Optimal exact designs of experiments via Mixed Integer Nonlinear Programming

Optimal exact designs are problematic to find and study because there is no unified theory for determining them and studying their properties. Each has its own challenges and when a method exists to confirm the design optimality, it is invariably applicable to the particular problem only. We propose a systematic approach to construct optimal exact designs by incorporating the Cholesky decomposition of the Fisher Information Matrix in a Mixed Integer Nonlinear Programming formulation. As examples, we apply the methodology to find D - and A -optimal exact designs for linear and nonlinear models using global or local optimizers. Our examples include design problems with constraints on the locations or the number of replicates at the optimal design points.

[1]  Radoslav Harman,et al.  Computing c-optimal experimental designs using the simplex method of linear programming , 2008, Comput. Stat. Data Anal..

[2]  Laird Ak.  DYNAMICS OF TUMOR GROWTH. , 1964 .

[3]  W. Welch.  Branch-and-Bound Search for Experimental Designs Based on D Optimality and Other Criteria , 1982 .

[4]  Ilya S. Molchanov,et al.  Steepest descent algorithms in a space of measures , 2002, Stat. Comput..

[5]  Nikolaos V. Sahinidis,et al.  Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming , 2002 .

[6]  Peter Goos,et al.  Constructing General Orthogonal Fractional Factorial Split-Plot Designs , 2015, Technometrics.

[7]  Tapio Westerlund,et al.  Comparison of Some High-performance MINLP Solvers , 2007 .

[8]  M. E. Johnson,et al.  Some Guidelines for Constructing Exact D-Optimal Designs on Convex Design Spaces , 1983 .

[9]  Peter Goos,et al.  D-Optimal Split-Plot Designs With Given Numbers and Sizes of Whole Plots , 2003, Technometrics.

[10]  Peter Goos,et al.  And by contacting: The MIMS Secretary , 2005 .

[11]  Paula Camelia Trandafir,et al.  Optimal designs for discriminating between some extensions of the Michaelis–Menten model , 2005 .

[12]  N. Gaffke.  On D-optimality of exact linear regression designs with minimum support , 1986 .

[13]  Guillaume Sagnol,et al.  Computing exact D-optimal designs by mixed integer second-order cone programming , 2013, 1307.4953.

[14]  Kenneth Sörensen,et al.  Optimal design of large-scale screening experiments: a critical look at the coordinate-exchange algorithm , 2016, Stat. Comput..

[15]  S. Berman.  An Extension of the Arc Sine Law , 1962 .

[16]  Jie Yang,et al.  Optimal Designs for Two-Level Factorial Experiments with Binary Response , 2010, 1003.1557.

[17]  Abhyuday Mandal,et al.  Optimal Designs for 2^k Factorial Experiments with Binary Response , 2011, 1109.5320.

[18]  W. G. Hunter,et al.  The Experimental Study of Physical Mechanisms , 1965 .

[19]  Anthony C. Atkinson,et al.  The construction of exact D optimum wxperimental designs with application to blocking response surface designs , 1989 .

[20]  Weng Kee Wong,et al.  Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming‐Based Approach , 2015, International statistical review = Revue internationale de statistique.

[21]  Arne Drud,et al.  CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems , 1985, Math. Program..

[22]  Christian Kirches,et al.  Mixed-integer nonlinear optimization*† , 2013, Acta Numerica.

[23]  A. Hill,et al.  The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves , 1910 .

[24]  Stephen P. Boyd,et al.  Applications of semidefinite programming , 1999 .

[25]  Toby J. Mitchell,et al.  An algorithm for the construction of “ D -optimal” experimental designs , 2000 .

[26]  Guillaume Sagnol,et al.  Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming , 2009, 0912.5467.

[27]  H. Wynn.  The Sequential Generation of $D$-Optimum Experimental Designs , 1970 .

[28]  Radoslav Harman,et al.  Computing efficient exact designs of experiments using integer quadratic programming , 2014, Comput. Stat. Data Anal..

