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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

1. Incidence matrices 2. Matrices and graphs 3. Matrices and digraphs 4. Matrices and bigraphs 5. Combinatorial matrix algebra 6. Existence theorems for combinatorially constrained matrices 7. Some special graphs 8. The permanent 9. Latin squares.

Combinatorial Matrix Theory

Preface. List of Contributors. An introduction to algebraic codes (V.S. Pless, W.C. Huffman, R.A. Brualdi). Coding constructions (V.S. Pless). Self-dual codes (E.M. Rains, N.J.A. Sloane). Bounds on the size of linear codes (A.E. Brouwer). An updated table of the best binary codes known (S. Litsyn). Universal bounds for codes and designs (V.I. Levenshtein). Complexity issues in coding theory (A. Barg). Covering radius (R.A. Brualdi, S. Litsyn, V.S. Pless). Quadratic residue codes and divisibility (H.N. Ward). Algebraic geometry codes (T. Hoholdt, J.H. van Lint, R. Pellikaan). Open problems on cyclic codes (P. Charpin). The algebraic theory of concolutional codes (R.J. McEliece). Author index. Subject index.

http://www.gbv.de/dms/ilmenau/toc/269730176.PDF

Handbook Of Coding Theory

Preface 1. Sign-solvability Bibliography 2. L-matrices Bibliography 3. Sign-solvability and digraphs Bibliography 4. S*-matrices Bibliography 5. Beyond S*-matrices Bibliography 6. SNS-matrices Bibliography 7. S2NS-matrices Bibliography 8. Extremal properties of L-matrices Bibliography 9. The inverse sign pattern graph Bibliography 10. Sign stability Bibliography 11. Related Topics Bibliography Master Bibliography Index.

Matrices of Sign-Solvable Linear Systems

A natural sequel to the author's previous book Combinatorial Matrix Theory written with H. J. Ryser, this is the first book devoted exclusively to existence questions, constructive algorithms, enumeration questions, and other properties concerning classes of matrices of combinatorial significance. Several classes of matrices are thoroughly developed including the classes of matrices of 0's and 1's with a specified number of 1's in each row and column (equivalently, bipartite graphs with a specified degree sequence), symmetric matrices in such classes (equivalently, graphs with a specified degree sequence), tournament matrices with a specified number of 1's in each row (equivalently, tournaments with a specified score sequence), nonnegative matrices with specified row and column sums, and doubly stochastic matrices. Most of this material is presented for the first time in book format and the chapter on doubly stochastic matrices provides the most complete development of the topic to date.

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Richard A. Brualdi

Papers

Combinatorial Matrix Theory

Handbook Of Coding Theory

Matrices of Sign-Solvable Linear Systems

Combinatorial Matrix Classes

Matrices of zeros and ones with fixed row and column sum vectors