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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.
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 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 $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, 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.
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.
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.
PICOS is a user friendly interface
to several conic and integer programming solvers,
very much like YALMIP
under MATLAB.
The main motivation for PICOS is to have the possibility to
enter an optimization problem as a high level model,
and to be able to solve it with several different solvers.
Multidimensional and matrix variables are handled in a natural fashion,
which makes it painless to formulate a SDP or a SOCP.
This is very useful for educational purposes,
and to quickly implement some models and
test their validity on simple examples.
Furthermore, with PICOS you can take advantage of the
python programming language to read and write data,
construct a list of constraints by using python list comprehensions,
take slices of multidimensional variables, etc.
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 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 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.