B. Hendrickson. The molecule problem: Exploiting structure in global optimization. SIAM J. Optim., 5:835-857, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....DG problem, we straightforwardly construct a constraint graph such that the nodes correspond directly to the atoms, and an edge between two nodes is added if the distance between the corresponding atoms is specified in the input. The unique realizability of such a graph is discussed in detail in [43] , where we find two necessary and one sufficient conditions for unique solution. For three dimensional structure computation, two necessary conditions are: The graph must be 4 connected. A graph is d connected if, after the removal of any (d 1) vertices and all incident edges, it remains ....

.... This method can be used to produce initial structures which may be subsequently refined by other techniques, such as those described in [22, 27, 38] While the above diagonalization method converts a sparse set of input distance bounds into a dense matrix problem, the approach presented in [43] maintains the sparsity structure throughout by exploiting the combinatorial properties of the induced constraint graph. The graph used in this case is the straightforward version: a point in the molecule maps directly to a node in the graph, and an input distance constraint between two points of ....

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B. Hendrickson, "The Molecule Problem: Exploiting Structure in Global Optimization", SIAM J. Optimization, vol. 5, no. 4, pp. 835-857, 1995.

....closely related and, thus, their associated completion problems can often be treated in an analogous manner. These matrix completion problems have many applications, e.g. to multidimensional scaling problems in statistics (cf. 28] to the molecule conformation problem in chemistry (cf. 10] [17]) to moment problems in analysis (cf. 5] 1.2. An excursion to semidefinite programming. The psd matrix completion problem is obviously an instance of the general semidefinite programming feasibility problem (F) Given integral n Theta n symmetric matrices Q 0 ; Q 1 ; Qm , determine ....

B. Hendrickson. The molecule problem: exploiting structure in global optimization. SIAM Journal on Optimization, 5:835--857, 1995.

....ij ) 2 : Hence, f( is zero precisely when the v i s provide a realization of the partial matrix A. This optimization problem is hard to solve (as it may have many local optimum solutions) Several algorithms have been proposed in the literature; see, in particular, CH88] GHR93] Ha91] [He90, He95], KTO93] MW97] PL97] They are based on general techniques for global optimization like tabu and pattern search [PL97] the continuation approach (which consists of transforming the original function f( into a smoother function having fewer local optimizers, MW96, MW97] or ....

.... and pattern search [PL97] the continuation approach (which consists of transforming the original function f( into a smoother function having fewer local optimizers, MW96, MW97] or divide and conqueer strategies aiming to break 6 the problem into a sequence of smaller or easier subproblems [CH88, He90, He95]. In [He90, He95] the basic step consist of finding principal submatrices having a unique realization, treating each of them separately and then trying to combine the solutions. Thus arises the problem of identifying principal submatrices having a unique realization, which turns out to be NP hard ....

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B. Hendrickson. The molecule problem: exploiting structure in global optimization. SIAM Journal on Optimization, 5:835--857, 1995.

.... problems (Crippen and Havel [30] or, from a graph theoretic point of view, 10 they are a class of NP complete graph embedding problems (Hendrickson [41] Saxe [79, 80] Mor e and Wu [64] Recent attempts to solve these problems on parallel high performance architectures are by (Hendrickson [42], Mor e and Wu [66] Byrd, Schnabel et al. [97] etc) A simple version of the distance geometry problem is to find a set of points to realize a given set of distances between some of the points. A more general version is to satisfy a given set of bounds on the distances. Mathematically, the ....

Bruce A. Hendrickson. The molecule problem: Exploiting structure in global optimization. SIAM J. Optimization, 5:835--857, 1995.

....between the points. There are surprisingly many applications for this problem, sometimes called the molecule problem. These applications include NMR data, determination of protein structure, surveying, satellite ranging, and molecular conformation; e.g. the survey [23] and the discussion in [50] and the related papers [49, 89, 114, 117, 44] We now consider the approximate EDMCP and follow the approach in [1] where the reader will nd all the proofs and details omitted here. Let A be a pre distance matrix and let H be an n n symmetric matrix with nonnegative elements (weights) ....

B. HENDRICKSON. The molecule problem: exploiting structure in global optimization. SIAM J. Optim., 5(4):835-857, 1995.

.... in general because it has been shown to be strongly NP complete in the one dimensional case [6, 13] and strongly NP hard in the higher dimension case [14] A large number of such methods for solving distance geometry problems have been proposed, such as Crippen and Havel[6] Havel[8] Hendrickson[9], Glunt, Hayden, Raydan[7] and Mor e and Wu[12, 13] The method we present in this paper is based on the stochastic perturbation global optimization approach[4, 2, 3, 5] with several new features. The purpose of this paper is to describe this approach and to demonstrate its capabilities on some ....

....space. Both the selection of small dimensional subproblems and some other important algo2 rithmic features are specific to the distance geometry problem. We experimented with our algorithm on Mor e and Wu s artificial problems [12] and on the protein fragment problems from Hendrickson[9]. For the artificial problems, even our first phase can find the exact solutions with great success. For the protein fragment problems, which are considerably more difficult, we have found the exact solutions solutions for problems with up to 377 atoms (1131 parameters) Another important issue ....

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B.A.Hendrickson, The Molecule Problem: Exploiting Structure in Global Optimization, SIAM J. Optimization, Vol. 5(1995), No.4, pp. 835-857.

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B. Hendrickson. The molecule problem: Exploiting structure in global optimization. SIAM J. Optim., 5:835-857, 1995.

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B. Hendrickson. The molecule problem: Exploiting structure in global optimization. SIAM J. on Optimization, 5:835--857, 1995.

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Hendrickson, B.A. (1995), The molecule problem: exploiting structure in global optimization, SIAM Journal on Optimization, 5:835-857. 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 900 1000 'table.gnuplot' using 1:2 0 0.5 1 1.5 2 2.5 3 0 100 200 300 400 500 600 700 800 900 1000 'table.gnuplot' using 1:4

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Hendrickson, B.A. (1995), The molecule problem: exploiting structure in global optimization, SIAM Journal on Optimization, 5:835-857.

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Hendrickson, B.A. (1995). The molecule problem: exploiting structure in global optimization. SIAM Journal on Optimization, 5:835--857.

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Bruce Hendrickson, "The molecule problem: Exploiting structure in global optimization," SIAM Journal on Optimization, vol. 5, no. 4, pp. 835--857, 1995.

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Bruce Hendrickson, "The molecule problem: Exploiting structure in global optimization," SIAM Journal on Optimization, vol. 5, no. 4, pp. 835--857, 1995.

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B. Hendrickson, The molecule problem: Exploiting structure in global optimization, SIAM J. Optim., 5 (1995), pp. 835--857.

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