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Computing a Celis-Dennis-Tapia trust-region step for equality constrained optimization

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Abstract

We study an approach for minimizing a convex quadratic function subject to two quadratic constraints. This problem stems from computing a trust-region step for an SQP algorithm proposed by Celis, Dennis and Tapia (1985) for equality constrained optimization. Our approach is to reformulate the problem into a univariate nonlinear equationφ(μ)=0 where the functionφ(μ) is continuous, at least piecewise differentiable and monotone. Well-established methods then can be readily applied. We also consider an extension of our approach to a class of non-convex quadratic functions and show that our approach is applicable to reduced Hessian SQP algorithms. Numerical results are presented indicating that our algorithm is reliable, robust and has the potential to be used as a building block to construct trust-region algorithms for small-sized problems in constrained optimization.

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References

  1. M.R. Celis, J.E. Dennis, J.M. Martínez, R.A. Tapia and K. Williamson, “An algorithm based on a convenient trust-region subproblem for nonlinear programming,” Technical report, Department of Mathematical Sciences, Rice University (Houston, TX, 1990), in preparation.

    Google Scholar 

  2. M.R. Celis, J.E. Dennis and R.A. Tapia, “A trust region strategy for nonlinear equality constrained optimization,” in: P.T. Boggs, R.H. Byrd and R.B. Schnabel, eds.,Numerical Optimization (SIAM, Philadelphia, 1985) pp. 71–82.

    Google Scholar 

  3. T.F. Coleman and A.R. Conn, “On the local convergence of quasi-Newton methods for the nonlinear programming problems,”SIAM Journal of Numerical Analysis 21 (1984) 755–769.

    Google Scholar 

  4. A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968).

    Google Scholar 

  5. D.M. Gay, “Computing optimal locally constrained steps,”SIAM Journal Scientific and Statistical Computing 2 (1981) 186–197.

    Google Scholar 

  6. M.D. Hebden, “An algorithm for minimization using exact second derivatives,” Technical Report TP 515, A.E.R.E. (Harwell, England, 1973).

    Google Scholar 

  7. J.J. Moré, “Recent developments in algorithms and software for trust region methods,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming Study, the State of the Art (Springer, Berlin, 1983) pp. 258–287.

    Google Scholar 

  8. J.J. Moré and D.C. Sorensen, “Newton's method,” in: Gene H. Golub, ed.,Studies in Numerical Analysis, MAA Studies in Mathematics No. 24 (The Mathematical Association of America, 1984) pp. 29–82.

  9. J. Nocedal and M. Overton, “Projected Hessian updating algorithms for nonlinear constrained optimization,”SIAM Journal of Numerical Analysis 22 (1985) 821–850.

    Google Scholar 

  10. J.M. Otega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).

    Google Scholar 

  11. M.J.D. Powell, “Variable metric methods for constrained optimization,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming Study, the State of the Art (Springer, Berlin, 1983) pp. 288–311.

    Google Scholar 

  12. M.J.D. Powell and Y. Yuan, “A trust region algorithm for equality constrained optimization,” Technical Report NA2, DAMPT, University of Cambridge (Cambridge, UK, 1986).

    Google Scholar 

  13. C.H. Reinsch, “Smoothing by spline functions, II,”Numerische Mathematik 16 (1971) 451–454.

    Google Scholar 

  14. D.C. Sorensen, “Newton's method with a model trust region modification,”SIAM Journal of Numerical Analysis 19 (1982) 409–426.

    Google Scholar 

  15. Y. Yuan, “On a subproblem of trust region algorithms for constrained optimization,”Mathematical Programming 47 (1990) 53–63.

    Google Scholar 

  16. Y. Yuan, “A dual algorithm for minimizing a quadratic function with two quadratic constraints,” Technical Report NA3, DAMPT, University of Cambridge (Cambridge, UK, 1988).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maryland, 21228, Baltimore, MD, USA

    Yin Zhang

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  1. Yin Zhang
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Additional information

This research was performed while the author was on a postdoctoral appointment in the Department of Mathematical Sciences, Rice University, Houston, TX, USA and was supported in part by AFOSR 85-0243 and DOE DEFG05-86ER 25017.

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Zhang, Y. Computing a Celis-Dennis-Tapia trust-region step for equality constrained optimization. Mathematical Programming 55, 109–124 (1992). https://doi.org/10.1007/BF01581194

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  • DOI: https://doi.org/10.1007/BF01581194

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