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Perturbation analysis of the euclidean distance matrix optimization problem and its numerical implications

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Abstract

Euclidean distance matrices have lately received increasing attention in applications such as multidimensional scaling and molecular conformation from nuclear magnetic resonance data in computational chemistry. In this paper, we focus on the perturbation analysis of the Euclidean distance matrix optimization problem (EDMOP). Under Robinson’s constraint qualification, we establish a number of equivalent characterizations of strong regularity and strong stability at a locally optimal solution of EDMOP. Those results extend the corresponding characterizations in Semidefinite Programming and are tailored to the special structure in EDMOP. As an application, we demonstrate a numerical implication of the established results on an alternating direction method of multipliers (ADMM) to a stress minimization problem, which is an important instance of EDMOP. The implication is that the ADMM method converges to a strongly stable solution under reasonable assumptions.

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References

  1. Alfakih, A.Y., Khandani, A., Wolkowicz, H.: Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12, 13–30 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Alfakih, A.Y., Wolkowicz, H.: Matrix completion problems. In: Handbook of Semidefinite programminP. International Series in Operations Research and Management Science, vol. 27, pp. 533–545. Kluwer Academic Publication, Boston (2002)

    Google Scholar 

  3. Al-Homidan, S., Wolkowicz, H.: Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming. Linear Algebra Appl. 406, 109–141 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    MATH  Google Scholar 

  5. Bonnans, J.F., Ramirez, C.H.: Perturbation analysis of second-order cone programming problems. Math. Program. Ser. B 104, 205–227 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer Ser. Statist, Springer, New York (2005)

    MATH  Google Scholar 

  7. Chan, Z.X., Sun, D.F.: Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming. SIAM J. Optim. 19, 370–396 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Dattorro, J.: Convex Optimization and Euclidean Distance Geometry. Meboo Publishing USA, Palo Alto, CA (2005)

    MATH  Google Scholar 

  9. Ding, C., Qi, H.-D.: Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction. Math. Program. 164, 341–381 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Ding, C., Sun, D.F., Toh, K.C.: An introduction to a class of matrix cone programming. Maths. Prog. 144, 141–179 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Ding, C., Sun, D.F., Zhang, L.W.: Characterization of the robust isolated calmness for a class of conic programming problems. SIAM J. Optim. 27, 67–90 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Dokmanic, I., Parhizkar, R., Ranieri, J., Vetterli, M.: Euclidean distance matrices: essential theory, algorithms, and applications. IEEE Signal Process. Mag. 32, 12–30 (2015)

    Google Scholar 

  13. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, New York (2009)

    MATH  Google Scholar 

  14. Drusvyatskiy, D., Krislock, N., Voronin, T.-L., Wolkowicz, H.: Noisy Euclidean distance realization: robust facial reduction and the pareto frontier. SIAM J. Optim. 27, 2301–2331 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)

    MATH  Google Scholar 

  16. Fazel, M., Pong, T.K., Sun, D., Tseng, P.: Hankel matrix rank mininization with applications to system identification and realization. SIAM J. Matrix Anal. Appl. 34, 946–977 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Glunt, W., Hayden, T.L., Hong, S., Wells, J.: An alternating projection algorithm for computing the nearest Euclidean distance matrix. SIAM J. Matrix Anal. Appl. 11, 589–600 (1990)

    MathSciNet  MATH  Google Scholar 

  18. Gaffke, N., Mathar, R.: A cyclic projection algorithm via duality. Metrika 36, 29–54 (1989)

    MathSciNet  MATH  Google Scholar 

  19. Hayden, T.L., Wells, J.: Approximation by matrices positive semidefinite on a subspace. Linear Algebra Appl. 109, 115–130 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Johnson, C.R., Kroschel, B., Wolkowicz, H.: An interior-point method for approximate positive semidefinite completion. Comput. Optim. Appl. 9, 175–190 (1998)

    MathSciNet  MATH  Google Scholar 

  21. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Kluwer Academic Publishers, Boston (2002)

    MATH  Google Scholar 

  22. Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)

