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.
Article PDF
Similar content being viewed by others
References
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.
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.
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.
A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968).
D.M. Gay, “Computing optimal locally constrained steps,”SIAM Journal Scientific and Statistical Computing 2 (1981) 186–197.
M.D. Hebden, “An algorithm for minimization using exact second derivatives,” Technical Report TP 515, A.E.R.E. (Harwell, England, 1973).
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.
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.
J. Nocedal and M. Overton, “Projected Hessian updating algorithms for nonlinear constrained optimization,”SIAM Journal of Numerical Analysis 22 (1985) 821–850.
J.M. Otega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
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.
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).
C.H. Reinsch, “Smoothing by spline functions, II,”Numerische Mathematik 16 (1971) 451–454.
D.C. Sorensen, “Newton's method with a model trust region modification,”SIAM Journal of Numerical Analysis 19 (1982) 409–426.
Y. Yuan, “On a subproblem of trust region algorithms for constrained optimization,”Mathematical Programming 47 (1990) 53–63.
Y. Yuan, “A dual algorithm for minimizing a quadratic function with two quadratic constraints,” Technical Report NA3, DAMPT, University of Cambridge (Cambridge, UK, 1988).
Author information
Authors and Affiliations
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01581194