14 research outputs found
Robust Feasibility of Systems of Quadratic Equations Using Topological Degree Theory
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.Comment: Added new Lemma 3.1, Figure 2, and Table 1. Improved writing in a few
place
On Computability and Triviality of Well Groups
The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1.
Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set.
For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact.
For the second part, we find examples of maps f, f\u27 from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status
LIPIcs
The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status
Persistence of Zero Sets
We study robust properties of zero sets of continuous maps
. Formally, we analyze the family
of all zero sets of all continuous maps
closer to than in the max-norm. The fundamental geometric property
of is that all its zero sets lie outside of .
We claim that once the space is fixed, is \emph{fully} determined
by an element of a so-called cohomotopy group which---by a recent result---is
computable whenever the dimension of is at most . More explicitly,
the element is a homotopy class of a map from or into a sphere.
By considering all simultaneously, the pointed cohomotopy groups form a
persistence module---a structure leading to the persistence diagrams as in the
case of \emph{persistent homology} or \emph{well groups}. Eventually, we get a
descriptor of persistent robust properties of zero sets that has better
descriptive power (Theorem A) and better computability status (Theorem B) than
the established well diagrams. Moreover, if we endow every point of each zero
set with gradients of the perturbation, the robust description of the zero sets
by elements of cohomotopy groups is in some sense the best possible (Theorem
C)
On Computability and Triviality of Well Groups
The concept of well group in a special but important case captures
homological properties of the zero set of a continuous map on a
compact space K that are invariant with respect to perturbations of f. The
perturbations are arbitrary continuous maps within distance r from f
for a given r>0. The main drawback of the approach is that the computability of
well groups was shown only when dim K=n or n=1.
Our contribution to the theory of well groups is twofold: on the one hand we
improve on the computability issue, but on the other hand we present a range of
examples where the well groups are incomplete invariants, that is, fail to
capture certain important robust properties of the zero set.
For the first part, we identify a computable subgroup of the well group that
is obtained by cap product with the pullback of the orientation of R^n by f. In
other words, well groups can be algorithmically approximated from below. When f
is smooth and dim K<2n-2, our approximation of the (dim K-n)th well group is
exact.
For the second part, we find examples of maps with all well
groups isomorphic but whose perturbations have different zero sets. We discuss
on a possible replacement of the well groups of vector valued maps by an
invariant of a better descriptive power and computability status.Comment: 20 pages main paper including bibliography, followed by 22 pages of
Appendi
Computing simplicial representatives of homotopy group elements
A central problem of algebraic topology is to understand the homotopy groups
of a topological space . For the computational version of the
problem, it is well known that there is no algorithm to decide whether the
fundamental group of a given finite simplicial complex is
trivial. On the other hand, there are several algorithms that, given a finite
simplicial complex that is simply connected (i.e., with
trivial), compute the higher homotopy group for any given .
%The first such algorithm was given by Brown, and more recently, \v{C}adek et
al.
However, these algorithms come with a caveat: They compute the isomorphism
type of , as an \emph{abstract} finitely generated abelian
group given by generators and relations, but they work with very implicit
representations of the elements of . Converting elements of this
abstract group into explicit geometric maps from the -dimensional sphere
to has been one of the main unsolved problems in the emerging field
of computational homotopy theory.
Here we present an algorithm that, given a~simply connected space ,
computes and represents its elements as simplicial maps from a
suitable triangulation of the -sphere to . For fixed , the
algorithm runs in time exponential in , the number of simplices of
. Moreover, we prove that this is optimal: For every fixed , we
construct a family of simply connected spaces such that for any simplicial
map representing a generator of , the size of the triangulation of
on which the map is defined, is exponential in
Computing all maps into a sphere
Given topological spaces X and Y, a fundamental problem of algebraic topology
is understanding the structure of all continuous maps X -> Y . We consider a
computational version, where X, Y are given as finite simplicial complexes, and
the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve
this problem in the stable range, where for some d >= 2, we have dim X <= 2d -
2 and Y is (d - 1)-connected; in particular, Y can be the d-dimensional sphere
S^d. The algorithm combines classical tools and ideas from homotopy theory
(obstruction theory, Postnikov systems, and simplicial sets) with algorithmic
tools from effective algebraic topology (locally effective simplicial sets and
objects with effective homology). In contrast, [X,Y] is known to be
uncomputable for general X,Y, since for X = S^1 it includes a well known
undecidable problem: testing triviality of the fundamental group of Y. In
follow-up papers, the algorithm is shown to run in polynomial time for d fixed,
and extended to other problems, such as the extension problem, where we are
given a subspace A of X and a map A -> Y and ask whether it extends to a map X
-> Y, or computing the Z_2-index---everything in the stable range. Outside the
stable range, the extension problem is undecidable.Comment: 42 pages; a revised and substantially updated version (referring to
follow-up papers and results