Part Four

1. Part Four Optimization

2. Motivation Given a function f(x), find an x0 where f(x) is the maximum or minimum.Noncomputer method:derive the exact formula of the first derivative f’(x) using calculus and then solve the equation f’(x) = 0. Question: How about find (x1, x2, x3, x4, x5) such that the following function is minimized?

3. Example: Optimization of Parachute Cost For each chute: A = 2πr2l = √2rc = kcA m = Mt/n Cost = c0 + c1l + c2A2 Task: Given the total weight Mt, the impact speed limit vc, and the initial height z0, determine the size (r) and the number of chutes (n) such that the total costn(c0 + c1l + c2A2) is the minimum while the impact speed of the chutes when reaching the ground is less than vc. MinimizeC = n(c0 + c1l + c2A2) Subject to v≤ v0 and n ≥ 1 where A = 2πr2l = √2r c = kcA m = Mt/n t = root [ z0 – gmt/c + (gm2/c2)(1-e-(c/m)t)] v = (gm/c)(1-e-(c/m)t)

4. Mathematical Background Optimization or mathematical programming problem Find x, which minimizes or maximizes f(x) Subject to di(x) ≤ ai i = 1, 2, …, m ci(x) = bi i = 1, 2, …, p where x: an n-dimensional design vector f(x): the objective function di(x): inequality constraints ci(x): equality constraints • If f(x) and the constraints are linear, it is linear programming. • If f(x) is quadratic and the constraints are linear, it is quadratic programming. • Otherwise, it is non-linear programming.

5. Overall Structure