Question 10.4: Consider the objective function f(C) = e^-c cos C defined on......
Consider the objective function\ f(C) = e^{-c} cos C defined on the uncertainty interval 0 ≤ C ≤ π. We know from Example 10.3 that the objective is convex on the given initial interval, so the convexity theorem guarantees a global minimum somewhere on the interval [0, π]. Find\ C_{min} using the thirds algorithm.
Applying the algorithm in Listing 10.2 gives the data listed in Table 10.2. It should be noted that the intervals of uncertainty are indeed nested and that the lengths of uncertainty also decrease monotonically. As expected, the sequence converges to the actual minimum at\ C_{min} = \frac{3}{4}\pi \mu \approx 2.356.
TABLE 10.2 Results of the thirds Sequential Line Search of Example 10.3
\ \begin{array}{c} \hline i&&a_i&&b_i&&\lambda_i&&\mu_i&&b_i-a_i\\\hline 0&&0.0000&&3.1416&&1.0472&&2.0944&&3.1416\\1&&1.0472&&3.1416&&1.7453&&2.4435&&2.0944\\2&&1.7453&&3.1416&&2.2107&&2.6762&&1.3963\\3&&1.7453&&2.6762&&2.0556&&2.3659&&0.9308\\4&&2.0556&&2.6762&&2.2625&&2.4693&&0.6206\\5&&2.0556&&2.4693&&2.1935&&2.3314&&0.4137\\6&&2.1935&&2.4693&&2.2854&&2.3774&&0.2758\\7&&2.2854&&2.4693&&2.3467&&2.4080&&0.1839\\8&&2.2854&&2.4080&&2.3263&&2.3672&&0.1226\\9&&2.3263&&2.4080&&2.3535&&2.3808&&0.0817 \end{array} \\ \begin{array}{c} \hline &C(approximate)=\frac{1}{2}(a+b)=2.3672&&&C(actual)=\frac{3}{4}\pi \approx 2.3562\\\hline \end{array}
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-given formulas and data f (C) = “objective function” -optimization procedure \ \lambda = \frac{1}{3}(2a+b) if \ f_{left} < f_{right} then b = μ end while LISTING 10.2 The thirds algorithm for a sequential line search. |