This study considers the set of homtopies between homotopic continuous maps
from compact submanifolds of Rd into Rd+1 s.t. the closed neighborhood around
any point in the range can be defined as the intersection of bounded d-manifolds.
Namely homotopies between such maps that agree along their non-empty boundaries
and are homotopic relative to their boundaries. We further restrict our study
to such maps that individually and jointly have only “simple” crossings or intersection
sets that are d 1-manifolds.
The goal of this study is to develop a rigorous framework for describing the
properties a “minimal homotopy” within this set of homotopies would possess,
should one exist. Informally we can think of a homotopy between two such continuous
maps with equal boundary as minimal if it sweeps “minimal volume” in
comparison to all other valid homotopies of its kind. This study develops a number
of such structural properties including those concerning existence.
We consider the problem of measuring the margin of robust feasibility of solutions
to a system of nonlinear equations. We study the special case of a system of
quadratic equations, which shows up in many practical applications such as the
power grid and other infrastructure networks. This problem is a generalization of
quadratically constrained quadratic programming (QCQP), which is NP-Hard in
the general setting. We develop approaches based on topological degree theory
to estimate bounds on the robustness margin of such systems. Our methods use
tools from convex analysis and optimization theory to cast the problems of checking
the conditions for robust feasibility as a nonlinear optimization problem. We
then develop inner bound and outer bound procedures for this optimization problem,
which could be solved efficiently to derive lower and upper bounds, respectively,
for the margin of robust feasibility. We evaluate our approach numerically on standard
instances taken from the MATPOWER database of AC power flow equations
that describe the steady state of the power grid. The results demonstrate that our
approach can produce tight lower and upper bounds on the margin of robust feasibility
for such instances.
|Commitee:||Vixie, Kevin, Cooper, Kevin|
|School:||Washington State University|
|School Location:||United States -- Washington|
|Source:||DAI-B 81/9(E), Dissertation Abstracts International|
|Keywords:||Degree theory, Homotopy, Topology|
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