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ODE versus SQP methods for constrained optimization. (English) Zbl 0655.65092
We review some methods which are designed to solve equality constrained minimization problems by following the trajectory defined by a system of ordinary differential equations. The numerical performance of a number of these methods is compared with that of some popular sequential quadratic programming (SQP) algorithms. On a set of eighteen “difficult” test problems, we observe that several of the ODE methods are more successful than any of the SQP techniques. We suggest that these experimental results indicate the need for research both to analyze and develop new ODE techniques and also to strengthen the currently available SQP algorithms.
Reviewer: A.A.Brown

MSC:
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
Software:
dverk; NPSOL; VMCWD
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[1] Arrow, K. J., andHurwicz, L.,Reduction of Constrained Maxima to Saddle-Point Problems, Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, Edited by J. Neyman, University of California Press, Berkeley, California, pp. 1-20, 1966.
[2] Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York, New York, 1968. · Zbl 0193.18805
[3] Evtushenko, Ju. G.,Two Numerical Methods for Solving Nonlinear Programming Problems, Soviet Mathematical Doklady, Vol. 15, pp. 420-423, 1974. · Zbl 0314.90083
[4] Tanabe, K.,A Geometric Method in Nonlinear Programming, Report No. 23343-AMD-780, Brookhaven National Laboratory, 1977. · Zbl 0336.65023
[5] Tanabe, K.,Differential Geometric Approach to Extended GRG Methods with Enforced Feasibility in Nonlinear Programming: Global Analysis, Report No. 24497-AMD-797, Brookhaven National Laboratory, 1978. · Zbl 0507.90079
[6] Tanabe, K.,Differential Geometric Methods in Nonlinear Programming, Report No. 26730-AMD-831, Brookhaven National Laboratory, 1979. · Zbl 0445.90069
[7] Evtushenko, Yu. G., andZhadan, V. G.,A Relaxation Method for Solving Problems of Nonlinear Programming, USSR Computational Mathematics and Mathematical Physics, Vol. 17, pp. 73-87, 1977. · Zbl 0397.90089 · doi:10.1016/0041-5553(77)90105-7
[8] Botsaris, C. A.,A Newton-Type Curvilinear Search Method for Constrained Optimization, Journal of Mathematical Analysis and Applications, Vol. 69, pp. 372-397, 1979. · Zbl 0408.65040 · doi:10.1016/0022-247X(79)90150-1
[9] Botsaris, C. A.,Constrained Optimization along Geodesics, Journal of Mathematical Analysis and Applications, Vol. 79, pp. 295-306, 1981. · Zbl 0477.49018 · doi:10.1016/0022-247X(81)90026-3
[10] Rosen, J. B.,Discussion Session on Unconstrained Minimization, Nonlinear Optimization, Edited by M. J. D. Powell, Academic Press, London, England, pp. 59-60, 1982.
[11] Brown, A. A., andBartholomew-Biggs, M. C.,Some Effective Methods for Unconstrained Optimization Based on the Solution of Systems of Ordinary Differential Equations, Technical Report No. 178, Numerical Optimization Centre, Hatfield Polytechnic, 1986. · Zbl 0651.90067
[12] Maany, Z. A.,The Optimization of Low Thrust Satellite Trajectories, PhD Thesis, Hatfield Polytechnic, 1986. · Zbl 0711.70034
[13] Courant, R.,Variational Methods for the Solution of Problems of Equilibrium and Vibration, Bulletin of the American Mathematical Society, Vol. 49, pp. 1-23, 1943. · Zbl 0063.00985 · doi:10.1090/S0002-9904-1943-07818-4
[14] Powell, M. J. D.,Variable Metric Methods for Constrained Optimization, Mathematical Programming (The State of the Art), Edited by A. Bachem, M. Grotschel, and B. Korte, Springer-Verlag, New York, New York, pp. 288-311, 1983.
[15] Abadie, J., andCarpentier, J.,Generalization of the Wolfe Reduced Gradient Method to the Case of Nonlinear Constraints, Optimization, Edited by R. Fletcher, Academic Press, London, England, pp. 37-48, 1969.
[16] Brown, A. A.,Optimization Methods Involving the Solution of Ordinary Differential Equations, PhD Thesis, Hatfield Polytechnic, 1986.
[17] Hindmarsh, A. C.,LSODE and LSODI: Two New Initial-Value Ordinary Differential Equation Solvers, ACM Signum Newsletter, Vol. 15, pp. 10-11, 1980.
[18] Hull, T. E., Enright, W. H., andJackson, K. R.,User’s Guide to DVERK: A Subroutine for Solving Nonstiff ODEs, Technical Report No. 100, Department of Computer Science, University of Toronto, 1976.
[19] Hock, W., andSchittkowski, K.,Test Examples for Nonlinear Programming Codes, Springer-Verlag, New York, New York, 1981. · Zbl 0452.90038
[20] Brown, A. A., andBartholomew-Biggs, M. C.,ODE vs SQP Methods for Constrained Optimization, Technical Report No. 179, Numerical Optimization Centre, Hatfield Polytechnic, 1987. · Zbl 0655.65092
[21] Powell, M. J. D.,A Fast Algorithm for Nonlinearly Constrained Optimization Calculations, Report No. DAMTP-77-NA2, University of Cambridge, England, 1977. · Zbl 0374.65032
[22] Biggs, M. C.,Constrained Minimization Using Recursive Equality Quadratic Programming, Toward Global Optimization, Edited by L. C. W. Dixon and G. P. Szego, North-Holland, Amsterdam, Holland, pp. 341-349, 1975.
[23] Gill, P. E., Murray, W., Saunders, M. A., andWright, M. H.,User’s Guide for SOL/NPSOL: A FORTRAN Package for Nonlinear Programming, Report No. SOL-83-12, Department of Operations Research, Stanford University, 1983.
[24] Powell, M. J. D.,VMCWD: A FORTRAN Subroutine for Constrained Optimization, Report No. DAMTP-82-NA4, University of Cambridge, 1982.
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