/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

Multicolored Trees in Complete Graphs

Exchange systems, matchings, and transversals

Combinatorial batch codes were defined by Paterson, Stinson, and
Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers each of which stores a subset of the items. It is required that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Only the case $t=1$ is considered here. An optimal combinatorial batch code is one in which the total storage required is a minimum. We establish an important connection between combinatorial batch codes and transversal matroids, and exploit this connection to characterize optimal combinatorial batch codes if $n=m+1$ and $n=m+2$.

Combinatorial batch codes and transversal matroids

Combinatorial Matrix Theory: Some Special Graphs

It was observed by Dulmage and Mendelsohn in their work on matrix reducibility that there is a one-to-one correspondence between bigraphs and digraphs determined by the utilization of the adjacency matrix. In this semiexpository paper we explore the interaction between this correspondence and a theory of matrix decomposability that is developed in several different articles. These results include: (a) a characterization of those bipartite graphs that can be labeled so that the resulting digraph is symmetric; (b) a criterion for the bigraph of a symmetric digraph to be connected; (c) a necessary and sufficient condition for a square binary matrix to be fully indecomposable in terms of its associated bigraph, and (d) matrix criteria for a digraph to be strongly, unilaterally, or weakly connected. We close with an unsolved extermal problem on the number of components of the bigraph of various orientations of a given graph. This leads to new amusing characterizations of trees and bigraphs.



Dedicated to the graph-theoretic partnership of Lloyd Dulmage and Nathan Mendelsohn.

Richard A. Brualdi

Papers

Multicolored Trees in Complete Graphs

Exchange systems, matchings, and transversals

Combinatorial batch codes and transversal matroids

Combinatorial Matrix Theory: Some Special Graphs

Bigraphs versus digraphs via matrices