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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.

At point 1, f(x) is greater than f at points .2 or 3. fixed for a given size simplex. Let us use a function of two variables to illustrate the procedure. At each iteration, to minimize f(x), f(x) is evaluated at each of three vertices of the triangle. The direction of search is oriented away from the point with the high-

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