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Pseudotransient Continuation for Solving Systems of Nonsmooth Equations with Inequality Constraints

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Abstract

This paper investigates a pseudotransient continuation algorithm for solving a system of nonsmooth equations with inequality constraints. We first transform the inequality constrained system of nonlinear equations to an augmented nonsmooth system, and then employ the pseudotransient continuation algorithm for solving the corresponding augmented nonsmooth system. The method gets its global convergence properties from the dynamics, and inherits its local convergence properties from the semismooth Newton method. Finally, we illustrate the behavior of our approach by some numerical experiments.

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References

  1. Bellavia, S., Macconi, M., Morini, B.: An affine scaling trust-region approach to bounded-constrained nonlinear systems. Appl. Numer. Math. 44, 257–280 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bellavia, S., Macconi, M., Morini, B.: STRSCNE: A scaled trust-region solver for constrained nonlinear equations. Comput. Optim. Appl. 28, 31–50 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Heinkenschloss, M., Ulbrich, M., Ulbrich, S.: Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption. Math. Program. 86, 615–635 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ulbrich, M., Ulbrich, S.: Superlinear convergence of affine-scaling interior-point newton methods for infinite-dimensional nonlinear problems with pointwise bounds. SIAM J. Control Optim. 38, 1938–1984 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418–445 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kanzow, C., Klug, A.: On affine-scaling interior-point Newton methods for nonlinear minimization with bound constraints. Comput. Optim. Appl. 35, 177–197 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kanzow, C., Klug, A.: An interior-point affine-scaling trust-region method for semismooth equations with box constraints. Comput. Optim. Appl. 37, 329–353 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg-Marquardt methods for constrained non-linear equations with strong local convergence properties. J. Comput. Appl. Math. 172, 375–397 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ulbrich, M.: Nonmonotone trust-region method for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. SIAM J. Optim. 11, 889–917 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Qi, L., Tong, X., Li, D.: An active-set projected trust region algorithm for box constrained nonsmooth equations. J. Optim. Theory Appl. 120, 601–625 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tong, X., Qi, L.: On the convergence of a trust region method for solving constrained nonlinear equations with degenerate solutions. J. Optim. Theory Appl. 123, 187–212 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sellami, H., Robinson, S.M.: Homotopies based on nonsmooth equations for solving nonlinear variational inequalities. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Applications. Plenum Press, New York (1996)

    Google Scholar 

  13. Sellami, H., Robinson, S.M.: Implementation of a continuation method for normal maps. Math. Program. 76, 563–578 (1997)

    MathSciNet  Google Scholar 

  14. Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction. Springer, Berlin (1990)

    MATH  Google Scholar 

  15. Daniel, J.W.: Newton’s method for nonlinear inequalities. Numer. Math. 6, 381–387 (1973)

    Article  MathSciNet  Google Scholar 

  16. Burke, J.V., Ferris, M.C.: A Gauss-Newton method for convex composite optimization. Math. Program. 71, 179–194 (1995)

    MathSciNet  Google Scholar 

  17. Garcia-Palomares, U.M., Restuccia, A.: A global quadratic algorithm for solving a system of mixed equalities and inequalities. Math. Program. 21, 290–300 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dennis, J.E., El-Alem, M., Williamson, K.: A trust-region approach to nonlinear systems of equalities and inequalities. SIAM J. Optim. 9, 291–315 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fletcher, R., Leyffer, S.: Filter-type algorithms for solving systems of algebraic equations and inequalities. In: Di Pillo, G., Murli, A. (eds.) High Performance Algorithms and Software for Nonlinear Optimization, pp. 259–278. Kluwer Academic, Dordrecht (2003)

    Google Scholar 

  20. Macconi, M., Morini, B., Porcelli, M.: Trust-region quadratic methods for nonlinear systems of mixed equalities. Preprint (2007)

  21. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations. SIAM, Philadelphia (1998)

    MATH  Google Scholar 

  22. Shampine, L.F.: Numerical Solution of Ordinary Differential Equations. Chapman and Hall, New York (1994)

    MATH  Google Scholar 

  23. Coffey, T.S., Kelley, C.T., Keyes, D.E.: Pseudo-transient continuation and differential-algebraic equations. SIAM J. Sci. Comput. 25, 553–569 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fowler, K.R., Kelley, C.T.: Pseudo-transient continuation for nonsmooth nonlinear equations. SIAM J. Numer. Anal. 43, 1385–1406 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Higham, D.J.: Trust region algorithms and time step selection. SIAM J. Numer. Anal. 37, 194–210 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kelley, C.T., Keyes, D.E.: Convergence analysis of pseudo-transient continuation. SIAM J. Numer. Anal. 35, 508–523 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kelley, C.T., Liao, L.-Z., Qi, L., Chu, M.T., Reese, J.P., Winton, C.: Projected pseudotransient continuation. SIAM J. Numer. Anal. 46, 3071–3083 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rheinboldt, W.C.: Numerical Analysis of Parametrized Nonlinear Equations. Wiley, New York (1986)

