Using mathematical programming to compute singular multivariate normal probabilities
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This paper describes two new, mathematical programming-based approaches for evaluating general, one-and two-sided, p-variate normal probabilities where the variance-covariance matrix (of arbitrary structure) is singular with rank r(r < p), and r and p can be of unlimited dimensions. In both cases, principal components are used to transform the original, ill-defined, p-dimensional integral into a well-defined, r-dimensional integral over a convex polyhedron. The first algorithm that is presented uses linear programming coupled with a Gauss-Legendre quadrature scheme to compute this integral, while the second algorithm uses multi-parametric programming techniques in order to significantly reduce the number of optimization problems that need to be solved. The application of the algorithms is demonstrated and aspects of computational performance are discussed through a number of examples, ranging from a practical problem that arises in chemical engineering to larger, numerical examples.