to cite

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Molchanov, Ilya, and Zuyev, Sergei. "Optimisation in space of measures and optimal design." ESAIM: Probability and Statistics 8 (2010): 12-24. <http://eudml.org/doc/104314>.

@article{Molchanov2010,
abstract = { The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes it possible to use the whole arsenal of descent algorithms from the general optimisation literature for finding optimal designs. The corresponding steepest descent involves adding a signed measure at every step and converges faster than the conventional sequential algorithms used to construct optimal designs. We study a new class of design problems when the observation points are distributed according to a Poisson point process arising in the situation when the total control on the placement of measurements is impossible. },
author = {Molchanov, Ilya, Zuyev, Sergei},
journal = {ESAIM: Probability and Statistics},
keywords = {Optimal experimental design; generalized equivalence theorem; constrained optimal design; Poisson design; optimization on measures; gradient methods.; gradient methods},
language = {eng},
month = {3},
pages = {12-24},
publisher = {EDP Sciences},
title = {Optimisation in space of measures and optimal design},
url = {http://eudml.org/doc/104314},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Molchanov, Ilya
AU - Zuyev, Sergei
TI - Optimisation in space of measures and optimal design
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 12
EP - 24
AB - The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes it possible to use the whole arsenal of descent algorithms from the general optimisation literature for finding optimal designs. The corresponding steepest descent involves adding a signed measure at every step and converges faster than the conventional sequential algorithms used to construct optimal designs. We study a new class of design problems when the observation points are distributed according to a Poisson point process arising in the situation when the total control on the placement of measurements is impossible.
LA - eng
KW - Optimal experimental design; generalized equivalence theorem; constrained optimal design; Poisson design; optimization on measures; gradient methods.; gradient methods
UR - http://eudml.org/doc/104314
ER -

