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Institute
- Mathematical Optimization (26) (remove)
18-20
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.
18-19
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.
18-01
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.
18-02
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.
17-32
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.
16-33
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.
16-23
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.
16-18
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.
16-16
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.
15-32
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.