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In this paper we present the problem of computing optimal tours of toll inspectors on German motorways. This problem is a special type of vehicle routing problem and builds up an integrated model, consisting of a tour planning and a duty rostering part. The tours should guarantee a network-wide control whose intensity is proportional to given spatial and time dependent traffic distributions. We model this using a space-time network and formulate the associated optimization problem by an integer program (IP). Since sequential approaches fail, we integrated the assignment of crews to the tours in our model. In this process all duties of a crew member must fit in a feasible roster. It is modeled as a Multi-Commodity Flow Problem in a directed acyclic graph, where specific paths correspond to feasible rosters for one month. We present computational results in a case-study on a German subnetwork which documents the practicability of our approach.
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.
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 propose a game theoretic model for the spatial distribution of inspectors on a transportation network. The problem is to spread out the controls so as to enforce the payment of a transit toll. We formulate a linear program to find the control distribution which maximizes the expected toll revenue, and a mixed integer program for the problem of minimizing the number of evaders. Furthermore, we show that the problem of finding an optimal mixed strategy for a coalition of $N$ inspectors can be solved efficiently by a column generation procedure. Finally, we give experimental results from an application to the truck toll on German motorways.
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.
This paper proposes the first model for toll enforcement optimization on German motorways. The enforcement is done by mobile control teams and our goal is to produce a schedule achieving network-wide control, proportional to spatial and time-dependent traffic distributions. Our model consists of two parts. The first plans control tours using a vehicle routing approach with profits and some side constraints. The second plans feasible rosters for the control teams. Both problems can be modeled as Multi-Commodity Flow Problems. Adding additional coupling constraints produces a large-scale integrated integer programming formulation. We show that this model can be solved to optimality for real world instances associated with a control area in East Germany.
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.
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.
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.
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 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.
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.
Network spot-checking games: Theory and application to toll enforcing in transportation networks
(2015)
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.
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 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 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 present the problem of planning mobile tours of inspectors on German motorways to enforce the payment of the toll for heavy good trucks. This is a special type of vehicle routing problem with the objective to conduct as good inspections as possible on the complete network. In addition, we developed a personalized crew rostering model, to schedule the crews of the tours. The planning of daily tours and the rostering are combined in a novel integrated approach and formulated as a complex and large scale Integer Program. The main focus of this paper extends our previous publications on how different requirements for the rostering can be modeled in detail. The second focus is on a bi-criteria analysis of the planning problem to find the balance between the control quality and the roster acceptance. Finally, computational results on real-world instances show the practicability of our method and how different input parameters influence the problem complexity.
We present a game-theoretic approach to optimize the strategies of toll enforcement on a motorway network. In contrast to previous approaches, we consider a network with an arbitrary topology, and we handle the fact that users may choose their Origin-Destination path; in particular they may take a detour to avoid sections with a high control rate. We show that a Nash equilibrium can be computed with an LP (although the game is not zero-sum), and we give a MIP for the computation of a Stackelberg equilibrium. Experimental results based on an application to the enforcement of a truck toll on German motorways are presented.