We investigate several computational problems related to stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in R each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in the computation of some expected statistics of a S...

We formalize the notion of approximate GCD for univariate poly-nomials given with limited accuracy and then address the problem of its computation. Algebraic concepts are applied in order to provide a solid foundation for a numerical approach. We exhibit the limitations of the euclidean algorithm through experiments, show that existing methods only solve part of the problem and assert its worst...

By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that B ∪ R is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygo...

In this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case. This reduction simply consists of computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial. This approach turns out to be very...

An explicit description of the convex hull of solutions to the uncapacitated lot-sizing problem with backlogging, in its natural space of production, setup, inventory and backlogging variables, has been an open question for many years. In this paper, we identify valid inequalities that subsume all previously known valid inequalities for this problem. We show that these inequalities are enough t...

Fejér's principle is readily proved; if the zero ai of pn(z) lies exterior to the convex hull of E, if a is the point of the convex hull nearest ai, and if we set a{ =(a+ai)/2, then the polynomial qn(z) = (z — ce{ )pn(z)/(z — ái) is an underpolynomial of pn(z) on E, so pn(z) cannot minimize any monotonie norm on E. The object of the present note is to give what is essentially a generalization o...

We propose the moment cone relaxation for a class of polynomial optimization problems (POPs) to extend the results on the completely positive cone programming relaxation for the quadratic optimization (QOP) model by Arima, Kim and Kojima. The moment cone relaxation is constructed to take advantage of sparsity of the POPs, so that efficient numerical methods can be developed in the future. We es...

We propose the moment cone relaxation for a class of polynomial optimization problems (POPs) to extend the results on the completely positive cone programming relaxation for the quadratic optimization (QOP) model by Arima, Kim and Kojima. The moment cone relaxation is constructed to take advantage of sparsity of the POPs, so that efficient numerical methods can be developed in the future. We es...

A numerical optimization approach is presented to optimize passive broadband detection performance of hull arrays through the adjustment of array shading weights. The approach is developed for general hull arrays in low signal-to-noise ratio scenarios, and is shown to converge rapidly to optimal solutions that maximize the array's deflection coefficient. The beamformer is not redesigned in this...