A manufacturer of lawn furniture makes two types of lawn chairs, one with a wood frame and one with a tubular aluminum frame. The wood-frame model costs $18 per unit to manufacture, and the aluminum-frame model costs $10 per unit. The company operates in a market where the number of units that can be sold depends on the price. It is estimated that in order to sell x units per day of the wood-frame model and y units per day of the aluminum-frame model, the selling price cannot exceed 10+31x−0.5+1.3y−0.2 dollars per unit for wood-frame chairs, and 5+15y−0.4+0.8x−0.08 dollars per unit for the aluminum-frame chairs. Find the optimal production levels. Suppose we are in the optimization setting and wish to find the minimum of a real-valued function F(x1,x2,…,xn) over the domain (x1,x2,…,xn)∈R^n. If we have an initial guess, x0∈Rn, we can iteratively update this guess according to xj+1=xj−γ∇F(xj). Here γ>0 is a fixed constant that you get to pick, referred to as the step size. a) ing your knowledge of the geometric interpretation of the gradient of a function, explain how gradient descent is moving you towards a minimum of
F. b) How would you augment the gradient descent procedure if you wanted to maximize F instead of minimize it? (This would be called gradient ascent) c) ing your answer to part
(b), go back to Example 2.12 from the notes and e gradient descent to find the maximum of the given function. Perform the iteration with three different step sizes: γ=0.1,γ=1, and γ=10. Comment on what happens with different values of γ.