By Charles Audet, Pierre Hansen, Brigitte Jaumard

We current a department and reduce set of rules that yields in finite time, a globally ☼-optimal answer (with admire to feasibility and optimality) of the nonconvex quadratically restricted quadratic programming challenge. the assumption is to estimate all quadratic phrases by means of successive linearizations inside a branching tree utilizing Reformulation-Linearization recommendations (RLT). to take action, 4 sessions of linearizations (cuts), looking on one to 3 parameters, are distinctive. for every type, we express find out how to decide on the easiest member with recognize to an exact criterion. The cuts brought at any node of the tree are legitimate within the complete tree, and never simply in the subtree rooted at that node. for you to increase the computational velocity, the constitution created at any node of the tree is versatile adequate for use at different nodes. Computational effects are mentioned that come with usual try difficulties taken from the literature. a few of these difficulties are solved for the 1st time with an explanation of worldwide optimality.

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Therefore, if the size of minimum triangulation is small, it can be found efficiently. 2 Interval Graphs An interval graph is any graph that is isomorphic to the intersection graph of a family of intervals on a real line; the family of intervals is called an interval representation or sometimes an interval realizer of the graph. Not every graph has an interval representation; consider, for example, a chordless cycle of length four. The “invention” of interval graphs is commonly attributed to the Hungarian mathematician Gyorgy Hajos who in 1957 posed the problem of characterizing this family of graphs.

Optimal alignments in linear space. Comput Appl Biosci 1988;4:11. 50. Needleman SB, Wunsch CD. A general method applicable to the search for similarity in the amino acid sequence of two proteins. J Mol Biol. 1970;48:443. 51. Notredame C, Higgins DG, Heringa J. T-Coffee: a novel method for fast and accurate multiple sequence alignment. J Mol Biol 2000;302:205. 52. Novichkov PS, Gelfand MS, Mironov AA. Gene recognition in eukaryotic DNA by comparison of genomic sequences. Bioinformatics 2001:17:1011.

The definition of the stability criterion is phrased as a property of a graph closely related to the partition intersection graph and called a character overlap graph. A character overlap graph for a set of taxa is a graph G = (V, E), where V is a set of characters, and (u, v) ∈ E if there exists a taxon T in the set such that both u and v are present in T . ” To motivate the concept of stability, consider a set of characters A, B, C, D, and a set of four taxa described respectively by character pairs: (A, B), (B, C), (C, D), and (D, A).

### A branch and cut algorithm for nonconvex quadratically constrained quadratic programming by Charles Audet, Pierre Hansen, Brigitte Jaumard

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