/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

We introduce a class ofn×n matrices of 0’s and 1’s which can be regarded as generalizations of the classical Ferrers boards of rook theory. Assuming the matrices are fully indecomposable, we determine the minimum permanent and the minimum number of 1’s as a function ofn. We also characterize these matrices in terms of weighted, top-rooted trees.

Partial ferrers matrices

We study a class of perfect graphs which, because they generalize interval graphs, we call pseudo-interval graphs. Like interval graphs, their vertices correspond to intervals of a linearly ordered set, but a modified definition of intersection is used in order to determine edges. The complements of pseudo-interval graphs are comparability graphs but unlike interval graphs, pseudo-interval graphs are only weakly triangulated. We characterize trees and complements of trees which are pseudo-interval graphs. Finally we determine all minimal non-pseudointerval graphs on eight or fewer vertices whose complements are comparability graphs and which are weakly triangulated. © 1995 John Wiley & Sons; Inc.

Pseudo-interval graphs

where r denotes the maximal (positive) characteristic root of A. Here per (Al") denotes the permanent of Am. We assume that the reader is familiar with the terminology and results of the classical PerronFrobenius-Wielandt theory of nonnegative matrices, the requisite parts of which can be found in [1, Chapter XIII]. The purpose of this note is to prove the following extension of the result quoted above.

/pdf/on-the-permanent-and-maximal-characteristic-root-of-a-3q0vwioht7.pdf

Richard A. Brualdi

Papers

Partial ferrers matrices

Pseudo-interval graphs

On the permanent and maximal characteristic root of a nonnegative matrix

Prime Interchange Graphs of Classes of Matrices of Zeros and Ones

Preface to the 14th ILAS Conference Proceedings Shanghai 2007