/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

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

/pdf/transversal-matroids-and-hall-s-theorem-5a78j75i0s.pdf

Transversal matroids and Hall’s theorem

In this note we consider completions of n×n symmetric (0,−1)-matrices to symmetric alternating sign matrices by replacing certain 0s with +1s. In particular, we prove that any n×n symmetric (0,−1)-matrix that can be completed to an alternating sign matrix by replacing some 0s with +1s can be completed to a symmetric alternating sign matrix. Similarly, any n × n symmetric (0,+1)-matrix that can be completed to an alternating sign matrix by replacing some 0s with −1s can be completed to a symmetric alternating sign matrix.

/pdf/symmetric-alternating-sign-matrices-4ppv081d1v.pdf

Symmetric alternating sign matrices.

Combinatorics and Graphs

We consider the problem of determining the minimum and maximum of the sum of the elements of the square of a matrix for the following classes of matrices:the (0, l)-matrices of order n with a specified number of l's; the symmetric (0,1)-matrices of order n with zero trace and with a specified number of l's above the main diagonal (adjacency matrices of graphs with a specified number of edges); and the subset of the latter consisting of adjacency matrices of connected graphs.

Some extremal problems concerning the square of a (0,1)-matrix

We consider a class of matrices whose row and column sum vectors are majorized by given vectors b and c, and whose entries lie in the interval [0, 1]. This class generalizes the class of doubly stochastic matrices. We investigate the corresponding polytope Ω(b|c) of such matrices. Main results include a generalization of the Birkhoff–von Neumann theorem and a characterization of the faces, including edges, of Ω(b|c).

/pdf/an-extension-of-the-polytope-of-doubly-stochastic-matrices-l2dq8ahyvx.pdf

Richard A. Brualdi

Papers

Transversal matroids and Hall’s theorem

Symmetric alternating sign matrices.

Combinatorics and Graphs

Some extremal problems concerning the square of a (0,1)-matrix

An extension of the polytope of doubly stochastic matrices