# Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming

## Abstract

Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (EDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (EDM). In this paper, we follow the successful approach in [20] and solve the EDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primal-dual interior-point algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Alfakih and H. Wolkowicz, “On the embeddability of weighted graphs in Euclidean spaces,” Technical Report CORR Report 98–12, University of Waterloo, 1998. Submitted-URL: ftp://orion.uwaterloo.ca/pub/henry/reports/embedEDM.ps.gz.Google Scholar
- 2.S. Al-Homidan, “Hybrid methods for optimization problems with positive semidefinite matrix constraints,” Ph.D. Thesis, University of Dundee, 1993.Google Scholar
- 3.S. Al-Homidan and R. Fletcher, “Hybrid methods for finding the nearest Euclidean distance matrix,” in Recent Advances in Nonsmooth Optimization, World Sci. Publishing River Edge, NJ, 1995, pp. 1–17.Google Scholar
- 4.F. Alizadeh, “Combinatorial optimization with interior point methods and semidefinite matrices,” Ph.D. Thesis, University of Minnesota, 1991.Google Scholar
- 5.F. Alizadeh, “Interior point methods in semidefinite programming with applications to combinatorial optimization,” SIAM Journal on Optimization, vol. 5, pp. 13–51, 1995.Google Scholar
- 6.F. Alizadeh, J.-P. Haeberly, M.V. Nayakkankuppam, and M.L. Overton, “Sdppack user's guide—version 0.8 beta,” Technical Report TR1997–734, Courant Institute of Mathematical Sciences, NYU, New York, March 1997.Google Scholar
- 7.M. Bakonyi and C.R. Johnson, “The Euclidean distance matrix completion problem,” SIAM J. Matrix Anal. Appl., vol. 16,no. 2, pp. 646–654, 1995.Google Scholar
- 8.R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press: New York, 1991.Google Scholar
- 9.G.M. Crippen and T.F. Havel, Distance Geometry and Molecular Conformation, Wiley: New York, 1988.Google Scholar
- 10.F. Critchley, “On certain linear mappings between inner-product and squared distance matrices.” Linear Algebra Appl., vol. 105, pp. 91–107, 1988.Google Scholar
- 11.R.W. Farebrother, “Three theorems with applications to Euclidean distance matrices,” Linear Algebra Appl., vol. 95, pp. 11–16, 1987.Google Scholar
- 12.W. Glunt, T.L. Hayden, S. Hong, and J. Wells, “An alternating projection algorithm for computing the nearest Euclidean distance matrix.” SIAM J. Matrix Anal. Appl., vol. 11,no. 4, pp. 589–600, 1990.Google Scholar
- 13.M.X. Goemans, “Semidefinite programming in combinatorial optimization,” Mathematical Programmings, vol. 79, pp. 143–162, 1997.Google Scholar
- 14.E.G. Gol'stein, Theory of Convex Programming, American Mathematical Society: Providence, RI, 1972.Google Scholar
- 15.J.C. Gower, “Properties of Euclidean and non-Euclidean distance matrices,” Linear Algebra Appl., vol. 67, pp. 81–97, 1985.Google Scholar
- 16.T.L. Hayden, J. Wells, W-M. Liu, and P. Tarazaga, “The cone of distance matrices,” Linear Algebra Appl., vol. 144, pp. 153–169, 1991.Google Scholar
- 17.C. Helmberg, “An interior point method for semidefinite programming and max-cut bounds,” Ph.D. Thesis, Graz University of Technology, Austria, 1994.Google Scholar
- 18.C. Helmberg, F. Rendl, R.J. Vanderbei, and H. Wolkowicz, “An interior point method for semidefinite programming,” SIAM Journal on Optimization, pp. 342–361, 1996. URL: ftp://orion.uwaterloo.ca/pub/henry/reports/sdp.ps.gz.Google Scholar
- 19.R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press: New York, 1985.Google Scholar
- 20.C. Johnson, B. Kroschel, and H. Wolkowicz, “An interior-point method for approximate positive semidefinite completions,” Computational Optimization and Applications, vol. 9,no. 2, pp. 175–190, 1998.Google Scholar
- 21.C.R. Johnson and P. Tarazaga, “Connections between the real positive semidefinite and distance matrix completion problems,” Linear Algebra Appl., vol. 223/224, pp. 375–391, 1995.Google Scholar
- 22.E. De Klerk, “Interior point methods for semidefinite programming,” Ph.D. Thesis, Delft University, 1997.Google Scholar
- 23.S. Kruk, M. Muramatsu, F. Rendl, R.J. Vanderbei, and H. Wolkowicz, “The Gauss-Newton direction in linear and semidefinite programming,” Technical Report CORR 98–16, University of Waterloo, Waterloo, Canada, 1998. Detailed Web Version at URL: ftp://orion.uwaterloo.ca/pub/henry/reports/gnsdplong.ps.gz.Google Scholar
