1. bookVolumen 30 (2020): Heft 4 (December 2020)
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2083-8492
Erstveröffentlichung
05 Apr 2007
Erscheinungsweise
4 Hefte pro Jahr
Sprachen
Englisch
access type Uneingeschränkter Zugang

Construction of constrained experimental designs on finite spaces for a modified Ek-optimality criterion

Online veröffentlicht: 31 Dec 2020
Volumen & Heft: Volumen 30 (2020) - Heft 4 (December 2020)
Seitenbereich: 659 - 677
Eingereicht: 23 May 2020
Akzeptiert: 13 Oct 2020
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2083-8492
Erstveröffentlichung
05 Apr 2007
Erscheinungsweise
4 Hefte pro Jahr
Sprachen
Englisch
Abstract

A simple computational algorithm is proposed for minimizing sums of largest eigenvalues of the matrix inverse over the set of all convex combinations of a finite number of nonnegative definite matrices subject to additional box constraints on the weights of those combinations. Such problems arise when experimental designs aiming at minimizing sums of largest asymptotic variances of the least-squares estimators are sought and the design region consists of finitely many support points, subject to the additional constraints that the corresponding design weights are to remain within certain limits. The underlying idea is to apply the method of outer approximations for solving the associated convex semi-infinite programming problem, which reduces to solving a sequence of finite min-max problems. A key novelty here is that solutions to the latter are found using generalized simplicial decomposition, which is a recent extension of the classical simplicial decomposition to nondifferentiable optimization. Thereby, the dimensionality of the design problem is drastically reduced. The use of the algorithm is illustrated by an example involving optimal sensor node activation in a large sensor network collecting measurements for parameter estimation of a spatiotemporal process.

Atkinson, A.C., Donev, A.N. and Tobias, R.D. (2007). Optimum Experimental Designs, with SAS, Oxford University Press, Oxford.Search in Google Scholar

Beddiaf, S., Autrique, L., Perez, L. and Jolly, J.-C. (2016). Heating source localization in a reduced time, International Journal of Applied Mathematics and Computer Science26(3): 623–640, DOI: 10.1515/amcs-2016-0043.10.1515/amcs-2016-0043Search in Google Scholar

Bernstein, D.S. (2005). Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, NJ.Search in Google Scholar

Bertsekas, D.P. (1999). Nonlinear Programming, 2nd Edn, Optimization and Computation Series, Athena Scientific, Belmont, MA.Search in Google Scholar

Bertsekas, D.P. (2015). Convex Optimization Algorithms, Athena Scientific, Belmont, MA.Search in Google Scholar

Bertsekas, D. and Yu, H. (2011). A unifying polyhedral approximation framework for convex optimization, SIAM Journal on Optimization21(1): 333–360.10.1137/090772204Search in Google Scholar

Böhning, D. (1986). A vertex-exchange-method in D-optimal design theory, Metrika33(12): 337–347.10.1007/BF01894766Search in Google Scholar

Botkin, N.D. and Stoer, J. (2005). Minimization of convex functions on the convex hull of a point set, Mathematical Methods of Operations Research62(2): 167–18.10.1007/s00186-005-0018-4Search in Google Scholar

Boyd, S. and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press, Cambridge.10.1017/CBO9780511804441Search in Google Scholar

Burclová, K. and Pázman, A. (2016). Optimal design of experiments via linear programming, Statistical Papers57(4): 893–910.10.1007/s00362-016-0782-7Search in Google Scholar

Chepuri, S.P. and Leus, G. (2015). Sparsity-promoting sensor selection for non-linear measurement models, IEEE Transactions on Signal Processing63(3): 684–698.10.1109/TSP.2014.2379662Search in Google Scholar

Coll, C. and Sánchez, E. (2019). Parameter identification and estimation for stage-structured population models, International Journal of Applied Mathematics and Computer Science29(2): 327–336, DOI: 10.2478/amcs-2019-0024.10.2478/amcs-2019-0024Search in Google Scholar

Cook, D. and Fedorov, V. (1995). Constrained optimization of experimental design, Statistics26: 129–178.10.1080/02331889508802474Search in Google Scholar