References

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  1. A.C. Atkinson and A.N. Donev, Optimum Experimental Designs. Clarendon Press, Oxford (1992).  Zbl0829.62070
  2. C.L. Atwood, Sequences converging to D-optimal designs of experiments. Ann. Statist.1 (1973) 342-352.  Zbl0263.62047
  3. C.L. Atwood, Convergent design sequences, for sufficiently regular optimality criteria. Ann. Statist.4 (1976) 1124-1138.  Zbl0344.62064
  4. D. Böhning, A vertex-exchange-method in D-optimal design theory. Metrika33 (1986) 337-347.  Zbl0601.62091
  5. R. Cominetti, Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim.21 (1990) 265-287.  Zbl0692.49018
  6. D. Cook and V. Fedorov, Constrained optimization of experimental design. Statistics26 (1995) 129-178.  Zbl0812.62080
  7. D.J. Daley and D. Vere–Jones, An Introduction to the Theory of Point Processes. Springer, New York (1988).  Zbl0657.60069
  8. N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory. Wiley, New York (1988).  Zbl0635.47001
  9. V.V. Fedorov, Theory of Optimal Experiments. Academic Press, New York (1972).  
  10. V.V. Fedorov, Optimal design with bounded density: Optimization algorithms of the exchange type. J. Statist. Plan. Inf.22 (1989) 1-13.  Zbl0682.62044
  11. V.V. Fedorov and P. Hackl, Model-Oriented Design of Experiments. Springer, New York, Lecture Notes in Statist. 125 (1997).  Zbl0878.62052
  12. I. Ford, Optimal Static and Sequential Design: A Critical Review, Ph.D. Thesis. Department of Statistics, University of Glasgow, Glasgow (1976).  
  13. A. Gaivoronski, Linearization methods for optimization of functionals which depend on probability measures. Math. Progr. Study28 (1986) 157-181.  Zbl0596.90071
  14. N. Gaffke and R. Mathar, On a Class of Algorithms from Experimental Design Theory. Optimization24 (1992) 91-126.  Zbl0817.90075
  15. E. Hille and R.S. Phillips, Functional Analysis and Semigroups. American Mathematical Society, Providence, AMS Colloquium Publications XXXI (1957).  Zbl0078.10004
  16. J. Kiefer, General equivalence theory for optimum designs (approximate theory). Ann. Statist.2 (1974) 849-879.  Zbl0291.62093
  17. J. Kiefer and J. Wolfowitz, The equivalence of two extremal problems. Canad. J. Math.14 (1960) 363-366.  Zbl0093.15602
  18. P. Kumaravelu, L. Hook, A.M. Morrison, J. Ure, S. Zhao, S. Zuyev, J. Ansell and A. Medvinsky, Quantitative developmental anatomy of definitive haematopoietic stem cells/long-term repopulating units (HSC/RUs): Role of the aorta-gonad-mesonephros (AGM) region and the yolk sac in colonisation of the mouse embryonic liver. Development129 (2002) 4891-4899.  
  19. E.P. Liski, N.K. Mandal, K.R. Shah and B.K. Singha, Topics in Optimal Design. Springer, New York, Lect. Notes Statist. 163 (2002).  
  20. H. Maurer and J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Programming16 (1979) 98-110.  Zbl0398.90109
  21. I. Molchanov and S. Zuyev, Steepest descent algorithms in space of measures. Statist. and Comput.12 (2002) 115-123.  
  22. I. Molchanov and S. Zuyev, Tangent sets in the space of measures: With applications to variational calculus. J. Math. Anal. Appl.249 (2000) 539-552.  Zbl1053.49016
  23. I. Molchanov and S. Zuyev, Variational analysis of functionals of a Poisson process. Math. Oper. Res.25 (2000) 485-508.  Zbl1018.49022
  24. C.H. Müller and A. Pázman, Applications of necessary and sufficient conditions for maximin efficient designs. Metrika48 (1998) 1-19.  Zbl0990.62062
  25. A. Pázman, Hilbert-space methods in experimantal design. Kybernetika14 (1978) 73-84.  Zbl0385.62052
  26. E. Polak, Optimization. Algorithms and Consistent Approximations. Springer, New York (1997).  Zbl0899.90148
  27. F. Pukelsheim, Optimal Design of Experiments. Wiley, New York (1993).  Zbl0834.62068
  28. S.M. Robinson, First order conditions for general nonlinear optimization. SIAM J. Appl. Math.30 (1976) 597-607.  Zbl0364.90093
  29. S.D. Silvey, Optimum Design. Chapman & Hall, London (1980).  
  30. D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and its Applications, Second Edition. Wiley, Chichester (1995).  Zbl0838.60002
  31. P. Whittle, Some general points in the theory of optimal experimental design. J. Roy. Statist. Soc. Ser. B35 (1973) 123-130.  Zbl0282.62065
  32. G. Winkler, Extreme points of moment sets. Math. Oper. Res.13 (1988) 581-587.  Zbl0669.60009
  33. C.-F. Wu, Some algorithmic aspects of the theory of optimal design. Ann. Statist.6 (1978) 1286-1301.  Zbl0392.62058
  34. C.-F. Wu, Some iterative procedures for generating nonsingular optimal designs. Comm. Statist. Theory Methods A7 (1978) 1399-1412.  Zbl0399.62076
  35. C.-F. Wu and H.P. Wynn, The convergence of general step-length algorithms for regular optimum design criteria. Ann. Statist.6 (1978) 1273-1285.  Zbl0396.62059
  36. H.P. Wynn, The sequential generation of D-optimum experimental designs. Ann. Math. Statist.41 (1970) 1655-1664.  Zbl0224.62038
  37. H.P. Wynn, Results in the theory and construction of D-optimum experimental designs. J. Roy. Statist. Soc. Ser. B34 (1972) 133-147.  Zbl0248.62033
  38. J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim.5 (1979) 49-62.  Zbl0401.90104

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