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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

Algorithms for constructing (0,1)-matrices with prescribed row and column sum vectors

Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdiere number, which, for instance, characterises certain topological properties of the graph. This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.

The mutually beneficial relationship of graphs and matrices

Let $A$ be a $k \by n$ underdetermined matrix. The sparse basis problem for the row space $W$ of $A$ is to find a basis of $W$ with the fewest number of nonzeros. Suppose that all the entries of $A$ are nonzero, and that they are algebraically independent over the rational number field. Then every nonzero vector in $W$ has at least $n-k+1$ nonzero entries. Those vectors in $W$ with exactly $n-k+1$ nonzero entries are the elementary vectors of $W$. A simple combinatorial condition that is both necessary and sufficient for a set of $k$ elementary vectors of $W$ to form a basis of $W$ is presented here. A similar result holds for the null space of $A$ where the elementary vectors now have exactly $k+1$ nonzero entries. These results follow from a theorem about nonzero minors of order $m$ of the $(m-1)$st compound of an $m \by n$ matrix with algebraically independent entries, which is proved using multilinear algebra techniques. This combinatorial condition for linear independence is a first step towards the design of algorithms that compute sparse bases for the row and null space without imposing artificial structure constraints to ensure linear independence.

Richard A. Brualdi

Papers

Algorithms for constructing (0,1)-matrices with prescribed row and column sum vectors

The mutually beneficial relationship of graphs and matrices

The Sparse Basis Problem and Multilinear Algebra

Patterns of alternating sign matrices

Combinatorial verification of the elementary divisors of tensor products