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Convex Quadratic Approximation

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Abstract

For some applications it is desired to approximate a set of m data points in ℝn with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n + 1)(n + 2)/2 parameters, s ≤ m, so as to minimize the approximation error in the L 1 norm. The approximating function is q(p,x), where p ∈ ℝs is the vector of parameters, and x ∈ ∝n. The Hessian of q(p,x) with respect to x (for fixed p) is positive semi-definite, and its Hessian with respect to p (for fixed x) is shown to be positive semi-definite and of rank ≤n. An algorithm is described for computing an optimal p* for any specified set of m data points, and computational results (for n = 4,6,10,15) are presented showing that the optimal q(p*,x) can be obtained efficiently. It is shown that the approximation will usually interpolate s of the m data points.

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References

  1. F. Alizadeh, J.-P.A. Haeberly, and M.L. Overton, “Complementarity and nondegeneracy in semidefinite programming,” Math. Programming, vol. 77, pp. 111-128, 1997. Semidefinite programming.

    Google Scholar 

  2. I. Barrodale, “L1 approximation and the analysis of data,” Appl. Stat., vol. 17, pp. 51-57, 1968.

    Google Scholar 

  3. I. Barrodale and F.D.K. Roberts, “An improved algorithm for discrete L1 linear approximation,” SIAM J. Numer. Anal., vol. 10, pp. 839-848, 1973.

    Google Scholar 

  4. M.S. Bazaraa and C.M. Shetty, Nonlinear Programming, John Wiley & Sons: New York-Chichester-Brisbane, 1979. Theory and algorithms.

    Google Scholar 

  5. K.A. Dill, A.T. Phillips, and J.B. Rosen, “CGU: An Algorithm for Molecular Structure Prediction, in Large-Scale Optimization with Applications,” Part III Minneapolis, MN, 1995, vol. 94 of IMA Vol. Math. Appl., Springer: New York, pp. 1-21, 1997.

    Google Scholar 

  6. P.E. Gill, W. Murray, and M.A. Saunders, “SNOPT: An SQP algorithm for large-scale constrained optimization,” SIAM J. Optim., vol. 12, pp. 979-1006 (electronic), 2002.

    Google Scholar 

  7. P.E. Gill, W. Murray, M.A. Saunders, and M.H. Wright, “User's guide for NPSOL (version 4.0): A Fortran package for nonlinear programming,” Tech. Rep. SOL-86-2, Systems Optimization Laboratory, Stanford University, Stanford, CA, 1986.

    Google Scholar 

  8. J.G. Hayes (Ed.), “Numerical approximation to functions and data,” Conference of the Institute of Mathematics and its Applications. Held in Canterbury, 1967. The Athlone Press, University of London: London, 1970.

    Google Scholar 

  9. J.C. Mitchell, J.B. Rosen, A.T. Phillips, and L.F. Ten Eyck, “Coupled optimization in protein docking,” in Proceedings of the Third Annual International Conference on Computational Molecular Biology, ACM Press, 1999, pp. 280-284.

  10. M.J.D. Powell, Approximation Theory and Methods, Cambridge University Press: Cambridge, 1981.

    Google Scholar 

  11. M.J.D. Powell, “On the lagrange functions of quadratic models that are defined by interpolation,” Optim. Methods Softw., vol. 16, pp. 289-309, 2001. Dedicated to Professor Laurence C. W. Dixon on the occasion of his 65th birthday.

    Google Scholar 

  12. J.R. Rice, The Approximation of Functions. Vol. I: Linear Theory, Addison-Wesley Publishing Co.: Reading, MA, London, 1964.

    Google Scholar 

  13. T.J. Rivlin, An Introduction to the Approximation of Functions, Dover Publications Inc.: New York, 1981. Corrected reprint of the 1969 original, Dover Books on Advanced Mathematics.

    Google Scholar 

  14. R.T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press: Princeton, NJ, 1970.

    Google Scholar 

  15. J.B. Rosen, K.A. Dill, and A.T. Phillips, “Protein structure and energy landscape dependence on sequence using a continuous energy function,” J. Comp. Biol., vol. 4, pp. 227-239, 1997.

    Google Scholar 

  16. J.B. Rosen, H. Park, J. Glick, and L. Zhang, “Accurate solution to overdetermined linear equations with errors using L1 norm minimization,” Comput. Optim. Appl., vol. 17, pp. 329-341, 2000.

    Google Scholar 

  17. D. Winfield, “Function minimization by interpolation in a data table,” J. Inst. Math. Appl., vol. 12, pp. 339-347, 1973.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA, USA

    J. Ben Rosen

  2. San Diego Supercomputer Center, University of California, San Diego, La Jolla, CA, USA

    Roummel F. Marcia

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  1. J. Ben Rosen
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  2. Roummel F. Marcia
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Rosen, J.B., Marcia, R.F. Convex Quadratic Approximation. Computational Optimization and Applications 28, 173–184 (2004). https://doi.org/10.1023/B:COAP.0000026883.13660.84

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