368 Nonlinear Programming II: Unconstrained Optimization Techniques
6.30 A. R. Colville,A Comparative Study of Nonlinear Programming Codes, Report 320-2949,
IBM New York Scientific Center, 1968.
6.31 E. D. Eason and R. G. Fenton, A comparison of numerical optimization methods for
engineering design,ASME Journal of Engineering Design, Vol. 96, pp. 196–200, 1974.
6.32 R.W.H. Sargent and D. J. Sebastian, Numerical experience with algorithms for uncon-
strained minimization, pp. 45–113 inNumerical Methods for Nonlinear Optimization,
F. A. Lootsma, Ed., Academic Press, London, 1972.
6.33 D. F. Shanno, Recent advances in gradient based unconstrained optimization techniques
for large problems,ASME Journal of Mechanisms, Transmissions, and Automation in
Design, Vol. 105, pp. 155–159, 1983.
6.34 S. S. Rao,Mechanical Vibrations, 4th ed., Pearson Prentice Hall, Upper Saddle River,
NJ, 2004.
6.35 R. T. Haftka and Z. Gurdal, ̈ Elements of Structural Optimization, 3rd ed., Kluwer Aca-
demic, Dordrecht, The Netherlands, 1992.
6.36 J. Kowalik and M. R. Osborne,Methods for Unconstrained Optimization Problems, Amer-
ican Elsevier, New York, 1968.Review Questions
6.1 State the necessary and sufficient conditions for the unconstrained minimum of a function.
6.2 Give three reasons why the study of unconstrained minimization methods is important.
6.3 What is the major difference between zeroth-, first-, and second-order methods?
6.4 What are the characteristics of a direct search method?
6.5 What is a descent method?
6.6 Define each term:
(a)Pattern directions
(b)Conjugate directions
(c)Simplex
(d)Gradient of a function
(e)Hessian matrix of a function6.7 State the iterative approach used in unconstrained optimization.
6.8 What is quadratic convergence?
6.9 What is the difference between linear and superlinear convergence?
6.10 Define the condition number of a square matrix.
6.11 Why is the scaling of variables important?
6.12 What is the difference between random jumping and random walk methods?
6.13 Under what conditions are the processes of reflection, expansion, and contraction used in
the simplex method?
6.14 When is the grid search method preferred in minimizing an unconstrained function?
6.15 Why is a quadratically convergent method considered to be superior for the minimization
of a nonlinear function?