https://leeds-faculty.colorado.edu/glover/TS%20-%20Future%20Paths%20for%20Integer%20Programming.pdf

Future paths for integer programming and links to artificial intelligence

We consider least-squares problems where the coefficient matrices A,b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.

/pdf/robust-solutions-to-least-squares-problems-with-uncertain-4qeg33823y.pdf

Robust Solutions to Least-Squares Problems with Uncertain Data

Ecologists have long sought a way to explain how the remarkable biodiversity observed in nature is maintained. On the one hand, simple models of interacting competitors cannot produce the stable persistence of very large ecological communities. On the other hand, neutral models, in which species do not interact and diversity is maintained by immigration and speciation, yield unrealistically small fluctuations in population abundance, and a strong positive correlation between a species' abundance and its age, contrary to empirical evidence. Models allowing for the robust persistence of large communities of interacting competitors are lacking. Here we show that very diverse communities could persist thanks to the stabilizing role of higher-order interactions, in which the presence of a species influences the interaction between other species. Although higher-order interactions have been studied for decades, their role in shaping ecological communities is still unclear. The inclusion of higher-order interactions in competitive network models stabilizes dynamics, making species coexistence robust to the perturbation of both population abundance and parameter values. We show that higher-order interactions have strong effects in models of closed ecological communities, as well as of open communities in which new species are constantly introduced. In our framework, higher-order interactions are completely defined by pairwise interactions, facilitating empirical parameterization and validation of our models.

Higher-order interactions stabilize dynamics in competitive network models

Structural balance theory affirms that signed social networks (i.e., graphs whose signed edges represent friendly/hostile interactions among individuals) tend to be organized so as to avoid conflictual situations, corresponding to cycles of negative parity. Using an algorithm for ground-state calculation in large-scale Ising spin glasses, in this paper we compute the global level of balance of very large online social networks and verify that currently available networks are indeed extremely balanced. This property is explainable in terms of the high degree of skewness of the sign distributions on the nodes of the graph. In particular, individuals linked by a large majority of negative edges create mostly “apparent disorder,” rather than true “frustration.”

/pdf/computing-global-structural-balance-in-large-scale-signed-3wer45llyy.pdf

Computing global structural balance in large-scale signed social networks

Measuring supply chain resilience using a deterministic modeling approach

A detailed account of various determinantal formulas is presented in a graph-theoretic form involving paths and cycles in the digraph of the matrix. For cases in which the digraph has special local properties, for example, a cutpoint or a bridge, particular formulas are given that are more efficient for computing the determinant than simply using the matrix representation. Applications are also given to characteristic deter- minants, general minors, and cofactors.

/pdf/matrices-digraphs-and-determinants-1639p9z6zw.pdf

Matrices, digraphs, and determinants

The symposium covered structural modeling, qualitative stability analysis, qualitative determinancy and extension, embedded structures, and query systems. Abstracts of individual items from the symposium were prepared separately for the data base. (GHT)

Computer-assisted analysis and model simplification

Algebraic Multiplicity of the Eigenvalues of a Tournament Matrix

Combinatorially symmetric matrices

If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n − 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n∉ {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.

John S. Maybee

Papers

Matrices, digraphs, and determinants

Computer-assisted analysis and model simplification

Algebraic Multiplicity of the Eigenvalues of a Tournament Matrix

Combinatorially symmetric matrices

Tournament matrices and their generalizations, I.