ZIB-OptDes
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A design point is inessential when it does not contribute to an optimal design, and can therefore be safely discarded from the design space. We derive three inequalities for the detection of such inessential points in c-optimal design: the first two are direct consequences of the equivalence theorem for c-optimality; the third one is derived from a second-order cone programming formulation of c-optimal design. Elimination rules for A-optimal design are obtained as a byproduct. When implemented within an optimization algorithm, each inequality gives a screening test that may provide a substantial acceleration by reducing the size of the problem online. Several examples are presented with a multiplicative algorithm to illustrate the effectiveness of the approach.
Model-based optimal designs of experiments (M-bODE) for nonlinear models are typically hard to compute. The literature on the computation of M-bODE for nonlinear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Second Order Programming (MISOCP) formulations to find, respectively, approximate and exact A- and D-optimal designs for 2đ factorial experiments for Generalized Linear Models (GLMs). First, locally optimal (approximate and exact) designs for GLMs are addressed using the formulation of Sagnol (J Stat Plan Inference 141(5):1684â1708, 2011). Next, we consider the scenario where the parameters are uncertain, and new formulations are proposed to find Bayesian optimal designs using the A- and log det D-optimality criteria. A quasi Monte-Carlo sampling procedure based on the Hammersley sequence is used for computing the expectation in the parametric region of interest. We demonstrate the application of the algorithm with the logistic, probit and complementary logâlog models and consider full and fractional factorial designs.
We show that the A-optimal design optimization problem over m design points in R^n is equivalent to minimizing a quadratic function plus a group lasso sparsity inducing term over n x m real matrices. This observation allows to describe several new algorithms for A-optimal design based on splitting and block coordinate decomposition. These techniques are well known and proved powerful to treat large scale problems in machine learning and signal processing communities. The proposed algorithms come with rigorous convergence guarantees and convergence rate estimate stemming from the optimization literature. Performances are illustrated on synthetic benchmarks and compared to existing methods for solving the optimal design problem.
We propose (Mixed Integer) Second Order Cone Programming formulations to find approximate and exact $D-$optimal designs for $2^k$
factorial experiments for Generalized Linear Models (GLMs). Locally optimal designs are addressed with Second Order Cone Programming
(SOCP) and Mixed Integer Second Order Cone Programming (MISOCP) formulations.
The formulations are extended for scenarios of parametric uncertainty employing the Bayesian framework for
\emph{log det} $D-$optimality criterion. A quasi Monte-Carlo sampling procedure based
on the Hammersley sequence is used for integrating the optimality criterion in the parametric region. The problems are solved in \texttt{GAMS}
environment using \texttt{CPLEX} solver. We demonstrate the application of the algorithm with the logistic, probit and complementary log-log models
and consider full and fractional factorial designs.
An algorithm based on a delayed constraint generation method for solving semi-infinite programs
for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax
optimization problem is solved using a semidefinite programming based approach that requires
the design space be discretized. A nonlinear programming solver is then used to solve the inner program
to determine the combination of the parameters that yields the worst-case value of the design criterion.
The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter
Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it
(i) produces designs that compare well with minimax $D-$optimal designs obtained from semi-infinite programming method in the literature;
(ii) can be applied to semidefinite representable optimality criteria, that include the common A-, E-,G-, I- and D-optimality criteria;
(iii) can tackle design problems with arbitrary linear constraints on the weights; and
(iv) is fast and relatively easy to use.
Let the design of an experiment be represented by an $s-$dimensional vector $w$ of weights with nonnegative components. Let the quality of $w$ for the estimation of the parameters of the statistical model be measured by the criterion of $D-$optimality, defined as the $m$th root of the determinant of the information matrix $M(w)=\sum_{i=1}^s w_i A_i A_i^T$, where $A_i$,$i=1,\ldots,s$ are known matrices with $m$ rows.
In this paper, we show that the criterion of $D-$optimality is second-order cone representable. As a result, the method of second-order cone programming can be used to compute an approximate $D-$optimal design with any system of linear constraints on the vector of weights. More importantly, the proposed characterization allows us to compute an exact $D-$optimal design, which is possible thanks to high-quality branch-and-cut solvers specialized to solve mixed integer second-order cone programming problems. Our results extend to the case of the criterion of $D_K-$optimality, which measures the quality of $w$ for the estimation of a linear parameter subsystem defined by a full-rank coefficient matrix $K$.
