Optimisation in space of measures and optimal design

Ilya Molchanov; Sergei Zuyev

ESAIM: Probability and Statistics (2004)

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

Abstract

top
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

top

Molchanov, Ilya, and Zuyev, Sergei. "Optimisation in space of measures and optimal design." ESAIM: Probability and Statistics 8 (2004): 12-24. <http://eudml.org/doc/245136>.

@article{Molchanov2004,
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},
language = {eng},
pages = {12-24},
publisher = {EDP-Sciences},
title = {Optimisation in space of measures and optimal design},
url = {http://eudml.org/doc/245136},
volume = {8},
year = {2004},
}

TY - JOUR
AU - Molchanov, Ilya
AU - Zuyev, Sergei
TI - Optimisation in space of measures and optimal design
JO - ESAIM: Probability and Statistics
PY - 2004
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
UR - http://eudml.org/doc/245136
ER -

References

top
  1. [1] A.C. Atkinson and A.N. Donev, Optimum Experimental Designs. Clarendon Press, Oxford (1992). Zbl0829.62070
  2. [2] C.L. Atwood, Sequences converging to D -optimal designs of experiments. Ann. Statist. 1 (1973) 342-352. Zbl0263.62047MR356385
  3. [3] C.L. Atwood, Convergent design sequences, for sufficiently regular optimality criteria. Ann. Statist. 4 (1976) 1124-1138. Zbl0344.62064MR418352
  4. [4] D. Böhning, A vertex-exchange-method in D -optimal design theory. Metrika 33 (1986) 337-347. Zbl0601.62091MR868043
  5. [5] R. Cominetti, Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21 (1990) 265-287. Zbl0692.49018MR1036588
  6. [6] D. Cook and V. Fedorov, Constrained optimization of experimental design. Statistics 26 (1995) 129-178. Zbl0812.62080MR1318209
  7. [7] D.J. Daley and D. Vere–Jones, An Introduction to the Theory of Point Processes. Springer, New York (1988). Zbl0657.60069
  8. [8] N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory. Wiley, New York (1988). Zbl0635.47001MR1009162
  9. [9] V.V. Fedorov, Theory of Optimal Experiments. Academic Press, New York (1972). Zbl0261.62002MR403103
  10. [10] V.V. Fedorov, Optimal design with bounded density: Optimization algorithms of the exchange type. J. Statist. Plan. Inf. 22 (1989) 1-13. Zbl0682.62044MR996795
  11. [11] V.V. Fedorov and P. Hackl, Model-Oriented Design of Experiments. Springer, New York, Lecture Notes in Statist. 125 (1997). Zbl0878.62052MR1454123
  12. [12] I. Ford, Optimal Static and Sequential Design: A Critical Review, Ph.D. Thesis. Department of Statistics, University of Glasgow, Glasgow (1976). 
  13. [13] A. Gaivoronski, Linearization methods for optimization of functionals which depend on probability measures. Math. Progr. Study 28 (1986) 157-181. Zbl0596.90071MR836766
  14. [14] N. Gaffke and R. Mathar, On a Class of Algorithms from Experimental Design Theory. Optimization 24 (1992) 91-126. Zbl0817.90075MR1238646
  15. [15] E. Hille and R.S. Phillips, Functional Analysis and Semigroups. American Mathematical Society, Providence, AMS Colloquium Publications XXXI (1957). Zbl0078.10004MR89373
  16. [16] J. Kiefer, General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 (1974) 849-879. Zbl0291.62093MR356386
  17. [17] J. Kiefer and J. Wolfowitz, The equivalence of two extremal problems. Canad. J. Math. 14 (1960) 363-366. Zbl0093.15602MR117842
  18. [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. Development 129 (2002) 4891-4899. 
  19. [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). Zbl0985.62057MR1933941
  20. [20] H. Maurer and J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Programming 16 (1979) 98-110. Zbl0398.90109MR517762
  21. [21] I. Molchanov and S. Zuyev, Steepest descent algorithms in space of measures. Statist. and Comput. 12 (2002) 115-123. MR1897510
  22. [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. Zbl1053.49016MR1781240
  23. [23] I. Molchanov and S. Zuyev, Variational analysis of functionals of a Poisson process. Math. Oper. Res. 25 (2000) 485-508. Zbl1018.49022MR1855179
  24. [24] C.H. Müller and A. Pázman, Applications of necessary and sufficient conditions for maximin efficient designs. Metrika 48 (1998) 1-19. Zbl0990.62062MR1647889
  25. [25] A. Pázman, Hilbert-space methods in experimantal design. Kybernetika 14 (1978) 73-84. Zbl0385.62052MR478496
  26. [26] E. Polak, Optimization. Algorithms and Consistent Approximations. Springer, New York (1997). Zbl0899.90148MR1454128
  27. [27] F. Pukelsheim, Optimal Design of Experiments. Wiley, New York (1993). Zbl0834.62068MR1211416
  28. [28] S.M. Robinson, First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30 (1976) 597-607. Zbl0364.90093MR406529
  29. [29] S.D. Silvey, Optimum Design. Chapman & Hall, London (1980). Zbl0468.62070MR606742
  30. [30] D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and its Applications, Second Edition. Wiley, Chichester (1995). Zbl0838.60002MR895588
  31. [31] P. Whittle, Some general points in the theory of optimal experimental design. J. Roy. Statist. Soc. Ser. B 35 (1973) 123-130. Zbl0282.62065MR356388
  32. [32] G. Winkler, Extreme points of moment sets. Math. Oper. Res. 13 (1988) 581-587. Zbl0669.60009MR971911
  33. [33] C.-F. Wu, Some algorithmic aspects of the theory of optimal design. Ann. Statist. 6 (1978) 1286-1301. Zbl0392.62058MR523763
  34. [34] C.-F. Wu, Some iterative procedures for generating nonsingular optimal designs. Comm. Statist. Theory Methods A 7 (1978) 1399-1412. Zbl0399.62076
  35. [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. Zbl0396.62059MR523762
  36. [36] H.P. Wynn, The sequential generation of D -optimum experimental designs. Ann. Math. Statist. 41 (1970) 1655-1664. Zbl0224.62038MR267704
  37. [37] H.P. Wynn, Results in the theory and construction of D -optimum experimental designs. J. Roy. Statist. Soc. Ser. B 34 (1972) 133-147. Zbl0248.62033MR350987
  38. [38] J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5 (1979) 49-62. Zbl0401.90104MR526427

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.