Abstract
The trajectory following method is a relatively old concept for solving the nonlinear programming problem. It is based on the idea of solving continuous differential equations whose equilibrium solutions satisfy the necessary conditions for a minimum. The method is “trajectory following” in the sense that an initial point is moved along a trajectory generated by the differential equations to the solution point. The fact that this method has not been popular is understandable. It may require many function evaluations for a solution and it may suffer from the same slow convergence difficulties that the “gradient method” is known to have. However with the advent of fast computers and efficient integration solvers, these difficulties may no longer exist for many problems of practical interest. Here the focus is on ease of use rather than number of function evaluations as a performance criteria. It is shown that the trajectory following method, used in conjunction with a good “stiff” integration routine, can provide a tool that is easy to use for solving a wide variety of optimization problems including linear programming, quadratic programming and global optimization problems.
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References
Arrow, K.J. and Hurwicz, L. (1977), Gradient methods for constrained maxima, in Studies in Resource Allocation Processes, pp. 146–153, Cambridge University Press, Cambridge.
Courant, R., (1943), Variational methods for the solution of problems of equilibrium and vibrations, Bull. Am. Math. Soc., Vol. 49, pp. 1–23.
Fiacco, A.V. and McCormick, G.P. (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York.
Goh, B.S. (1978), Global convergence of some differential equations algorithms for solving equations involving positive variables, BIT, Vol. 18, pp. 84–90.
Grace, A. (1992), Optimization Toolbox for Use with MATLAB, The Mathworks, Inc., Matick, MA.
Kelly, H.J. (1962), Method of gradients, in Optimization Techniques, pp. 205–254, Academic Press, New York.
Liu, Y. and Teo, K.L. (1999), A bridging method for global optimization, In Review.
Reklaitis, G.V.A, Ravindran, V.A. and Ragsdell, K.M. (1983), Engineering Optimization Methods and Applications, Wiley, New York.
Yan, W.Y., Teo, K.L. and Moore, J.B. (1994), A gradient flow approach to computing (LQ) optimal output feedback gains, Optimal Control Applications and Methods, Vol. 15, pp. 67–75.
Vincent, T.L. and Grantham, W.J. (1981), Optimality in Parametric Systems, Wiley-Interscience, New York.
Vincent, T.L. and Grantham, W.J. (1997), Nonlinear and Optimal Control Systems, Wiley, New York.
Vincent, T.L. and Teo, K.L. (1999), Global optimization using trajectory following, In Review.
Zang, Z. and Vincent, T.L. (1999), Continuous-time envelope-constrained filter design, In Review.
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© 2000 Kluwer Academic Publishers
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Vincent, T.L. (2000). Optimization by Way of the Trajectory Following Method. In: Yang, X., Mees, A.I., Fisher, M., Jennings, L. (eds) Progress in Optimization. Applied Optimization, vol 39. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0301-5_16
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DOI: https://doi.org/10.1007/978-1-4613-0301-5_16
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