A Class of Semidefinite Programs with rank-one solutions

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.Comment: 16 page

Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming

Elfving's Theorem is a major result in the theory of optimal experimental design, which gives a geometrical characterization of $c-$optimality. In this paper, we extend this theorem to the case of multiresponse experiments, and we show that when the number of experiments is finite, $c-,A-,T-$ and $D-$optimal design of multiresponse experiments can be computed by Second-Order Cone Programming (SOCP). Moreover, our SOCP approach can deal with design problems in which the variable is subject to several linear constraints. We give two proofs of this generalization of Elfving's theorem. One is based on Lagrangian dualization techniques and relies on the fact that the semidefinite programming (SDP) formulation of the multiresponse $c-$optimal design always has a solution which is a matrix of rank $1$. Therefore, the complexity of this problem fades. We also investigate a \emph{model robust} generalization of $c-$optimality, for which an Elfving-type theorem was established by Dette (1993). We show with the same Lagrangian approach that these model robust designs can be computed efficiently by minimizing a geometric mean under some norm constraints. Moreover, we show that the optimality conditions of this geometric programming problem yield an extension of Dette's theorem to the case of multiresponse experiments. When the number of unknown parameters is small, or when the number of linear functions of the parameters to be estimated is small, we show by numerical examples that our approach can be between 10 and 1000 times faster than the classic, state-of-the-art algorithms

Improved Analysis of two Algorithms for Min-Weighted Sum Bin Packing

We study the Min-Weighted Sum Bin Packing problem, a variant of the classical Bin Packing problem in which items have a weight, and each item induces a cost equal to its weight multiplied by the index of the bin in which it is packed. This is in fact equivalent to a batch scheduling problem that arises in many fields of applications such as appointment scheduling or warehouse logistics. We give improved lower and upper bounds on the approximation ratio of two simple algorithms for this problem. In particular, we show that the knapsack-batching algorithm, which iteratively solves knapsack problems over the set of remaining items to pack the maximal weight in the current bin, has an approximation ratio of at most 17/10

A Case Study on Optimizing Toll Enforcements on Motorways

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

Restricted Adaptivity in Stochastic Scheduling

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 ?((log m)/(log log m)). 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 ?-delay and ?-shift policies whose degree of adaptivity can be controlled by a parameter. We present a policy - belonging to both classes - which is an ?(log log m)-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 ?-delay and ?-shift policy when both parameters, respectively, are in the order of the expected makespan of an optimal non-anticipatory policy

Surgery Scheduling in Flexible Operating Rooms by using a Convex Surrogate Model of Second-Stage Costs

We study the elective surgery planning problem in a hospital with operation rooms shared by elective and emergency patients. This problem can be split in two distinct phases. First, a subset of patients to be operated in the next planning period has to be selected, and the selected patients have to be assigned to a block and a tentative starting time. Then, in the online phase of the problem, a policy decides how to insert the emergency patients in the schedule and may cancel planned surgeries. The overall goal is to minimize the expectation of a cost function representing the assignment of patient to blocks, case cancellations, overtime, waiting time and idle time. We model the offline problem by a two-stage stochastic program, and show that the second-stage costs can be replaced by a convex piecewise linear surrogate model that can be computed in a preprocessing step. This results in a mixed integer program which can be solved in a short amount of time, even for very large instances of the problem. We also describe a greedy policy for the online phase of the problem, and analyze the performance of our approach by comparing it to either heuristic methods or approaches relying on sampling average approximation (SAA) on a large set of benchmarking instances. Our simulations indicate that our approach can reduce the expected costs by as much as 20% compared to heuristic methods and is able to solve problems with $1000$ patients in about one minute, while SAA-approaches fail to obtain near-optimal solutions within 30 minutes, already for $100$ patients

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