We prove that some other widely used criteria are also second-order cone representable, for instance, the criteria of $A-$, $A_K$-, $G-$ and $I-$optimality.
We present several numerical examples demonstrating the efficiency and general applicability of the proposed method. We show that in many cases the mixed integer second-order cone programming approach allows us to find a provably optimal exact design, while the standard heuristics systematically miss the optimum.
Model-based optimal design of experiments (M-bODE) is a crucial step in model parametrization since it encloses a framework that maximizes the amount of information extracted from a battery of lab experiments. We address the design of M-bODE for dynamic models considering a continuous representation of the design. We use Semidefinite Programming (SDP) to derive robust minmax formulations for nonlinear models, and extend the formulations to other criteria. The approaches are demonstrated for a CSTR where a two-step reaction occurs.
Model-based optimal design of experiments (M-bODE) is a crucial step in model parametrization since it encloses a framework
that maximizes the amount of information extracted from a battery of lab experiments.
We address the design of M-bODE for dynamic models considering a continuous representation of the design.
We use Semidefinite Programming (SDP) to derive robust minmax formulations for nonlinear models,
and extend the formulations to other criteria. The approaches are demonstrated for a CSTR where a two-step reaction occurs.
We consider a stationary discrete-time linear process that can be observed by a finite number of sensors. The experimental design for the observations consists of an allocation of available resources to these sensors. We formalize the problem of selecting a design that maximizes the information matrix of the steady-state of the Kalman filter, with respect to a standard optimality criterion, such as $D-$ or $A-$optimality. This problem generalizes the optimal experimental design problem for a linear regression model with a finite design space and uncorrelated errors. Finally, we show that under natural assumptions, a steady-state optimal design can be computed by semidefinite programming.
We prove a mathematical programming characterisation of approximate partial D-optimality under general linear constraints. We use this characterisation with a branch-and-bound method to compute a list of all exact D-optimal designs for estimating a pair of treatment contrasts in the presence of a nuisance time trend up to the size of 24 consecutive trials.
We prove a mathematical programming characterisation of approximate partial D-optimality under general linear constraints. We use this characterisation with a branch-and-bound method to compute a list of all exact D-optimal designs for estimating a pair of treatment contrasts in the presence of a nuisance time trend up to the size of 24 consecutive trials.
We consider a stationary discrete-time linear process that can be observed by a finite number of sensors.
The experimental design for the observations consists of an allocation of available resources to these sensors.
We formalize the problem of selecting a design that maximizes the information matrix of the steady-state of the Kalman filter,
with respect to a standard optimality criterion, such as $D-$ or $A-$optimality.
This problem generalizes the optimal experimental design problem for a linear regression model with a finite design space and uncorrelated errors.
Finally, we show that under natural assumptions, a steady-state optimal design can be computed by semidefinite programming.
In the past few years several applications of optimal experimental designs have emerged to optimize the measurements in communication networks. The optimal design problems arising from this kind of applications share three interesting properties: (i) measurements are only available at a small number of locations of the network; (ii) each monitor can simultaneously measure several quantities, which can be modeled by ``multiresponse experiments"; (iii) the observation matrices depend on the topology of the network. In this paper, we give an overview of these experimental design problems and recall recent results for the computation of optimal designs by Second Order Cone Programming (SOCP). New results for the network-monitoring of a discrete time process are presented. In particular, we show that the optimal design problem for the monitoring of an AR1 process can be reduced to the standard form and we give experimental results.
We study a family of combinatorial optimization problems defined by a parameter $p\in[0,1]$, which involves spectral functions applied to positive semidefinite matrices, and has some application in the theory of optimal experimental design. This family of problems tends to a generalization of the classical maximum coverage problem as $p$ goes to $0$, and to a trivial instance of the knapsack problem as $p$ goes to $1$. In this article, we establish a matrix inequality which shows that the objective function is submodular for all $p\in[0,1]$, from which it follows that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors. We prove an inequality which generalizes a classical result from the theory of optimal designs, and allows us to give a rounding procedure with an approximation factor which tends to $1$ as $p$ goes to $1$.
