2. (20 points) Wolfe conditions. Consider the following function f(x)=500-x(x-20) (a) Is the function f(x) concave?...

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2. (20 points) Wolfe conditions. Consider the following function f(x)=500-x(x-20)³ (a) Is the function f(x) concave? (One way to think about it is that if -f(r) is convex, then f(x) is concave). (b) Use steepest descent with line search to find an optimal solution to the following problem. max f(x)=500-r(x-20)³ Your implementation of line search should include the Wolfe conditions for finding a good step size. Report the optimal value of f(x) and the value z that achieves it. A few notes on this problem. This is a maximization problem, so you want to minimize -f(x). You should write a Julia/Python function called line search() to implement the Wolfe conditions for step size. This function is called from steepest.descent () and it will return a step size a. The signature for my implementation looks like this: function line_search(f, Vf, x, p; c1-1e-4, c2-0.9, rho=0.75) Also, the function f(x) has a saddle point, and steepest descent is attracted to saddle points. So, be sure to experiment with different starting values so that you get the minimum. It is a good idea to plot this function. 2. (20 points) Wolfe conditions. Consider the following function f(x)=500-x(x-20)³ (a) Is the function f(x) concave? (One way to think about it is that if -f(r) is convex, then f(x) is concave). (b) Use steepest descent with line search to find an optimal solution to the following problem. max f(x)=500-r(x-20)³ Your implementation of line search should include the Wolfe conditions for finding a good step size. Report the optimal value of f(x) and the value z that achieves it. A few notes on this problem. This is a maximization problem, so you want to minimize -f(x). You should write a Julia/Python function called line search() to implement the Wolfe conditions for step size. This function is called from steepest.descent () and it will return a step size a. The signature for my implementation looks like this: function line_search(f, Vf, x, p; c1-1e-4, c2-0.9, rho=0.75) Also, the function f(x) has a saddle point, and steepest descent is attracted to saddle points. So, be sure to experiment with different starting values so that you get the minimum. It is a good idea to plot this function.

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1 a To determine if the function fx is concave we can compute its second derivative fx 500 x x 20 3 ... View the full answer