    Google Scholar 

  23. Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Optim. 20, 2679–2708 (2010)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Micchelli, C.A.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11–22 (1986)

    MathSciNet  MATH  Google Scholar 

  26. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, New York (2018)

    MATH  Google Scholar 

  27. Qi, H.D.: A semismooth Newton method for the nearest Euclidean distance matrix problem. SIAM J. Matrix Anal. Appl. 34, 67–93 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Qi, H.-D.: Conditional quadratic semidefinite programming: examples and methods. J. Oper. Res. Soc. China 2, 143–170 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Qi, H.-D., Yuan, X.: Computing the nearest Euclidean distance matrix with low embedding dimensions. Math. Prog. 147, 351–389 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Rebonato, R.: Volatility and Correlation, 2nd edn. Wiley, New Jersey (2004)

    MATH  Google Scholar 

  31. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    MathSciNet  MATH  Google Scholar 

  32. Rockafellar, R.T., Wets, R.-B.: Variational Analysis. Springer Verlag, Berlin (1998)

    Book  MATH  Google Scholar 

  33. Schoenberg, I.J.: Remarks to Maurice Frechét’s article “Sur la définition axiomatique d’une classe d’espace distancés vectoriellement applicable sur l’espace de Hilbert’’. Ann. Math. 36, 724–732 (1935)

    MathSciNet  MATH  Google Scholar 

  34. Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Prog. 77, 301–320 (1997)

    MathSciNet  MATH  Google Scholar 

  35. Sun, D.F.: The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761–776 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Sun, D.F., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002)

    MathSciNet  MATH  Google Scholar 

  37. Sun, J., Boyd, S., Xiao, L., Diaconis, P.: The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. SIAM Rev. 48, 681–699 (2006)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Y., Zhang, L.W.: Nonsingularity in second-order cone programming via the smoothing metric projector. Sci. China Math. 53, 1025–1038 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Weinberger, K.Q., Lawrence, K.S.: An introduction to nonlinear dimensionality reduction by maximum variance unfolding. AAAI 6, 1683–1686 (2006)

    Google Scholar 

  40. Zhang, H., Liu, Y., Lei, H.: Localization from incomplete Euclidean distance matrix: performance analysis for the svd-mds approach. IEEE Trans. Sig. Process. 67, 2196–2209 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Zhang, Q., Zhao, X., Ding, C.: Matrix optimization based Euclidean embedding with outliers. Comput. Optim. Appl. 79, 235–271 (2021)

    MathSciNet  MATH  Google Scholar 

  42. Zhou, S., Xiu, N., Qi, H.-D.: Robust Euclidean embedding via EDM optimization. Math. Prog. Comput. 12, 337–387 (2020)

    MathSciNet  MATH  Google Scholar 

  43. Zhou, S., Xiu, N., Qi, H.-D.: A fast matrix majorization-projection method for penalized stress minimization with box constraints. IEEE Trans. Sig. Process. 66, 4331–4346 (2018)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the editor and the two referees for their detailed comments that have improved the quality of the paper. In particular, one referee points us to the relevant papers [2, 3] on SDP approach for EDM optimization.

Funding

The research of Guo was partially supported by NSFC 11801057 and Qi was partially supported by IEC/NSFC/191543 and P0045347.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Shaoyan Guo

  2. Department of Applied Mathematics and CAS AMSS-PolyU Joint Lab, The Hong Kong Polytechnic University, Hong Kong, China

    Hou-Duo Qi

  3. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Liwei Zhang

Authors
  1. Shaoyan Guo
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  2. Hou-Duo Qi
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  3. Liwei Zhang
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Correspondence to Hou-Duo Qi.

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Dedicated to Prof Asen L. Dontchev—a leader, a friend and a mentor.

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Guo, S., Qi, HD. & Zhang, L. Perturbation analysis of the euclidean distance matrix optimization problem and its numerical implications. Comput Optim Appl 86, 1193–1227 (2023). https://doi.org/10.1007/s10589-023-00505-z

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