an affine space with a cylinder.2 What is the new value of zmax and zmin, if the cylinder increases its radius in a small value, that is, if the radius changes from (r = 1) to (r = 1 + Δr) (Consider the interpretation of the Lagrange multipliers).
3 The following algebraic equation gives the mechanical power in a wind turbine: (2.59)where P is the power extracted from the wind; ρ is the air density; Cp is the performance coefficient or power coefficient; λ is the tip speed ratio; v is the wind velocity, and A is the area covered by the rotor (see [15] for details). Determine the value of λ that produce maximum efficiency if the performance coefficient is given by (Equation 2.60): (2.60)Use the gradient method, starting from λ = 10 and a step of t = 0.1. Hint: use the module SymPy to obtain the expression of the gradient.
4 Solve the following optimization problem using the gradient method: (2.61)Depart from the point (0, 0) and use a fixed step t = 0.8. Repeat the problem with a fixed step t = 1.1. Show a plot of convergence.
5 Solve the following optimization problem using the gradient method. (2.62)where 1n is a column vector of size n, with all entries equal to 1; b is a column vector such that bk = kn2; and H is a symmetric matrix of size n × n constructed in the following way: hkm = (m + k) / 2 if k ≠ m and hkm = n2 + n if k = m. Show the convergence of the method for different steps t and starting from an initial point x = 0. Use n = 10, n = 100, and n = 1000. All index k or m starts in zero.
6 Show that Euclidean, Manhattan, and uniform norms fulfill the four conditions to be considered a norm.
7 Consider a modified version of Example 2.6, where the position of the common point E must be such that xE = yE. Solve this optimization problem using Newton’s method.
8 Solve the problem of Item 4 with the following constraint (use Newton’s method): (2.63)
9 Solve problem of Item 5 including the following constraint (use Newton’s method): (2.64)
10 Newton’s method can be used to solve unconstrained optimization problems. Solve the following problem using Newton’s method and compare the convergence rate and the solution with the gradient method. (2.65)
Notes
1 1 At this point, the only tool we have to check these results is plotting the function and locating the optimum.
2 2 Notice P is defined in a line outside the function definition. Recall that x2 is represented as x**2 in Python (see Appendix C)
3 3 A complete discussion about the calculation of t is beyond this book’s objectives. Interested readers can consult the work of Nesterov and Nemirovskii, in [11] and [12].
4 4 Again, a classic method that became more important with the advent of the computer.
5 5 The jacobian matrix of S is equivalent to the hessian matrix of f.
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