UNCONSTRAINED MULTIVARIABLE OPTIMIZATION

UNCONSTRAINED MULTIVARIABLE OPTIMIZATION .. methods Using Function Values Only 183 .. methods That Use First derivatives 189 .. Newton's Method 197 .. Quasi-Newton methods 208 References .. 210 .. Supplementary References 211 Problems .. 211 182 PART I1 : OPTIMIZATION Theory and methods THE NUMERICAL OPTIMIZATION of general nonlinear MULTIVARIABLE objective func- tions requires efficient and robust techniques. Efficiency is important because these problems require an iterative solution procedure, and trial and error becomes impractical for more than three or four variables. Robustness (the ability to achieve a solution) is desirable because a general nonlinear function is unpredictable in its behavior; there may be relative maxima or minima, saddle points, regions of con- vexity, concavity, and so on. In some regions the OPTIMIZATION algorithm may progress very slowly toward the optimum, requiring excessive computer time.

this may require more computer time than finite differencing to get derivatives. For nonsrnooth functions, a function-values-only method may. be more successful than using a derivative-based method. We first describe some simple nonderivative methods and then present a series of methods that use derivative information. We

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  Series, Methods, Derivatives

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