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Maximum feasible subsystems of distance geometry constraints

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Abstract

We study the problem of satisfying the maximum number of distance geometry constraints with minimum experimental error. This models the determination of the shape of proteins from atomic distance data which are obtained from nuclear magnetic resonance experiments and exhibit experimental and systematic errors. Experimental errors are represented by interval constraints on Euclidean distances. Systematic errors occur from a misassignment of distances to wrong atomic pairs: we represent such errors by maximizing the number of satisfiable distance constraints. We present many mathematical programming formulations, as well as a “matheuristic” algorithm based on reformulations, relaxations, restrictions and refinement. We show that this algorithm works on protein graphs with hundreds of atoms and thousands of distances.

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References

  1. Achlioptas, D., Naor, A., Peres, Y.: Rigorous location of phase transitions in hard optimization problems. Nature 435(9), 759–764 (2005)

    Article  Google Scholar 

  2. Ahmadi, A., Hall, G.: Sum of squares basis pursuit with linear and second order cone programming. In: Harrington, H., Omar, M., Wright, M. (eds.) Algebraic and Geometric Methods in Discrete Mathematics. Contemporary Mathematics, vol. 685, pp. 27–54. AMS, Providence, RI (2017)

    Google Scholar 

  3. Ahmadi, A., Majumdar, A.: DSOS and SDSOS optimization: more tractable alternatives to sum of squares and semidefinite optimization. SIAM J. Appl. Algebra Geometry 3(2), 193–230 (2019)

    Article  MathSciNet  Google Scholar 

  4. Amaldi, E., Bruglieri, M., Casale, G.: A two-phase relaxation-based heuristic for the maximum feasible subsystem problem. Comput. Oper. Res. 35, 1465–1482 (2008)

    Article  Google Scholar 

  5. Amaldi, E., Pfetsch, M., Trotter, L.: On the maximum feasible subsystem problem, IISS and IIS-hypergraphs. Math. Program. 95, 533–554 (2003)

    Article  MathSciNet  Google Scholar 

  6. Barker, G., Carlson, D.: Cones of diagonally dominant matrices. Pac. J. Math. 57(1), 15–32 (1975)

    Article  MathSciNet  Google Scholar 

  7. Barvinok, A.: Measure concentration in optimization. Math. Program. 79, 33–53 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Berger, B., Kleinberg, J., Leighton, T.: Reconstructing a three-dimensional model with arbitrary errors. J. ACM 46(2), 212–235 (1999)

    Article  MathSciNet  Google Scholar 

  9. Berman, H., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T., Weissig, H., Shindyalov, I.N., Bourne, P.: The protein data bank. Nucleic Acid Res. 28, 235–242 (2000)

    Article  Google Scholar 

  10. Bhatia, R.: Matrix Analysis. New York (1997)

  11. COIN-OR. Introduction to IPOPT: A tutorial for downloading, installing, and using IPOPT (2006)

  12. D’Ambrosio, C., Vu, K., Lavor, C., Liberti, L., Maculan, N.: New error measures and methods for realizing protein graphs from distance data. Discrete Comput. Geom. 57(2), 371–418 (2017)

    Article  MathSciNet  Google Scholar 

  13. Dias, G., Liberti, L.: Diagonally dominant programming in distance geometry. In: Cerulli, R., Fujishige, S., Mahjoub, R. (eds.) International Symposium in Combinatorial Optimization. LNCS, vol. 9849, pp. 225–236. Springer, New York (2016)

  14. Gerschgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix. Izvestia Akademii Nauk USSR 6, 749–754 (1931)

    MATH  Google Scholar 

  15. Gonçalves, D., Mucherino, A., Lavor, C., Liberti, L.: Recent advances on the interval distance geometry problem. J. Glob. Optim. 69, 525–545 (2017)

    Article  MathSciNet  Google Scholar 

  16. Goodall, C.: Procrustes methods in the statistical analysis of shape. J. R. Stat. Soc. B 53(2), 285–339 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Greer, R.: Trees and hills: methodology for maximizing functions of systems of linear relations. Annals of Discrete Mathematics, vol. 22. Elsevier, Amsterdam (1984)