    MATH  Google Scholar 

  29. Mulder, W., Leer, B.V.: Experiments with implicit upwind methods for the Euler equations. J. Comput. Phys. 59, 232–246 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  30. Coffey, T.S., McMullan, R.J., Kelley, C.T., McRae, D.S.: Globally convergent algorithms for nonsmooth nonlinear equations in computational fluid dynamics. J. Comput. Appl. Math. 152, 69–81 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Knoll, D.A., McHugh, P.: Enhanced nonlinear iterative techniques applied to a nonequilibrium plasma flow. SIAM J. Sci. Comput. 19, 291–301 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Farthing, M.W., Kees, C.E., Coffey, T., Kelley, C.T., Miller, C.T.: Efficient steady-state solution techniques for variably saturated groundwater flow. Adv. Water Resour. 26, 833–849 (2003)

    Article  Google Scholar 

  33. Gherman, I., Schulz, V.: Preconditioning of one-shot pseudo-timestepping methods for shape optimization. Proc. Appl. Math. Mech. 5, 741–742 (2005)

    Article  Google Scholar 

  34. Hazra, S.B., Schulz, V.: Simultaneous pseudo-timestepping for PDE-model based optimization problems. BIT 44, 457–472 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Hazra, S.B., Schulz, V., Brezillon, J.: Simultaneous pseudo-time stepping for 3D aerodynamic shape optimization. J. Numer. Math. 16, 139–161 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. Hazra, S.B., Schulz, V., Brezillon, J., Gauger, N.: Aerodynamic shape optimization using simultaneous pseudo-timestepping. J. Comput. Phys. 204, 46–64 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. Hazra, S.B., Schulz, V.: Simultaneous pseudo-timestepping for aerodynamic shape optimization problems with constraints. SIAM J. Sci. Comput. 28, 1078–1099 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  38. Boggs, P.T.: The solution of nonlinear systems of equations by A-stable integration techniques. SIAM J. Numer. Anal. 8, 767–785 (1971)

    Article  MathSciNet  Google Scholar 

  39. Keller, H.B.: Global homotopies and Newton methods. In: de Boor, C., Golub, G.H. (eds.) Recent Advances in Numerical Analysis, pp. 73–94. Academic Press, New York (1979)

    Google Scholar 

  40. Klopfenstein, R.W.: Zeros of nonlinear functions. J. Assoc. Comput. Mach. 8, 366–373 (1961)

    MATH  MathSciNet  Google Scholar 

  41. Incerti, S., Parisi, V., Zirilli, F.: A new method for solving nonlinear simultaneous equations. SIAM J. Numer. Anal. 16, 779–789 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  42. Kubíček, M.: Algorithm 502: Dependence of solution of nonlinear systems on a parameter. ACM Trans. Math. Softw. 2, 98–107 (1976)

    Article  MATH  Google Scholar 

  43. Smale, S.: A convergent process of price adjustment and global Newton methods. J. Math. Econ. 3, 1–14 (1976)

    Article  MathSciNet  Google Scholar 

  44. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  45. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  Google Scholar 

  46. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  47. Pang, J.-S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  49. Greenbaum, A.: Iterative Methods for Solving Linear Systems. Frontiers in Applied Mathematics, vol. 17. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  50. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics, vol. 16. SIAM, Philadelphia (1995)

    MATH  Google Scholar 

  51. Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method. Fundamentals of Algorithms. SIAM, Philadelphia (2003)

    MATH  Google Scholar 

  52. Outrata, J.V., Zowe, J.: A Newton method for a class of quasi-variational inequalities. Comput. Optim. Appl. 4, 5–21 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  53. Jiang, H., Fukushima, M., Qi, L., Sun, D.: A trust region method for solving generalized complementarity problems. SIAM J. Optim. 8, 140–157 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  54. Andreani, R., Friedlander, A., Santos, S.A.: On the resolution of the generalized nonlinear complementarity problem. SIAM J. Optim. 12, 303–321 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  55. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  56. Bullard, L.G., Biegler, L.T.: Iterative linear programming strategies for constrained simulation. Comput. Chem. Eng. 15, 239–254 (1991)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Colorado Denver, Campus Box 170, P.O. Box 173364, Denver, CO, 80217-3364, USA

    J. Chen

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

    L. Qi

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  1. J. Chen
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  2. L. Qi
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Correspondence to J. Chen.

Additional information

Communicated by G. Di Pillo.

This work was supported by the Hong Kong Research Grant Council.

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Chen, J., Qi, L. Pseudotransient Continuation for Solving Systems of Nonsmooth Equations with Inequality Constraints. J Optim Theory Appl 147, 223–242 (2010). https://doi.org/10.1007/s10957-010-9719-9

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  • DOI: https://doi.org/10.1007/s10957-010-9719-9

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