We present a new semidefinite representation for the trace of
a real function f applied to symmetric matrices, when a
semidefinite representation of the convex function f is known. Our construction
is intuitive, and yields a representation that is more compact than the previously known one.
We also show with the help of matrix geometric means and the Riemannian metric of the set of positive definite matrices
that for a rational number p in the interval (0,1],
the matrix X raised to the exponent p is the largest element
of a set represented by linear matrix inequalities.
We give numerical results for a problem inspired from the theory
of experimental designs, which show that the new semidefinite programming formulation
yields a speed-up factor in the order of 10.
We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP. The optimization problems of this class are semidefinite packing problems, which are the SDP analogs to vector packing problems. Of particular interest is the case in which our result guarantees the existence of a solution of rank one: we show that the computation of this solution actually reduces to a Second Order Cone Program (SOCP). We point out an application in statistics, in the optimal design of experiments.
Let the design of an
experiment be represented by an $s$-dimensional vector
$\vec{w}$ of weights with non-negative components. Let the quality of
$\vec{w}$ for the estimation of the parameters of the statistical model
be measured by the criterion of $D$-optimality defined as the $m$-th root
of the determinant of the information matrix $M(\vec{w})=\sum_{i=1}^s
w_iA_iA_i^T$, where $A_i$, $i=1,...,s$, are known matrices with $m$ rows.
In the paper, we show that the criterion of $D$-optimality is second-order
cone representable. As a result, the method of second order cone
programming can be used to compute an approximate $D$-optimal design with
any system of linear constraints on the vector of weights. More
importantly, the proposed characterization allows us to compute an
\emph{exact} $D$-optimal design, which is possible thanks to high-quality
branch-and-cut solvers specialized to solve mixed integer second order cone
problems.
We prove that some other widely used criteria are also second order cone
representable, for instance the criteria of $A$-, and $G$-optimality, as
well as the criteria of $D_K$- and $A_K$-optimality, which are extensions
of $D$-, and $A$-optimality used in the case when only a specific system of
linear combinations of parameters is of interest.
We present several numerical examples demonstrating the efficiency and
universality of the proposed method. We show that in many cases the mixed
integer second order cone programming approach allows us to find a provably
optimal exact design, while the standard heuristics systematically miss the
optimum.
PICOS is a user friendly interface
to several conic and integer programming solvers,
very much like YALMIP
under MATLAB.
The main motivation for PICOS is to have the possibility to
enter an optimization problem as a high level model,
and to be able to solve it with several different solvers.
Multidimensional and matrix variables are handled in a natural fashion,
which makes it painless to formulate a SDP or a SOCP.
This is very useful for educational purposes,
and to quickly implement some models and
test their validity on simple examples.
Furthermore, with PICOS you can take advantage of the
python programming language to read and write data,
construct a list of constraints by using python list comprehensions,
take slices of multidimensional variables, etc.
In the past few years several applications of optimal
experimental designs have emerged to optimize the measurements
in communication networks. The optimal design problems arising from
this kind of applications share three interesting properties:
(i) measurements are only available at a small number of locations of the network;
(ii) each monitor can simultaneously measure several quantities, which
can be modeled by ``multiresponse experiments";
(iii) the observation matrices depend on the topology of the network.
In this paper, we give an overview of these experimental design
problems and recall recent results for the computation of optimal
designs by Second Order Cone Programming (SOCP). New results for the
network-monitoring of a discrete time process are presented. In particular, we show
that the optimal design problem for the monitoring of an AR1 process can be reduced
to the standard form and we give experimental results.
We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite
matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP.
The optimization problems of this class are semidefinite packing problems,
which are the SDP analogs to vector packing problems.
Of particular interest is the case in which our result guarantees the existence of a solution
of rank one: we show that the computation of this solution actually reduces to a
Second Order Cone Program (SOCP).
We point out an application in statistics, in the optimal design of experiments.