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Convex Analysisの二,三の進展について

In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.

Semidefinite programming

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

https://www.math.ucdavis.edu/~saito/data/graphlap/merris-graphlap-eigvals.pdf

Laplacian matrices of graphs: a survey

In a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents, expresses by a communication graph with nonnegative weights. The question we ask in this paper is the following: is it possible to achieve a form of agreement also in presence of antagonistic interactions, modeled as negative weights on the communication graph? The answer to this question is affirmative: on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign. Necessary and sufficient conditions are obtained to describe cases in which this is possible. These conditions have strong analogies with the theory of monotone systems. Linear and nonlinear Laplacian feedback designs are proposed.

Consensus Problems on Networks With Antagonistic Interactions

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

The nested-subset hypothesis of Patterson and Atmar states that species composition on islands with less species richness is a proper subset of those on islands with greater species richness The sum of species absences, referred to as gaps, was suggested as a metric for nestedness, and null models have been used to test whether or not island species exhibited nestedness Simberloff and Martin stated that finding examples of non-nested faunas was difficult We revisit previous analyses of nested faunas and introduce a new metric we call “discrepancy” which we recommend as a measure for nestedness We also recommend that the sample spaces conserve both row sums (number of species per site) and column sums (number of sites per species) derived from the incidence matrix We compare our results to previous analyses

Richard A. Brualdi

Papers

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

Nested species subsets, gaps, and discrepancy

Codes with a poset metric

The diagonal equivalence of a nonnegative matrix to a stochastic matrix

On the spectral radius of (0,1)-matrices