Minimal Homotopies

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.

Robust Feasibility

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.