# Optimization and Flow-Invariance via High Order Tangent Cones

Abstract (Summary)

The goals of this dissertation are: 1) to present some results on the flow-invariance of a closed set Sof a Banach space with respect to a differential equation, and to discuss optimization problems on S, as well; 2) to point out their unifying effect in the theory of differential equations and optimization. For the following optimization problem, one establishes necessary conditions of extremum in terms of the high order tangential directions to the constraint set at the extremum point: F(x0)=Local Minimum F(x), subject to x ? S, where X is a normed space, F: X? IR is a function of class Cpin a neighborhood of x0 ? S ? X, S?Ø, p? 1. It is analyzed in detail the case when S is the kernel D G of a function G : X?IRm, m? 1. To this aim, one describes the high order tangent cones to the set DGat x ? DG, and then derives some sufficient conditions for the optimality of F on DG. The characterizations of the high order tangent cones are also used to obtain some necessary and sufficient conditions for the flow-invariance of a subset DG= {x ? X; G (x)= 0}, of a Banach space X with respect to the differential equation u(n)(t) = F (u(t)),t ? 0, where G : U? IRm, m? 1, is a n-times Fréchet differentiable mapping on an open subset U of X, n ? 3, and F : U ? X is locally Lipschitz. The examples discussed illustrate some applications of the results presented.
Bibliographical Information:

Advisor:

School:Ohio University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:necessary conditions and sufficient of optimality fréchet differentiality constrained optimalization problems bouligand s tempest core high order tangent vectors flow invariance problem

ISBN:

Date of Publication:01/01/2005