Algorithms for Minimization Without Derivatives
This outstanding text for graduate students and researchers proposes improvements to existing algorithms, extends their related mathematical theories, and offers details on new algorithms for approximating local and global minima. None of the algorithms requires an evaluation of derivatives; all depend entirely on sequential function evaluation, a highly practical scenario in the frequent event of difficult-to-evaluate derivatives.
Topics include the use of successive interpolation for finding simple zeros of a function and its derivatives; an algorithm with guaranteed convergence for finding a minimum of a function of one variation; global minimization given an upper bound on the second derivative; and a new algorithm for minimizing a function of several variables without calculating derivatives. Many numerical examples augment the text, along with a complete analysis of rate of convergence for most algorithms and error bounds that allow for the effect of rounding errors.
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ALGOL 60 ALGOL procedures ALGOL W approximation bisection Chapter Comp computed condition number Corollary defined Dekker's algorithm described in Section divided differences effect of rounding eigenvalues equation example Fibonacci search finding a zero Fletcher floating-point FMIN function evaluations required given in Section gives global minimum glomin2d golden section search Golub guaranteed Hessian matrix holds i5-unimodal interval iteration Jarratt Kowalik and Osborne Lemma linear searches Lipschitz Lipschitz continuous LONG REAL ARRAY macheps Math nonlinear number of function numerical results OO BEGIN order at least order of convergence Ostrowski 1966 polynomial Powell problem procedure glomin procedure zero quadratic convergence quadratic function real procedure result follows Rosenbrock rounding errors satisfies second derivative sequence shows simple zero singular value decomposition stopping criterion successive linear interpolation successive parabolic interpolation sufficiently superlinear convergence Suppose Table TEST Theorem 5.1 tion unimodal unimodal function upper bound variables weak order zero2
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