14 research outputs found

    Robust Feasibility of Systems of Quadratic Equations Using Topological Degree Theory

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    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

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    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

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    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 &gt; 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 &lt; 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

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    We study robust properties of zero sets of continuous maps f:XRnf:X\to\mathbb{R}^n. Formally, we analyze the family Zr(f)={g1(0):gf<r}Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\} of all zero sets of all continuous maps gg closer to ff than rr in the max-norm. The fundamental geometric property of Zr(f)Z_r(f) is that all its zero sets lie outside of A:={x:f(x)r}A:=\{x:\,|f(x)|\ge r\}. We claim that once the space AA is fixed, Zr(f)Z_r(f) is \emph{fully} determined by an element of a so-called cohomotopy group which---by a recent result---is computable whenever the dimension of XX is at most 2n32n-3. More explicitly, the element is a homotopy class of a map from AA or X/AX/A into a sphere. By considering all r>0r>0 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

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    The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f:KRnf: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 LL_\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:KRnf,f': 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.Comment: 20 pages main paper including bibliography, followed by 22 pages of Appendi

    Computing simplicial representatives of homotopy group elements

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    A central problem of algebraic topology is to understand the homotopy groups πd(X)\pi_d(X) of a topological space XX. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group π1(X)\pi_1(X) of a given finite simplicial complex XX is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex XX that is simply connected (i.e., with π1(X)\pi_1(X) trivial), compute the higher homotopy group πd(X)\pi_d(X) for any given d2d\geq 2. %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 πd(X)\pi_d(X), d2d\geq 2 as an \emph{abstract} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of πd(X)\pi_d(X). Converting elements of this abstract group into explicit geometric maps from the dd-dimensional sphere SdS^d to XX 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 XX, computes πd(X)\pi_d(X) and represents its elements as simplicial maps from a suitable triangulation of the dd-sphere SdS^d to XX. For fixed dd, the algorithm runs in time exponential in size(X)size(X), the number of simplices of XX. Moreover, we prove that this is optimal: For every fixed d2d\geq 2, we construct a family of simply connected spaces XX such that for any simplicial map representing a generator of πd(X)\pi_d(X), the size of the triangulation of SdS^d on which the map is defined, is exponential in size(X)size(X)

    Computing all maps into a sphere

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    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
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