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Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints

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Abstract

This paper presents two new trust-region methods for solving nonlinear optimization problems over convex feasible domains. These methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates. The algorithms are proved to be convergent to critical points of the problem from any starting point. Extensive numerical experiments show that this approach is competitive with the LANCELOT package.

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References

  1. I. Bongartz, A.R. Conn, N.I.M. Gould and Ph.L. Toint, CUTE: Constrained and unconstrained testing environment,ACM Transactions on Mathematical Software 21 (1995) 123–160.

    Article  MATH  Google Scholar 

  2. J.F. Bonnans, E. Panier, A. Tits and J.L. Zhou, Avoiding the Maratos effect by means of a nonmonotone linesearch II: Inequality constrained problems—feasible iterates,SIAM Journal on Numerical Analysis 29 (1992) 1187–1202.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.V. Burke and J.J. Moré, On the identification of active constraints,SIAM Journal on Numerical Analysis 25 (1988) 1197–1211.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.V. Burke, J.J. Moré and G. Toraldo, Convergence properties of trust region methods for linear and convex constraints,Mathematical Programming, Series A 47 (1990) 305–336.

    Article  MATH  Google Scholar 

  5. P.H. Calamai and J.J. Moré, Projected gradient methods for linearly constrained problems,Mathematical Programming 39 (1987) 93–116.

    Article  MathSciNet  MATH  Google Scholar 

  6. R.M. Chaniberlain, C. Lemaréchal, H.C. Pedersen and M.J.D. Powell, The watchdog technique for forcing convergence in algorithms for constrained optimization,Mathematical Programming Studies 16 (1982) 1–17.

    Google Scholar 

  7. A.R. Conn, N.I.M. Gould and Ph.L. Toint, Global convergence of a class of trust region algorithms for optimization with simple bounds,SIAM Journal on Numerical Analysis 25 (1988) 433–460; 26 (1989) 764–767.

    Article  MathSciNet  MATH  Google Scholar 

  8. A.R. Conn, N.I.M. Gould and Ph.L. Toint,LANCELOT: a Fortran package for large-scale nonlinear optimization (Release A) (Springer Verlag, Heidelberg, Berlin, New York, 1992).

    MATH  Google Scholar 

  9. A.R. Conn, Nick Gould, A. Sartenaer and Ph.L. Toint, Global convergence of a class of trust region algorithms for optimization using inexact projections on convex constraints,SIAM Journal on Optimization 3 (1993) 164–221.

    Article  MathSciNet  MATH  Google Scholar 

  10. A.R. Conn, Nick Gould, A. Sartenaer and Ph.L. Toint, Convergence properties of an augmented Lagrangian algorithms for optimization with a combination of general equality and linear constraints,SIAM Journal on Optimization 33 (1993) 61–75.

    MathSciNet  Google Scholar 

  11. N.Y. Deng, Y. Xiao and F.J. Zhou, Nonmonotonic trust region algorithms,Journal of Optimization Theory and Applications 76 (1993) 259–285.

    Article  MathSciNet  MATH  Google Scholar 

  12. J.E. Dennis and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).

    MATH  Google Scholar 

  13. F. Facchinei and S. Lucidi, Nonmonotone bundle-type scheme for convex nonsmooth minimization,Journal of Optimization Theory and Applications 76 (1993) 241–257.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Fletcher,Practical Methods of Optimization (Wiley, Chichester, UK, 2nd edn., 1987).

    MATH  Google Scholar 

  15. L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton’s method,SIAM Journal on Numerical Analysis 23 (1986) 707–716.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Grippo, F. Lampariello and S. Lucidi, A truncated Newton method with nonmonotone line search for unconstrained optimization,Journal of Optimization Theory and Applications 60 (1989) 401–419.

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Grippo, F. Lampariello and S. Lucidi, A class of nonmonotone stabilization methods in unconstrained optimization,Numerische Mathematik 59 (1991) 779–805.

    Article  MathSciNet  MATH  Google Scholar 

  18. J.J. More, Recent developments in algorithms and software for trust region methods, in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art (Springer Verlag, Heidelberg, 1983) 258–287.

    Google Scholar 

  19. J.J. More, Trust regions and projected gradients, in: M. Iri and K. Yajima, eds.,System Modelling and Optimization (Springer Verlag, Heidelberg, 1988) 1–13.

    Chapter  Google Scholar 

  20. E. Panier and A. Tits, Avoiding the Maratos effect by means of a nonmonotone linesearch I: general constrained problems,SIAM Journal on Numerical Analysis 28 (1991) 1183–1195.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. B. Schnabel and E. Eskow, A new modified Cholesky factorization,SIAM Journal on Scientific and Statistical Computing 11 (1991) 1136–1158.

    Article  MathSciNet  Google Scholar 

  22. T. Steihaug, The conjugate gradient method and trust regions in large scale optimization,SIAM Journal on Numerical Analysis 20 (1983) 626–637.

    Article  MathSciNet  MATH  Google Scholar 

  23. Ph.L. Toint, Towards an efficient sparsity exploiting Newton method for minimization, in: I.S. Duff, ed.,Sparse Matrices and Their Uses (Academic Press, London, 1981) 57–88.

    Google Scholar 

  24. Ph.L. Toint, Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space,IMA Journal of Numerical Analysis 8 (1988) 231–252.

    Article  MathSciNet  MATH  Google Scholar 

  25. Ph.L. Toint, A non-monotone trust-region algorithm for nonlinear optimization subject to convex constraints, the complete numerical results, Technical Report 94/26. Department of Mathematics (FUNDP), Namur, Belgium, 1994).

    Google Scholar 

  26. Ph.L. Toint, An assessment of non-monotone linesearch techniques for unconstrained optimization,SIAM Journal on Scientific Computing 17 (1996) 725–739.

    Article  MathSciNet  MATH  Google Scholar 

  27. Y. Xiao and F.J. Zhou, Nonmonotone trust region methods with curvilinear path in unconstrained optimization,Computing 48 (1992) 303–317.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Facultés Universitaires ND de la Paix, B-5000, Namur, Belgium

    Philippe L. Toint

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  1. Philippe L. Toint
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Toint, P.L. Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints. Mathematical Programming 77, 69–94 (1997). https://doi.org/10.1007/BF02614518

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  • DOI: https://doi.org/10.1007/BF02614518

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