Djelassi, H., Glass, M. and Mitsos, A. (2019). Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints, Journal of Global Optimization75(2): 341–392.10.1007/s10898-019-00764-3Search in Google Scholar

Duarte, B.P.M., Granjo, J.F.O. and Wong, W.K. (2020). Optimal exact designs of experiments via mixed integer nonlinear programming, Statistics and Computing30(1): 93–112.10.1007/s11222-019-09867-zSearch in Google Scholar

Duarte, B.P.M. and Wong, W.K. (2014). A semi-infinite programming based algorithm for finding minimax optimal designs for nonlinear models, Statistics and Computing24(6): 1063–1080.10.1007/s11222-013-9420-6Search in Google Scholar

Esteban-Bravo, M., Leszkiewicz, A. and Vidal-Sanz, J.M. (2017). Exact optimal experimental designs with constraints, Statistics and Computing27(3): 845–863.10.1007/s11222-016-9658-xSearch in Google Scholar

Fedorov, V.V. (1989). Optimal design with bounded density: Optimization algorithms of the exchange type, Journal of Statistical Planning and Inference22: 1–13.10.1016/0378-3758(89)90060-8Search in Google Scholar

Fedorov, V.V. and Leonov, S.L. (2014). Optimal Design for Nonlinear Response Models, CRC Press, Boca Raton, FL.10.1201/b15054Search in Google Scholar

Harman, R. (2004). Minimal efficiency of designs under the class of orthogonally invariant information criteria, Metrika60(2): 137–153.10.1007/s001840300301Search in Google Scholar

Harman, R. and Benková, E. (2017). Barycentric algorithm for computing d-optimal size- and cost-constrained designs of experiments, Metrika80(2): 201–225.10.1007/s00184-016-0599-3Search in Google Scholar

Harman, R., Filová, L. and Richtárik, P. (2020). A randomized exchange algorithm for computing optimal approximate designs of experiments, Journal of the American Statistical Association115(529): 348–361.10.1080/01621459.2018.1546588Search in Google Scholar

Harman, R. and Pronzato, L. (2007). Improvements on removing nonoptimal support points in d-optimum design algorithms, Statistics & Probability Letters77(1): 90–94.10.1016/j.spl.2006.05.014Search in Google Scholar

Harville, D.A. (1997). Matrix Algebra From a Statistician’s Perspective, Springer-Verlag, New York, NY.10.1007/b98818Search in Google Scholar

Herzog, R., Riedel, I. and Uciński, D. (2018). Optimal sensor placement for joint parameter and state estimation problems in large-scale dynamical systems with applications to thermo-mechanics, Optimization and Engineering19(3): 591–627.10.1007/s11081-018-9391-8Search in Google Scholar

Hettich, R. and Kortanek, K.O. (1993). Semi-infinite programming: Theory, methods and applications, SIAM Review35(3): 380–429.10.1137/1035089Search in Google Scholar

Jacobson, M.Z. (1999). Fundamentals of Atmospheric Modeling, Cambridge University Press, Cambridge.Search in Google Scholar

Joshi, S. and Boyd, S. (2009). Sensor selection via convex optimization, IEEE Transactions on Signal Processing57(2): 451–462.10.1109/TSP.2008.2007095Search in Google Scholar

Katoh, N. (2001). Combinatorial optimization algorithms in resource allocation problems, in C.A. Floudas and P.M. Pardalos (Eds), Encyclopedia of Optimization, Vol. 1, Kluwer Academic Publishers, Dordrecht, pp. 259–264.10.1007/0-306-48332-7_56Search in Google Scholar

Khapalov, A.Y. (2010). Source localization and sensor placement in environmental monitoring, International Journal of Applied Mathematics and Computer Science20(3): 445–458, DOI: 10.2478/v10006-010-0033-3.10.2478/v10006-010-0033-3Search in Google Scholar

Langtangen, H.P. and Logg, A. (2016). Solving PDEs in Python. The FEniCS Tutorial I, Springer-Verlag, Cham.10.1007/978-3-319-52462-7Search in Google Scholar

