Minimal Homotopies And Robust Feasibility Using Topological Degree Theory
Rapone, Benjamin Joseph
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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.