[29]  Peter Goos,et al.  Symmetry breaking in mixed integer linear programming formulations for blocking two-level orthogonal experimental designs , 2016, Comput. Oper. Res..

[30]  L. Imhof.  A-Optimum Exact Designs for Quadratic Regression , 1998 .

[31]  N. Higham,et al.  Stability of methods for matrix inversion , 1992 .

[32]  F. Pukelsheim.  Optimal Design of Experiments , 1993 .

[33]  David M. Steinberg,et al.  Fast Computation of Designs Robust to Parameter Uncertainty for Nonlinear Settings , 2009, Technometrics.

[34]  Gene H. Golub,et al.  Matrix computations , 1983 .

[35]  Sven Leyffer,et al.  Numerical Experience with Lower Bounds for MIQP Branch-And-Bound , 1998, SIAM J. Optim..

[36]  Jesús López-Fidalgo,et al.  Efficiencies of Rounded Optimal Approximate Designs for Small Samples , 2001 .

[37]  C. Pantelides.  The consistent intialization of differential-algebraic systems , 1988 .

[38]  K. Kortanek,et al.  Equivalence Theorems and Cutting Plane Algorithms for a Class of Experimental Design Problems , 1977 .

[39]  W. J. Studden,et al.  Theory Of Optimal Experiments , 1972 .

[40]  T. Westerlund,et al.  An extended cutting plane method for solving convex MINLP problems , 1995 .

[41]  Holger Dette,et al.  Adaptive grid semidefinite programming for finding optimal designs , 2017, Statistics and Computing.

[42]  Weng Kee Wong,et al.  A Semi-Infinite Programming based algorithm for determining T-optimum designs for model discrimination , 2015, J. Multivar. Anal..

[43]  Valerii V. Fedorov,et al.  Optimal Design for Nonlinear Response Models , 2013 .

[44]  Abhyuday Mandal,et al.  Algorithmic Searches for Optimal Designs , 2015 .

[45]  D. Mcneil,et al.  The distribution by age of the frequency of first marriage in a female cohort. , 1972, Journal of the American Statistical Association.

[46]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1986, Math. Program..

[47]  R. D. Cook,et al.  A Comparison of Algorithms for Constructing Exact D-Optimal Designs , 1980 .

[48]  Laird Ak.  Dynamics of Tumour Growth , 1964 .

[49]  F. Chang,et al.  EXACT A-OPTIMAL DESIGNS FOR QUADRATIC REGRESSION , 1998 .

[50]  Mercedes Esteban-Bravo,et al.  Exact optimal experimental designs with constraints , 2017, Stat. Comput..

[51]  Eligius M. T. Hendrix,et al.  Replicationfree optimal designs in regression analysis , 1996 .

[52]  P. Goos,et al.  An integer linear programing approach to find trend-robust run orders of experimental designs , 2019, Journal of Quality Technology.

[53]  Peter Goos,et al.  D -optimal response surface designs in the presence of random block effects , 2001 .

[54]  R. K. Meyer,et al.  The Coordinate-Exchange Algorithm for Constructing Exact Optimal Experimental Designs , 1995 .

[55]  Anthony C. Atkinson,et al.  Optimum Experimental Designs, with SAS , 2007 .

[56]  Peter Goos,et al.  Blocking Orthogonal Designs With Mixed Integer Linear Programming , 2015, Technometrics.

[57]  Eligius M. T. Hendrix,et al.  Global Optimization Problems in Optimal Design of Experiments in Regression Models , 2000, J. Glob. Optim..

[58]  A. Donev,et al.  Crossover designs with correlated observations. , 1998, Journal of biopharmaceutical statistics.

[59]  J. Kiefer.  General Equivalence Theory for Optimum Designs (Approximate Theory) , 1974 .

[60]  H. Chernoff.  Locally Optimal Designs for Estimating Parameters , 1953 .

[61]  L. Imhof,et al.  Exact $$D$$-optimal designs for first-order trigonometric regression models on a partial circle , 2013 .