TY - JOUR A1 - Sagnol, Guillaume A1 - Pauwels, Edouard T1 - An unexpected connection between Bayes A-optimal designs and the group lasso JF - Statistical Papers N2 - 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. Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-73059 VL - 60 IS - 2 SP - 215 EP - 234 ER - TY - CHAP A1 - Sagnol, Guillaume A1 - Schmidt genannt Waldschmidt, Daniel A1 - Tesch, Alexander T1 - The Price of Fixed Assignments in Stochastic Extensible Bin Packing T2 - WAOA 2018: Approximation and Online Algorithms N2 - We consider the stochastic extensible bin packing problem (SEBP) in which n items of stochastic size are packed into m bins of unit capacity. In contrast to the classical bin packing problem, the number of bins is fixed and they can be extended at extra cost. This problem plays an important role in stochastic environments such as in surgery scheduling: Patients must be assigned to operating rooms beforehand, such that the regular capacity is fully utilized while the amount of overtime is as small as possible. This paper focuses on essential ratios between different classes of policies: First, we consider the price of non-splittability, in which we compare the optimal non-anticipatory policy against the optimal fractional assignment policy. We show that this ratio has a tight upper bound of 2. Moreover, we develop an analysis of a fixed assignment variant of the LEPT rule yielding a tight approximation ratio of (1+e−1)≈1.368 under a reasonable assumption on the distributions of job durations. Furthermore, we prove that the price of fixed assignments, related to the benefit of adaptivity, which describes the loss when restricting to fixed assignment policies, is within the same factor. This shows that in some sense, LEPT is the best fixed assignment policy we can hope for. Y1 - 2018 U6 - https://doi.org/10.1007/978-3-030-04693-4_20 VL - 11312 SP - 327 EP - 347 ER - TY - GEN A1 - Duarte, Belmiro P.M. A1 - Sagnol, Guillaume A1 - Wong, Weng Kee T1 - An algorithm based on Semidefinite Programming for finding minimax optimal designs N2 - 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. T3 - ZIB-Report - 18-01 KW - Cutting plane algorithm KW - Design efficiency KW - Equivalence theorem KW - Model-based optimal design KW - Nonlinear programming Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-66249 SN - 1438-0064 ER - TY - GEN A1 - Sagnol, Guillaume A1 - Blanco, Marco A1 - Sauvage, Thibaut T1 - The Cone of Flow Matrices: Approximation Hierarchies and Applications N2 - Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone generated by the matrices $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, where $\vec{1}_P\in\RR^n$ is the incidence vector of the (s,t)-path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over $\mathcal{K}$. This cone is intractable: we prove that the membership problem associated to $\mathcal{K}$ is NP-complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial-time the decomposition of a matrix $X\in \operatorname{span} \mathcal{K}$ as a linear combination of some $\vec{1}_P\vec{1}_P^T$'s. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of~K. We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities. T3 - ZIB-Report - 17-32 KW - Flows in graphs KW - Approximation Hierarchies KW - Copositive Programming KW - Semidefinite Programming Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-64399 SN - 1438-0064 ER - TY - GEN A1 - Duarte, Belmiro P.M. A1 - Sagnol, Guillaume T1 - Approximate and exact D-optimal designs for $2^k$ factorial experiments for Generalized Linear Models via SOCP N2 - 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. T3 - ZIB-Report - 18-02 Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-66256 SN - 1438-0064 ER - TY - JOUR A1 - Duarte, Belmiro A1 - Sagnol, Guillaume T1 - Approximate and exact optimal designs for 2^k factorial experiments for generalized linear models via second order cone programming JF - Statistical Papers N2 - 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 2𝑘 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):1684–1708, 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 log–log models and consider full and fractional factorial designs. Y1 - 2020 U6 - https://doi.org/10.1007/s00362-018-01075-7 VL - 61 SP - 2737 EP - 2767 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Tesch, Alexander A1 - Sagnol, Guillaume T1 - Algorithmen unterstützen OP-Planung T2 - Management & Krankenhaus N2 - Mathematische Algorithmen können durch Vorhersage von Unsicherheiten optimierte OP-Pläne berechnen, sodass mehrere Zielkriterien wie Überstunden, Wartezeit und Ausfälle im OP minimiert werden. Y1 - 2019 IS - 12 SP - 20 PB - Wiley ER - TY - GEN A1 - Sagnol, Guillaume A1 - Schmidt genannt Waldschmidt, Daniel A1 - Tesch, Alexander T1 - The Price of Fixed Assignments in Stochastic Extensible Bin Packing N2 - We consider the stochastic extensible bin packing problem (SEBP) in which $n$ items of stochastic size are packed into $m$ bins of unit capacity. In contrast to the classical bin packing problem, bins can be extended at extra cost. This problem plays an important role in stochastic environments such as in surgery scheduling: Patients must be assigned to operating rooms beforehand, such that the regular capacity is fully utilized while the amount of overtime is as small as possible. This paper focuses on essential ratios between different classes of policies: First, we consider the price of non-splittability, in which we compare the optimal non-anticipatory policy against the optimal fractional assignment policy. We show that this ratio has a tight upper bound of $2$. Moreover, we develop an analysis of a fixed assignment variant of the LEPT rule yielding a tight approximation ratio of $1+1/e \approx 1.368$ under a reasonable assumption on the distributions of job durations. Furthermore, we prove that the price of fixed assignments, which describes the loss when restricting to fixed assignment policies, is within the same factor. This shows that in some sense, LEPT is the best fixed assignment policy we can hope for. T3 - ZIB-Report - 18-19 KW - Approximation Algorithms KW - Stochastic Scheduling KW - Extensible Bin Packing Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-68415 SN - 1438-0064 ER - TY - JOUR A1 - Sagnol, Guillaume A1 - Blanco, Marco A1 - Sauvage, Thibaut T1 - The Cone of Flow Matrices: Approximation Hierarchies and Applications JF - Networks N2 - Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone generated by the matrices $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, where $\vec{1}_P\in\RR^n$ is the incidence vector of the (s,t)-path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over $\mathcal{K}$. This cone is intractable: we prove that the membership problem associated to $\mathcal{K}$ is NP-complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial-time the decomposition of a matrix $X\in \operatorname{span} \mathcal{K}$ as a linear combination of some $\vec{1}_P\vec{1}_P^T$'s. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of~K. We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities. Y1 - 2018 U6 - https://doi.org/10.1002/net.21820 VL - 72 IS - 1 SP - 128 EP - 150 ER - TY - GEN A1 - Sagnol, Guillaume A1 - Blanco, Marco A1 - Sauvage, Thibaut T1 - Approximation Hierarchies for the cone of flow matrices N2 - Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where $1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$. Several combinatorial problems reduce to a linear optimization problem over $K$. This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a completely positive representation of $K$. We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities. T3 - ZIB-Report - 18-20 KW - Flows in graphs KW - Approximation hierarchies KW - Copositive programming Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-68424 SN - 1438-0064 ER -