- 24.M. Laurent, “A tour d'horizon on positive semidefinite and Euclidean distance matrix completion problems,” in Topics in Semidefinite and Interior-Point Methods, The Fields Institute for Research in Mathematical Sciences, Communications Series, vol. 18, American Mathematical Society: Providence, RI, 1998.Google Scholar
- 25.J. De Leeuw and W. Heiser, “Theory of multidimensional scaling,” in Handbook of Statistics, P.R. Krishnaiah and L.N. Kanal (Eds.), North-Holland, 1982, vol. 2, pp. 285–316.Google Scholar
- 26.S. Lele, “Euclidean distance matrix analysis (EDMA): Estimation of mean form and mean form difference,” Math. Geol., vol. 25,no. 5, pp. 573–602, 1993.Google Scholar
- 27.F. Lempio and H. Maurer, “Differential stability in infinite-dimensional nonlinear programming,” Appl. Math. Optim., vol. 6, pp. 139–152, 1980.Google Scholar
- 28.J.J. Moré and Z. Wu, “Global continuation for distance geometry problems,” Technical Report MCS-P505–0395, Applied Mathematics Division, Argonne National Labs, Chicago, IL, 1996.Google Scholar
- 29.J.J. Moré and Z. Wu, “Distance geometry optimization for protein structures,” Technical Report MCS-P628–1296, Applied Mathematics Division, Argonne National Labs, Chicago, IL, 1997.Google Scholar
- 30.Y.E. Nesterov and A.S. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming, SIAM Publications, SIAM: Philadelphia, USA, 1994.Google Scholar
- 31.P.M. Pardalos, D. Shalloway, and G. Xue (Eds.), “Global minimization of nonconvex energy functions: Molecular conformation and protein folding,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 23, American Mathematical Society: Providence, RI, 1996. Papers from the DIMACS Workshop held as part of the DIMACS Special Year on Mathematical Support for Molecular Biology at Rutgers University, New Brunswick, New Jersey, March 1995.Google Scholar
- 32.G. Pataki, “Cone programming and eigenvalue optimization: Geometry and algorithms,” Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1996.Google Scholar
- 33.M.V. Ramana, “An algorithmic analysis of multiquadratic and semidefinite programming problems,” Ph.D. Thesis, Johns Hopkins University, Baltimore, MD, 1993.Google Scholar
- 34.I.J. Schoenberg, “Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert,” Ann. Math., vol. 36, pp. 724–732, 1935.Google Scholar
- 35.J. Sturm, “Primal-dual interior point approach to semidefinite programming,” Ph.D. Thesis, Erasmus University Rotterdam, 1997.Google Scholar
- 36.P. Tarazaga, T.L. Hayden, and J. Wells, “Circum-Euclidean distance matrices and faces,” Linear Algebra Appl., vol. 232, pp. 77–96, 1996.Google Scholar
- 37.M. Todd, “On search directions in interior-point methods for semidefinite programming,” Technical Report TR1205, School of OR and IE, Cornell University, Ithaca, NY, 1997.Google Scholar
- 38.W.S. Torgerson, “Multidimensional scaling. I. Theory and method,” Psychometrika, vol. 17, pp. 401–419, 1952.Google Scholar
- 39.M.W. Trosset, “Applications of multidimensional scaling to molecular conformation,” Technical Report, Rice University, Houston, Texas, 1997.Google Scholar
- 40.M.W. Trosset, “Computing distances between convex sets and subsets of the positive semidefinite matrices,” Technical Report, Rice University, Houston, Texas, 1997.Google Scholar
- 41.M.W. Trosset, “Distance matrix completion by numerical optimization,” Technical Report, Rice University, Houston, Texas, 1997.Google Scholar
- 42.L. Vandenberghe and S. Boyd, “Positive definite programming,” in Mathematical Programming: State of the Art, 1994, The University of Michigan, 1994, pp. 276–308.Google Scholar
- 43.L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review, vol. 38, pp. 49–95, 1996.Google Scholar
- 44.H. Wolkowicz, “Some applications of optimization in matrix theory,” Linear Algebra and its Applications, vol. 40, pp. 101–118, 1981.Google Scholar
- 45.S. Wright, Primal-Dual Interior-Point Methods, SIAM: Philadelphia, PA, 1996.Google Scholar
- 46.Y. Ye, Interior Point Algorithms: Theory and Analysis, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons: New York, 1997.Google Scholar
- 47.F.Z. Zhang, “On the best Euclidean fit to a distance matrix,” Beijing Shifan Daxue Xuebao, vol. 4, pp. 21–24, 1987.Google Scholar
- 48.Z. Zou, R.H. Byrd, and R.B. Schnabel, “A stochastic/perturbation global optimization algorithm for distance geometry problems,” Technical Report, Department of Computer Science, University of Colorado, Boulder, CO, 1996.Google Scholar