Sequential detection of convexity from noisy function evaluations

Sequential detection of convexity from noisy function evaluations. Jian, N., Henderson, S. G., & Hunter, S. R. In Tolk, A., Diallo, S. D., Ryzhov, I. O., Yilmaz, L., Buckley, S., & Miller, J. A., editors, Proceedings of the 2014 Winter Simulation Conference, pages 3892–3903, Piscataway, NJ, 2014. Institute of Electrical and Electronics Engineers, Inc..

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Consider a function that can only be evaluated with noise. Given estimates of the function values from simulation on a finite set of points, we seek a procedure to detect convexity or non-convexity of the true function on those points. We review an existing frequentist hypothesis test, and introduce a sequential Bayesian test. Our Bayesian test applies for both independent sampling and sampling with common random numbers, with known or unknown sampling variance. In each iteration, we collect a set of samples and update a posterior distribution on the true function values, and use that as the prior belief in our next iteration. We then approximate the probability that the function is convex based on the posterior using Monte Carlo simulation.

@inproceedings{2014jiahenhunWSC,
	Year = {2014},
	Author = {N. Jian and S. G. Henderson and S. R. Hunter},
	Title = {Sequential detection of convexity from noisy function evaluations},
	Booktitle = {Proceedings of the 2014 Winter Simulation Conference},
	Editor = {A. Tolk and S. D. Diallo and I. O. Ryzhov and L. Yilmaz and S. Buckley and J. A. Miller},
	Publisher = {Institute of Electrical and Electronics Engineers, Inc.},
        Address = {Piscataway, NJ},
        Pages = {3892--3903},
        doi = {10.1109/WSC.2014.7020215},
        url_Paper = {http://web.ics.purdue.edu/~hunter63/PAPERS/2014jiahenhunWSC.pdf},
        abstract = {Consider a function that can only be evaluated with noise. Given estimates of the function values from simulation on a finite set of points, we seek a procedure to detect convexity or non-convexity of the true function on those points. We review an existing frequentist hypothesis test, and introduce a sequential Bayesian test. Our Bayesian test applies for both independent sampling and sampling with common random numbers, with known or unknown sampling variance. In each iteration, we collect a set of samples and update a posterior distribution on the true function values, and use that as the prior belief in our next iteration.  We then approximate the probability that the function is convex based on the posterior using Monte Carlo simulation.},
	keywords = {structure detection}}

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