Larsson, T., Migdalas, A. and Patriksson, M. (2015). A generic column generation principle: Derivation and convergence analysis, Operational Research15(2): 163–198.10.1007/s12351-015-0171-3Search in Google Scholar

Larsson, T., Patriksson, M. and Strömberg, A. (1998). Ergodic convergence in subgradient optimization, Optimization Methods and Software9(1–3): 93–120.10.1080/10556789808805688Search in Google Scholar

Lu, Z. and Pong, T.K. (2013). Computing optimal experimental designs via interior point method, SIAM Journal on Matrix Analysis and Applications34(4): 1556–1580.10.1137/120895093Search in Google Scholar

Maculan, N., Santiago, C.P., Macambira, E.M. and Jardim, M.H.C. (2003). An O(n) algorithm for projecting a vector on the intersection of a hyperplane and a box in ℝn, Journal of Optimization Theory and Applications117(3): 553–574.10.1023/A:1023997605430Search in Google Scholar

Marshall, A.W., Olkin, I. and Arnold, B.C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd Edn, Springer-Verlag, New York, NY.10.1007/978-0-387-68276-1Search in Google Scholar

Melas, V. (2006). Functional Approach to Optimal Experimental Design, Springer-Verlag, New York, NY.Search in Google Scholar

Patan, M. and Kowalów, D. (2018). Distributed scheduling of measurements in a sensor network for parameter estimation of spatio-temporal systems, International Journal of Applied Mathematics and Computer Science28(1): 39–54, DOI: 10.2478/amcs-2018-0003.10.2478/amcs-2018-0003Search in Google Scholar

Patan, M. and Uciński, D. (2008). Configuring a sensor network for fault detection in distributed parameter systems, International Journal of Applied Mathematics and Computer Science18(4): 513–524, DOI: 10.2478/v10006-008-0045-4.10.2478/v10006-008-0045-4Search in Google Scholar

Patan, M. and Uciński, D. (2019). Generalized simplicial decomposition for optimal sensor selection in parameter estimation of spatiotemporal processes, 2019 American Control Conference (ACC), Philadelphia, PA, USA, pp. 2546–2551.Search in Google Scholar

Patriksson, M. (2001). Simplicial decomposition algorithms, in C.A. Floudas and P.M. Pardalos (Eds), Encyclopedia of Optimization, Vol. 5, Kluwer Academic Publishers, Dordrecht, pp. 205–212.10.1007/0-306-48332-7_468Search in Google Scholar

Pázman, A. (1986). Foundations of Optimum Experimental Design, Mathematics and Its Applications, D. Reidel Publishing Company, Dordrecht.Search in Google Scholar

Polak, E. (1987). On the mathematical foundations of nondifferentiable optimization in engineering design, SIAM Review29(1): 21–89.10.1137/1029002Search in Google Scholar

Polak, E. (1997). Optimization. Algorithms and Consistent Approximations, Applied Mathematical Sciences, Springer-Verlag, New York, NY.10.1007/978-1-4612-0663-7Search in Google Scholar

Pronzato, L. (2003). Removing non-optimal support points in D-optimum design algorithms, Statistics & Probability Letters63: 223–228.10.1016/S0167-7152(03)00081-6Search in Google Scholar

Pronzato, L. and Pàzman, A. (2013). Design of Experiments in Nonlinear Models. Asymptotic Normality, Optimality Criteria and Small-Sample Properties, Springer-Verlag, New York, NY.10.1007/978-1-4614-6363-4Search in Google Scholar

Pronzato, L. and Zhigljavsky, A.A. (2014). Algorithmic construction of optimal designs on compact sets for concave and differentiable criteria, Journal of Statistical Planning and Inference154: 141–155.10.1016/j.jspi.2014.04.005Search in Google Scholar

Pukelsheim, F. (1993). Optimal Design of Experiments, Probability and Mathematical Statistics, John Wiley & Sons, New York, NY.Search in Google Scholar

Reemtsen, R. and Görner, S. (1998). Numerical methods for semi-infinite programming: A survey, in R. Reemtsen and J.-J. Rückmann (Eds), Semi-Infinite Programming,Kluwer Academic Publishers, Boston, MA, pp. 195–275.10.1007/978-1-4757-2868-2_7Search in Google Scholar

