On systems-theoretic problems with symmetric multilinear parameter dependence.

摘要

This dissertation concentrates on systems whose mathematical description involves parameters entering into various equations and set descriptions both multilinearly and symmetrically. The systems under consideration are described by a so-called symmetric pair (f,X). Much of the research herein is devoted to demonstrating how results from the literature can be improved if symmetry is brought into play. In particular, when a multilinear function f(x) is maximized or minimized on X via extreme point function evaluations, we see that symmetry often leads to a "drastic" reduction in computation versus that required for a generic multilinear function. For example, for a symmetric pair with f(x) being multilinear and X in Rn being a symmetric hypercube, finding the maximum or minimum of f(x) on X requires considering only a linear number of extreme points in n versus 2n for the non-symmetric case. For applications, we develop results exploiting symmetry in the areas of probability, optimization and finance.;Generalizing in later chapters, we define a new concept called multi-group symmetry where subgroups of variables may enter into f( x) a symmetric way which can be exploited. In this framework, we have m groups each of size N with n = mN and we provide theorems involving the maximization and minimization of multilinear functions on hyperrectangles and polytopes.;Finally, one highlight of this dissertation is a result for a class of resource allocation problems involving n suppliers, a single resource and the inventory carrying costs for m identical warehouses. We seek to minimize a separable, concave sum of symmetric functions over a polytope representing the constraints. For any instance of this problem, when finding a solution via extreme point function evaluation, we show that because of symmetry and Schur concavity, for cases where there are a small number of warehouses in comparison to suppliers, the number of extreme points one needs to check can be much less than the number of extreme points of the constraint polytope.