    MATH  Google Scholar 

  18. Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24(6), 417–441 (1933)

    Article  Google Scholar 

  19. IBM. ILOG CPLEX 12.9 User’s Manual. IBM (2019)

  20. Lavor, C., Liberti, L., Maculan, N.: Computational experience with the molecular distance geometry problem. In: Pintér, J. (ed.) Global Optimization: Scientific and Engineering Case Studies, pp. 213–225. Springer, Berlin (2006)

    Chapter  Google Scholar 

  21. Lavor, C., Liberti, L., Mucherino, A.: The interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem with inexact distances. J. Glob. Optim. 56, 855–871 (2013)

    Article  MathSciNet  Google Scholar 

  22. Liberti, L.: Undecidability and hardness in mixed-integer nonlinear programming. RAIRO Oper. Res. 53, 81–109 (2019)

    Article  MathSciNet  Google Scholar 

  23. Liberti, L.: Distance geometry and data science. TOP 28, 271–339 (2020)

    Article  MathSciNet  Google Scholar 

  24. Liberti, L., Iommazzo, G., Lavor, C., Maculan, N.: A cycle-based formulation of the Distance Geometry Problem. In C. Gentile et al., (ed.), Proceedings of 18th Cologne-Twente Workshop, volume 4 of AIRO, Springer, New York (2020)

  25. Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)

    Article  MathSciNet  Google Scholar 

  26. Liberti, L., Marinelli, F.: Mathematical programming: Turing completeness and applications to software analysis. J. Combin. Optim. 28(1), 82–104 (2014)

    Article  MathSciNet  Google Scholar 

  27. Liberti, L., Vu, K.: Barvinok’s Naive algorithm in distance geometry. Oper. Res. Lett. 46, 476–481 (2018)

    Article  MathSciNet  Google Scholar 

  28. Luisi, P.: Molecular conformational rigidity: an approach to quantification. Naturwissenschaften 64, 569–574 (1977)

    Article  Google Scholar 

  29. Malliavin, T., Mucherino, A., Lavor, C., Liberti, L.: Systematic exploration of protein conformational space using a distance geometry approach. J. Chem. Inf. Model. 59, 4486–4503 (2019)

    Article  Google Scholar 

  30. Mucherino, A., Gonçalves, D.S., Liberti, L., Lin, J.-H., Lavor, C., Maculan, N., MD-JEEP: a new release for discretizable distance geometry problems with interval data. Annals of Computer Science and Information Systems, Sofia, Bulgaria 1–7, 2020 (2020)

  31. Nilges, M., Macias, M., O’Donoghue, S., Oschkinat, H.: Automated NOESY interpretation with ambiguous distance restraints: the refined NMR solution structure of the Pleckstrin homology domain from \(\beta \)-spectrin. J. Mol. Biol. 269, 408–422 (1997)

    Article  Google Scholar 

  32. Sahinidis, N.V., Tawarmalani, M.: BARON 7.2.5: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual (2005)

  33. Saxe, J.: Embeddability of weighted graphs in \(k\)-space is strongly NP-hard. In: Proceedings of 17th Allerton Conference in Communications, Control and Computing, pp. 480–489 (1979)

  34. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Design, Politecnico di Milano, Milan, Italy

    Maurizio Bruglieri

  2. Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy

    Roberto Cordone

  3. LIX CNRS Ecole Polytechnique, Institut Polytechnique de Paris, 91128, Palaiseau, France

    Leo Liberti

Authors
  1. Maurizio Bruglieri
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  2. Roberto Cordone
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  3. Leo Liberti
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Correspondence to Leo Liberti.

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One of the authors (LL) was partly funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement no. 764759.

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Bruglieri, M., Cordone, R. & Liberti, L. Maximum feasible subsystems of distance geometry constraints. J Glob Optim 83, 29–47 (2022). https://doi.org/10.1007/s10898-021-01003-4

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