Sagnol, G. (2011). Computing optimal designs of multiresponse experiments reduces to second-order cone programming, Journal of Statistical Planning and Inference141(5): 1684–1708.10.1016/j.jspi.2010.11.031Search in Google Scholar

Sagnol, G. and Harman, R. (2015). Computing exact D-optimal designs by mixed integer second-order cone programming, The Annals of Statistics43(5): 2198–2224.10.1214/15-AOS1339Search in Google Scholar

Sahm, M. and Schwabe, R. (2001). A note on optimal bounded designs, in A. Atkinson et al. (Eds), Optimum Design 2000, Kluwer Academic Publishers, Dordrecht, Chapter 13, pp. 131–140.10.1007/978-1-4757-3419-5_13Search in Google Scholar

Seber, G.A.F. and Wild, C.J. (1989). Nonlinear Regression, John Wiley & Sons, New York, NY.10.1002/0471725315Search in Google Scholar

Shimizu, K. and Aiyoshi, E. (1980). Necessary conditions for min-max problems and algorithms by a relaxation procedure, IEEE Transactions on Automatic ControlAC-25(1): 62–66.10.1109/TAC.1980.1102226Search in Google Scholar

Silvey, S.D., Titterington, D.M. and Torsney, B. (1978). An algorithm for optimal designs on a finite design space, Communications in Statistics—Theory and Methods14: 1379–1389.10.1080/03610927808827719Search in Google Scholar

Torsney, B. and Mandal, S. (2001). Construction of constrained optimal designs, in A. Atkinson et al. (Eds), Optimum Design 2000, Kluwer Academic Publishers, Dordrecht, Chapter 14, pp. 141–152.10.1007/978-1-4757-3419-5_14Search in Google Scholar

Uciński, D. (2005). Optimal Measurement Methods for Distributed-Parameter System Identification, CRC Press, Boca Raton, FL.10.1201/9780203026786Search in Google Scholar

Uciński, D. (2012). Sensor network scheduling for identification of spatially distributed processes, International Journal of Applied Mathematics and Computer Science22(1): 25–40, DOI: 10.2478/v10006-012-0002-0.10.2478/v10006-012-0002-0Search in Google Scholar

Uciński, D. (2015). An algorithm for construction of constrained D-optimum designs, in A. Steland et al. (Eds), Stochastic Models, Statistics and Their Applications, Springer Proceedings in Mathematics & Statistics, Springer-Verlag, Cham, pp. 461–468.10.1007/978-3-319-13881-7_51Search in Google Scholar

Uciński, D. (2020). D-optimal sensor selection in the presence of correlated measurement noise, Measurement164: 107873.10.1016/j.measurement.2020.107873Search in Google Scholar

Uciński, D. and Patan, M. (2007). D-optimal design of a monitoring network for parameter estimation of distributed systems, Journal of Global Optimization39(2): 291–322.10.1007/s10898-007-9139-zSearch in Google Scholar

Wu, C.-F. (1978). Some algorithmic aspects of the theory of optimal designs, The Annals of Statistics6(6): 1286–1301.10.1214/aos/1176344374Search in Google Scholar

Yu, Y. (2010). Monotonic convergence of a general algorithm for computing optimal designs, The Annals of Statistics38(3): 1593–1606.10.1214/09-AOS761Search in Google Scholar

Yu, Y. (2011). D-optimal designs via a cocktail algorithm, Statistics and Computing21(3): 475–481.10.1007/s11222-010-9183-2Search in Google Scholar

Zarrop, M.B. and Goodwin, G.C. (1975). Comments on “Optimal inputs for system identification”, IEEE Transactions on Automatic ControlAC-20(2): 299–300.10.1109/TAC.1975.1100891Search in Google Scholar

Zhang, L., Wu, S.-Y. and López, M.A. (2010). A new exchange method for convex semi-infinite programming, SIAM Journal on Optimization20(6): 2959–2977.10.1137/090767133Search in Google Scholar

Empfohlene Artikel von Trend MD

Planen Sie Ihre Fernkonferenz mit Scienceendo