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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.
We present an extension of Taylor's Theorem for the piecewise polynomial expansion of non-smooth evaluation procedures involving absolute value operations. Evaluation procedures are computer programs of mathematical functions in closed form expression and allow a different treatment of smooth operations or calls to the absolute value function. The well known classical Theorem of Taylor defines polynomial approximations of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential- or differential-algebraic equations, for the construction of solvers for continuous non-linear optimization of finite dimensional objective functions and for root solving of non-linear systems of equations. The long term goal is the stabilization and acceleration of already known methods and the derivation of new methods by incorporating piecewise polynomial Taylor expansions. The herein provided proof of the higher order approximation quality of the new generalized expansions is constructive and allows efficiently designed algorithms for the execution and computation of the piecewise polynomial expansions. As a demonstration towards the ultimate goal we will derive a prototype of a {\$}{\$}k{\$}{\$}k-step method on the basis of polynomial interpolation and the proposed generalized expansions.
Urban transportation systems are subject to a high level of variation and fluctuation in demand over the day. When this variation and fluctuation are observed in both time and space, it is crucial to develop line plans that are responsive to demand. A multi-period line planning approach that considers a changing demand during the planning horizon is proposed. If such systems are also subject to limitations of resources, a dynamic transfer of resources from one line to another throughout the planning horizon should also be considered. A mathematical modelling framework is developed to solve the line planning problem with a cost-oriented approach considering transfer of resources during a finite length planning horizon of multiple periods. We use real-life public transportation network data for our computational results. We analyze whether or not multi-period solutions outperform single period solutions in terms of feasibility and relevant costs. The importance of demand variation on multi-period solutions is investigated. We evaluate the impact of resource transfer constraints on the effectiveness of solutions. We also study the effect of period lengths along with the problem parameters that are significant for and sensitive to the optimality of solutions.
Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order $\bar d-1$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function $|...|$ apart from arithmetic operations and $\bar d$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of $\mathcal O({\bar d}^2)$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in $\bar d$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains.
Recently, Kronqvist et al. (J Global Optim 64(2):249–272, 2016) rediscovered the supporting hyperplane algorithm of Veinott (Oper Res 15(1):147–152, 1967) and demonstrated its computational benefits for solving convex mixed integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley’s cutting plane algorithm (J Soc Ind Appl Math 8(4):703–712, 1960) applied to a particular reformulation of the problem. As a result, we extend the applicability of the supporting hyperplane algorithm to convex problems represented by a class of general, not necessarily convex nor differentiable, functions.
Two essential ingredients of modern mixed-integer programming (MIP) solvers are diving heuristics that simulate a partial depth-first search in a branch-and-bound search tree and conflict analysis of infeasible subproblems to learn valid constraints. So far, these techniques have mostly been studied independently: primal heuristics under the aspect of finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. Here, we combine both concepts in two different ways. First, we develop a diving heuristic that targets the generation of valid conflict constraints from the Farkas dual. We show that in the primal this is equivalent to the optimistic strategy of diving towards the best bound with respect to the objective function. Secondly, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classical coefficient diving. The computational performance of both methods is evaluated using an implementation in the source-open MIP solver SCIP. Experiments are carried out on publicly available test sets including Miplib 2010 and Cor@l.
A Physarum-Inspired Algorithm for Minimum-Cost Relay Node Placement in Wireless Sensor Networks
(2020)
A Polyhedral Study of Event-Based Models for the Resource-Constrained Project Scheduling Problem
(2020)
We consider event-based Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP) that represent an alternative to the common time-indexed model (DDT) of Pritsker et al. (1969) for the case where the underlying time horizon is large or job processing times are subject to huge variations. In contrast to the time-indexed model, the size of event-based models does not depend on the time horizon. For two event-based formulations OOE and SEE of Koné et al. (2011) we present new valid inequalities that dominate the original formulation. Additionally, we introduce a new event-based model: the Interval Event-Based Model (IEE). We deduce linear transformations between all three models that yield the strict domination order IEE > SEE > OOE for their linear programming (LP) relaxations, meaning that IEE has the strongest linear relaxation among the event-based models. We further show that the popular DDT formulation can be retrieved from IEE by certain polyhedral operations, thus giving a unifying view on a complete branch of MIP formulations for the RCPSP. In addition, we analyze the computational performance of all presented models on test instances of the PSPLIB (Kolisch and Sprecher 1997).
As the natural gas market is moving towards short-term planning, accurate and robust short-term forecasts of the demand and supply of natural gas is of fundamental importance for a stable energy supply, a natural gas control schedule, and transport operation on a daily basis. We propose a hybrid forecast model, Functional AutoRegressive and Convolutional Neural Network model, based on state-of-the-art statistical modeling and artificial neural networks. We conduct short-term forecasting of the hourly natural gas flows of 92 distribution nodes in the German high-pressure gas pipeline network, showing that the proposed model provides nice and stable accuracy for different types of nodes. It outperforms all the alternative models, with an improved relative accuracy up to twofold for plant nodes and up to fourfold for municipal nodes. For the border nodes with rather flat gas flows, it has an accuracy that is comparable to the best performing alternative model.
We consider the problem of verifying linear properties of neural networks. Despite their success in many classification and prediction tasks, neural networks may return unexpected results for certain inputs. This is highly problematic with respect to the application of neural networks for safety-critical tasks, e.g. in autonomous driving. We provide an overview of algorithmic approaches that aim to provide formal guarantees on the behavior of neural networks. Moreover, we present new theoretical results with respect to the approximation of ReLU neural networks. On the other hand, we implement a solver for verification of ReLU neural networks which combines mixed integer programming (MIP) with specialized branching and approximation techniques. To evaluate its performance, we conduct an extensive computational study. For that we use test instances based on the ACAS Xu System and the MNIST handwritten digit data set. Our solver is publicly available and able to solve the verification problem for instances which do not have independent bounds for each input neuron.