We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids.Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.

Corner cut, Design ideal, Factorial design, Latin hypercube sampling, Linear aberration, State polytope,

5Computer simulators of real world processes are often computationally expensive 6 and require many inputs. The problem of the computational expense can be handled 7 using emulation technology; however, highly-multidimensional input spaces may re-8 quire more simulator runs to train and validate the emulator. We aim to reduce the 9 dimensionality of the problem by screening the simulator's inputs for non-linear effects 10 on the output rather than distinguishing between negligible and active effects. Our 11 proposed method is built upon the elementary effects method for screening (Morris, 12 1991) and utilises a threshold value to separate the inputs with linear and non-linear Variance-based sensitivity analyses methods (Saltelli et al., 2000) offer more precise results 39in terms of the percentage of variance explained by each input factor and their interactions. 40However, these methods require many more model runs and are employed at a later stage 41 of simulator analysis than screening algorithms. 42Our aim in this paper is to develop a method that identifies inputs with non-linear Consider a deterministic simulator Y (·) with k input variables and design region [0, 1] k . 87The simulator is assumed to be a smooth real valued function with a domain containing 88 the design region. Computation of elementary effects starts from a point x from which a 89 trajectory is constructed with k random moves of size ∆. One-at-a-time (OAT) moves are 90 performed along each single coordinate axis in turn to end at point x + ∆(e 1 + · · · + e k ). 91The elementary effect for the i-th input variable for the trajectory starting atwhere ∆ > 0 is fixed. Here i = 1, . . . , k indexes input factors and e i is the unit vector in 93 the direction of the i-th axis where e 0 is defined as 0. A total of k + 1 evaluations of Y (·) 94are performed, ending with effects EE 1 (x), . . . , EE k (x). Each EE i (x) is a measure of the 95 variation in the output with respect to a change in input i at point x. 96Consider R starting points x r , r = 1, . . . , R. From each point x r , we perform k OAT 97 moves and compute elementary effects EE i (x r ) for every input factor so that the total 98 number of runs used in the EE method is (k + 1) × R. The following sample moments are 99 computed for each input factor:The moment µ i is an average effect measure, and high values suggest dominant contribution of R between 10 and 50 is mentioned in recent literature (Campolongo et al., 2004(Campolongo et al., , 2007. 111A larger value of R will improve the quality of the estimations, but at the price of extra 112 simulator runs. 113The step size ∆ is selected in such a way that all the simulator runs lie in the input space where Z * is a stochastic process whose covariance structure 155 depends only on the variables with non-linear effects; that is, the x i with i ∈ {1, . . . , k} \ A. 156The residual process Z * is therefore placed in a lower dimensional space simplifying the 157 design and inference tasks. 158For our algorithm t...

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We examine the optimal design of measurements on queues with particular reference to the M/M/1 queue. Using the statistical theory of design of experiments, we calculate numerically the Fisher information matrix for an estimator of the arrival rate and the service rate to find optimal times to measure the queue when the number of measurements are limited for both interfering and non-interfering measurements.We prove that in the non-interfering case, the optimal design is equally spaced. For the interfering case, optimal designs are not necessarily equally spaced. We compute optimal designs for a variety of queuing situations and give results obtained under the D-and D s -optimality criteria.