TY - CHAP A1 - Duarte, Belmiro P.M. A1 - Sagnol, Guillaume A1 - Oliveira, Nuno M.C. T1 - A Robust Minimax Semidefinite Programming Formulation for Optimal Design of Experiments for Model Parametrization T2 - 12th International Symposium on Process Systems Engineering and 25th European Symposium on Computer Aided Process Engineering N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1016/B978-0-444-63578-5.50146-8 VL - 37 SP - 905 EP - 910 ER - TY - JOUR A1 - Sagnol, Guillaume A1 - Harman, Radoslav T1 - Computing exact D-optimal designs by mixed integer second-order cone programming JF - The Annals of Statistics N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1214/15-AOS1339 VL - 43 IS - 5 SP - 2198 EP - 2224 ER - TY - GEN A1 - Harman, Radoslav A1 - Sagnol, Guillaume T1 - Computing D-optimal experimental designs for estimating treatment contrasts under the presence of a nuisance time trend N2 - 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. T3 - ZIB-Report - 15-02 KW - optimal design of experiments KW - nuisance trend KW - branch and bound Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-53640 SN - 1438-0064 ER - TY - CHAP A1 - Harman, Radoslav A1 - Sagnol, Guillaume ED - Steland, Ansgar ED - Rafajłowicz, Ewaryst ED - Szajowski, Krzysztof T1 - Computing D-optimal experimental designs for estimating treatment contrasts under the presence of a nuisance time trend T2 - Stochastic Models, Statistics and Their Applications N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1007/978-3-319-13881-7_10 VL - 122 SP - 83 EP - 91 PB - Springer ER - TY - GEN A1 - Duarte, Belmiro P.M. A1 - Sagnol, Guillaume A1 - Oliveira, Nuno M.C. T1 - A robust minimax Semidefinite Programming formulation for optimal design of experiments for model parametrization N2 - 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. T3 - ZIB-Report - 15-03 KW - Optimal design of experiments KW - Semidefinite Programming KW - Robust minmax designs Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-54626 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Sagnol, Guillaume A1 - Schwartz, Stephan T1 - An Extended Network Interdiction Problem for Optimal Toll Control N2 - We study an extension of the shortest path network interdiction problem and present a novel real-world application in this area. We consider the problem of determining optimal locations for toll control stations on the arcs of a transportation network. We handle the fact that drivers can avoid control stations on parallel secondary roads. The problem is formulated as a mixed integer program and solved using Benders decomposition. We present experimental results for the application of our models to German motorways. T3 - ZIB-Report - 15-32 KW - network interdiction Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-55405 SN - 1438-0064 ER - TY - CHAP A1 - Sagnol, Guillaume A1 - Harman, Radoslav ED - Steland, Ansgar ED - Rafajłowicz, Ewaryst ED - Szajowski, Krzysztof T1 - Optimal Designs for Steady-state Kalman filters T2 - Stochastic Models, Statistics and Their Applications N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1007/978-3-319-13881-7_17 VL - 122 SP - 149 EP - 157 PB - Springer ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Buwaya, Julia A1 - Sagnol, Guillaume A1 - Swarat, Elmar T1 - Network spot-checking games: Theory and application to toll enforcing in transportation networks JF - Networks N2 - We introduce the class of spot-checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. In an SC game, the pure strategies of network users correspond to paths in a graph, and the pure strategies of the inspectors are subset of arcs to be controlled. Although SC games are not zero-sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium (SSE) is more relevant for this problem and we give a mixed integer programming (MIP) formulation for this problem. We show that the computation of such an equilibrium is NP-hard. More generally, we prove that it is NP-hard to compute a SSE in a polymatrix game, even if the game is pairwise zero-sum. Then, we give some bounds on the price of spite, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we report computational experiments on instances constructed from real data, for an application to the enforcement of a truck toll in Germany. These numerical results show the efficiency of the proposed methods, as well as the quality of the bounds derived in this article. Y1 - 2015 U6 - https://doi.org/10.1002/net.21596 VL - 65 SP - 312 EP - 328 PB - Wiley Periodicals, Inc. ER -