• April 21, 2017
• Data Processing

By L. E. Scales

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Extra info for Introduction to Non-Linear Optimization

Sample text

The initial point is x 0 , the search vector is p0 and the linear minimum becomes the new point x 1 . 2 FUNCTION COMPARISON METHODS We assume that an interval [a, b] is known which brackets a local minimum x*, that is, a ~x* ~b. This is called an interval of uncertainty. 4. By evaluating the objective function at points over the interval it is possible to obtain a smaller interval of uncertainty. Continuing along these lines leads to an iterative process by which the interval of uncertainty may be made arbitrarily small.

3. 3: Quadratic interpolation 2 3 input a 1 , b 1 , c 1 , xtol,ftol set Fa= F(aJ), Fb = F(bJ), Fe= F(cJ) for k = 1 , 2, ... repeat se t xA*_ - -1 (bi-ci)Fa + (ci-ai)Fb + (ai-bl)Fe 2 (bk-ck)Fa + (ck-ak)Fb + (ak-bk)Fe 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ~~~~~~~~~~~~~~ Fx = F(x*) if x* < ck and Fx ck and Fx >Fe then set ak+l = ak, bk+l = x*' ck+l = ck Fb =- Fx else if x* < ck and Fx >Fe then set ak+l = x*' bk+l = bk, ck+l = ck Fa= Fx else set ak+l = ck, bk+l = bk, ck+l = x* Fa =Fe, Fe= Fx end end until bk+l - ak+t < xtol or (F(ck)- F(ck+d)/F(ck) <[to/ Line 1' 19 6-8 12-14 Comments The iterations terminate either when the interval of uncertainty has fallen below xtol or when the relative change in function value is less than [tal.

An algorithm can at best be shown to exhibit global convergence to a local minimum on a general, sufficiently smooth, non-linear function. Detailed proofs of convergence are omitted from this book, We shall imply convergence where possible by establishing that a sufficient reduction in function value can be obtained at each iteration until the solution is found. 2 Rate of convergence Given that a method is stable, it is important to have a theoretical indication of the rate of convergence to a solution since this may be so slow as to render the method of little practical value.