Devadatta M. Kulkarni

Hierarchical Overlapping Coordination for Large-Scale Optimization by Decomposition

Decomposition of large engineering design problems into smaller design subproblems enhances robustness and speed of numerical solution algorithms. Design subproblems can be solved in parallel, using the optimization technique most suitable for the underlying subproblem. This also reflects the typical multidisciplinary nature of system design problems and allows better interpretation of results. Hierarchical overlapping coordination (HOC) simultaneously uses two or more problem decompositions, each of them associated with different partitions of the design variables and constraints. Coordination is achieved by the exchange of information between decompositions. We present the HOC algorithm and a sufficient condition for global convergence of the algorithm to the solution of a convex optimization problem. The convergence condition involves the rank of a matrix derived from the Jacobian of the constraints. Computational results obtained by applying the HOC algorithm to problems of various sizes are also presented.

Generalized matrix tree theorem for mixed graphs

In this article we provide a combinatorial description of an arbitrary minor of the Laplacian matrix (L) of a mixed graph (a graph with some oriented and some unoriented edges). This is a generalized Matrix Tree Theorem. We also characterize the non-singular substructures of a mixed graph. The sign attached to a nonsingular substructure is described in terms of labeling and the number of unoriented edges included in certain paths. Nonsingular substructures may be viewed as generalized matchings, because in the case of disjoint vertex sets corresponding to the rows and columns of a minor of L, our generalized Matrix Tree Theorem provides a signed count over matchings between those vertex sets. A mixed graph is called quasi bipartite if it does not contain a non singular cycle (a cycle containing an odd number of un-oriented edges). We give several characterizations of quasi-bipartite graphs.

EIGENVALUES OF TRIDIAGONAL PSEUDO-TOEPLITZ MATRICES

Abstract In this article we determine the eigenvalues of sequences of tridiagonal matrices that contain a Toeplitz matrix in the upper left block.

Algebraic graph theory without orientation

Abstract Let G be an undirected graph with vertices {v 1 ,v 2 ,…,>;v ⋎ } and edges {e1,e2, …,eϵ}. Let M be the ⋎ × ϵ matrix whose ijth entry is 1 if ej is a link incident with vi, 2 if ej is a loop at vi, and 0 otherwise. The matrix obtained by orienting the edges of a loopless graph G (i.e., changing one of the 1s to a − 1 in each column of M) has been studied extensively in the literature. The purpose of this paper is to explore the substructures of G and the vector spaces associated with the matrix M without imposing such an orientation. We describe explicitly bases for the kernel and range of the linear transformation from Rϵ to R⋎ defined by M. Our main results are determinantal formulas, using the unoriented Laplacian matrix MMt, to count certain spanning substructures of G. These formulas may be viewed as generalizations of the matrix tree theorem. The point of view adopted in this paper also gives rise to a matroid structure on the edges of G analogous to the cycle matroid and its dual. In this setting, the analogue of a spanning forest can have components with one odd cycle, and the analogue of an edge cut has the property that its removal creates a new bipartite component.

On the minors of an incidence matrix and its Smith normal form

Abstract Consider the vertex-edge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of vertex-disjoint odd cycles (i.e., the odd tulgeity ) of the graph. The problem of determining this number is shown to be NP-hard. Turning to maximal minors, we determine the rank of the incidence matrix. This depends on the number of components of the graph containing no odd cycle. We then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix, as well as its Smith normal form. These results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs. Finally, we give a sufficient condition for a system of equations (whose coefficient matrix is an incidence matrix) to admit an integer solution.

On Hilbertian ideals

Abhyankar defined the index of a monomial in a matrix of indeterminates X to be the maximal size of any minor of X whose principal diagonal divides the given monomial. Using this concept, he characterized a free basis for general type of determinantal ideals formed by the minors coming from a saturated subset of X. In this paper, to a monomial in X of index p we associate a combinatorial object called a superskeleton of latitude p, which can loosely be described as a p-tuple of “almost nonintersecting paths” in a rectangular lattice of points. Using this map, we prove that the ideal generated by the p by p minors of a saturated set in X is hilbertian, i.e., the Hilbert polynomial of this ideal coincides with its Hilbert function for all nonnegative integers.

Convergence Criteria for Hierarchical Overlapping Coordination of Linearly Constrained Convex Design Problems

Decomposition of multidisciplinary engineering system design problems into smaller subproblems is desirable because it enhances robustness and understanding of the numerical results. Moreover, subproblems can be solved in parallel using the optimization technique most suitable for the underlying mathematical form of the subproblem. Hierarchical overlapping coordination (HOC) is an interesting strategy for solving decomposed problems. It simultaneously uses two or more design problem decompositions, each of them associated with different partitions of the design variables and constraints. Coordination is achieved by the exchange of information between decompositions. This article presents the HOC algorithm and several new sufficient conditions for convergence of the algorithm to the optimum in the case of convex problems with linear constraints. One of these equivalent conditions involves the rank of the constraint matrix that is computationally efficient to verify. Computational results obtained by applying the HOC algorithm to quadratic programming problems of various sizes are included for illustration.

Edge version of the matrix tree theorem for trees

We provide a combinatorial description of all the minors of the edge version of the Laplacian matrix of a mixed tree. The description involves the common SDRs for the forests obtained by deleting from the tree the edge sets corresponding to the row and column indices of the minor.

Hilbert Polynomial of a Certain Ladder-Determinantal Ideal

A ladder-shaped array is a subset of a rectangular array which looks like a Ferrers diagram corresponding to a partition of a positive integer. The ideals generated by the p-by-p minors of a ladder-type array of indeterminates in the corresponding polynomial ring have been shown to be hilbertian (i.e., their Hilbert functions coincide with Hilbert polynomials for all nonnegative integers) by Abhyankar and Kulkarni [3, p 53–76]. We exhibit here an explicit expression for the Hilbert polynomial of the ideal generated by the two-by-two minors of a ladder-type array of indeterminates in the corresponding polynomial ring. Counting the number of paths in the corresponding rectangular array having a fixed number of “turning points” above the path corresponding to the ladder is an essential ingredient of the combinatorial construction of the Hilbert polynomial. This gives a constructive proof of the hilbertianness of the ideal generated by the two-by-two minors of a ladder-type array of indeterminates.

Counting of paths and coefficients of the Hilbert polynomial of a determinantal ideal

Let X= [Xij]1⩽i⩽m1⩽i⩽n be an m × n of matrix of indeterminates over a field K. Abhyankar defines the index of a monomial in Xij to be the largest k such that the principal diagonal of some k × k minor of X divides the given monomial. Abhyankar has given a formula for counting the set of monomials in Xij of degree v of index at most p, satisfying a certain set of index conditions. This formula gives the Hilbert polynomial of a certain generalized determinantal ideal which can be viewed as a polynomial in v with rational coefficients. We develop a combinatorial map from this set of monomials to the set of p-tuples of nonintersecting paths in the m × n rectangular lattice of points. A path from (a, n) to (m, b) in a rectangular m × n array is obtained by moving either left or down at each point. The point where the path turns from down to left is called its node. Using the combinatorial map, we get formulae counting sets of p-tuples of nonintersecting paths having a fixed number of nodes in a rectangular lattice. This helps us to interpret ‘coefficients’ of Hilbert polynomials of generalized determinantal ideals combinatorially. This enables us to answer questions raised by Abhyankar in the monograph ‘Enumerative Combinatorics of Young Tableaux’.