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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 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.
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.
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.
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.
Network Spot Checking Games: Theory and Application to Toll Enforcing in Transportation Networks
(2014)
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 edges 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 is
more relevant for this problem, but we show that this is NP-hard.
However, we give some bounds on the \emph{price of spite},
which measures how the
payoff of the inspector
degrades when committing to a Nash equilibrium.
Finally, we demonstrate the quality of these bounds for a real-world application,
namely the enforcement of a truck toll on German motorways.
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 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 $1$.
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.
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.
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 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.
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.
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.
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.
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.
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.
We propose an algorithm to approximate the distribution of the completion time (makespan)
and the tardiness costs of a project, when durations are lognormally distributed. This problem arises naturally for the optimization of surgery scheduling,
where it is very common to assume lognormal procedure times. We present an analogous of Clark's formulas to compute the moments of the maximum of a set of
lognormal variables. Then, we use moment matching formulas to approximate the earliest starting time of each activity of the project by a shifted lognormal variable.
This approach can be seen as a lognormal variant of a state-of-the-art method used for the statistical static timing analysis (SSTA) of digital circuits.
We carried out numerical experiments with instances based on real data from the application to surgery scheduling. We obtained very
promising results, especially for the approximation of the mean overtime in operating rooms,
for which our algorithm yields results of a similar quality to Monte-Carlo simulations
requiring an amount of computing time several orders of magnitude larger.
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.
Statistical methods to design computer experiments usually rely on a Gaussian process (GP) surrogate model, and typically aim at selecting design points (combinations of algorithmic and model parameters) that minimize the average prediction variance, or maximize the prediction accuracy for the hyperparameters of the GP surrogate.
In many applications, experiments have a tunable precision, in the sense that one software parameter controls the tradeoff between accuracy and computing time (e.g., mesh size in FEM simulations or number of Monte-Carlo samples).
We formulate the problem of allocating a budget of computing time over a finite set of candidate points for the goals mentioned above. This is a continuous optimization problem, which is moreover convex whenever the tradeoff function accuracy vs. computing time is concave.
On the other hand, using non-concave weight functions can help to identify sparse designs. In addition, using sparse kernel approximations drastically reduce the cost per iteration of the multiplicative weights updates that can be used to solve this problem.
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.
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.
The problem of allocating operating rooms (OR) to surgical cases is a challenging task,
involving both combinatorial aspects and uncertainty handling. In this article,
we formulate this problem as a job shop scheduling problem, in which the job durations follow a lognormal distribution.
We propose to use a cutting-plane approach to solve a robust version of this optimization problem. To this end,
we develop an algorithm based on fixed-point iterations to solve the subproblems that
identify worst-case scenarios and generate cut inequalities. The procedure is illustrated with numerical experiments based
on real data from a major hospital in Berlin.
Robust Allocation of Operating Rooms: a Cutting Plane Approach to handle Lognormal Case Durations
(2016)
The problem of allocating operating rooms (OR) to surgical cases is a challenging task, involving both combinatorial aspects
and uncertainty handling. We formulate this problem as a parallel machines scheduling problem, in which job durations follow a lognormal distribution,
and a fixed assignment of jobs to machines must be computed.
We propose a cutting-plane approach to solve the robust counterpart of this optimization problem.
To this end, we develop an algorithm based on fixed-point iterations that identifies worst-case scenarios and generates cut inequalities.
The main result of this article uses Hilbert's projective geometry to prove the convergence of this procedure under mild conditions.
We also propose two exact solution methods for a similar problem, but with a polyhedral uncertainty set, for which
only approximation approaches were known. Our model can
be extended to balance the load over several planning periods in a rolling horizon.
We present extensive numerical experiments for instances based on real data from a major hospital in Berlin. In particular, we find that:
(i) our approach performs well compared to a previous model that ignored the distribution of case durations;
(ii) compared to an alternative stochastic programming approach, robust optimization yields solutions that are more robust against uncertainty, at a small price in terms of average cost;
(iii) the \emph{longest expected processing time first} (LEPT) heuristic performs well and efficiently protects against extreme scenarios, but only if a good prediction model for the durations is available.
Finally, we draw a number of managerial implications from these observations.
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, the crews of the tours have to be scheduled. Thus, we developed a personalized crew rostering model. 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 paper focuses first on different requirements for the rostering and how they can be modeled in detail. The second focus is on a bicriterion analysis of the planning problem to find the balance between the control quality and the roster acceptance. On the one hand the tour planning is a profit maximization problem and on the other hand the rostering should be made in a employee friendly way. Finally, computational results on real-world instances show the practicability of our method.
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.
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 is
more relevant for this problem, but we show that this is NP-hard.
However, we give some bounds on the \emph{price of spite},
which measures how the
payoff of the inspector
degrades when committing to a Nash equilibrium.
Finally, we demonstrate the quality of these bounds for a real-world application,
namely the enforcement of a truck toll on German motorways.
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.
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 is
more relevant for this problem, but we show that this is NP-hard.
However, we give some bounds on the \emph{price of spite},
which measures how the
payoff of the inspector
degrades when committing to a Nash equilibrium.
Finally, we demonstrate the quality of these bounds for a real-world application,
namely the enforcement of a truck toll on German motorways.
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.
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 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.
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.
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.
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.
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.
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.
Robust Allocation of Operating Rooms: a Cutting Plane Approach to handle Lognormal Case Durations
(2018)
The problem of allocating operating rooms (OR) to surgical cases is a challenging task, involving both combinatorial aspects
and uncertainty handling. We formulate this problem as a parallel machines scheduling problem, in which job durations follow a lognormal distribution,
and a fixed assignment of jobs to machines must be computed.
We propose a cutting-plane approach to solve the robust counterpart of this optimization problem.
To this end, we develop an algorithm based on fixed-point iterations that identifies worst-case scenarios and generates cut inequalities.
The main result of this article uses Hilbert's projective geometry to prove the convergence of this procedure under mild conditions.
We also propose two exact solution methods for a similar problem, but with a polyhedral uncertainty set, for which
only approximation approaches were known. Our model can
be extended to balance the load over several planning periods in a rolling horizon.
We present extensive numerical experiments for instances based on real data from a major hospital in Berlin. In particular, we find that:
(i) our approach performs well compared to a previous model that ignored the distribution of case durations;
(ii) compared to an alternative stochastic programming approach, robust optimization yields solutions that are more robust against uncertainty, at a small price in terms of average cost;
(iii) the \emph{longest expected processing time first} (LEPT) heuristic performs well and efficiently protects against extreme scenarios, but only if a good prediction model for the durations is available.
Finally, we draw a number of managerial implications from these observations.
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.
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.
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.
We consider the stochastic scheduling problem of minimizing the expected makespan on m parallel identical machines. While the (adaptive) list scheduling policy achieves an approximation ratio of 2, any (non-adaptive) fixed assignment policy has performance guarantee Ω(logm/loglogm). Although the performance of the latter class of policies are worse, there are applications in which non-adaptive policies are desired. In this work, we introduce the two classes of δ-delay and τ-shift policies whose degree of adaptivity can be controlled by a parameter. We present a policy - belonging to both classes - which is an O(loglogm)-approximation for reasonably bounded parameters. In other words, an exponential improvement on the performance of any fixed assignment policy can be achieved when allowing a small degree of adaptivity. Moreover, we provide a matching lower bound for any δ-delay and τ-shift policy when both parameters, respectively, are in the order of the expected makespan of an optimal non-anticipatory policy.
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.
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 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.