Problem-1

1.1. Write a program that will find the maxima of the following function using Golden section search algorithm starting from xl = -4 and x_u = 1. Your function should take x_l , x_u, and number of iterations as input parameters. Also plot the graph estimated maxima Vs. iteration number and true error Vs. iteration number.

f(x) = x³ - 10 * x

1.2. Develop a bracketing algorithm to find the maxima of the above function that will generate two random candidate points x₁ and x₂ with in the bound x_l and x_u, such that x₁ is in the upper half of the bound and x₂ is in the lower half of the bound. Then after the algorithm will choose new bounds (x_lower and x_upper) exactly like in Golden search. Then again two new random candidate points will be generated with in the bound new xl and new xu and the process keeps going till a defined number of iterations.Your function should take x_l , x_u, and number of iterations as input parameters. Plot the graph estimated maxima Vs. iteration number and true error Vs. iteration number. Also state for this problem was Golden search better or the random bracket search better?

2a. Steepest ascent

Write a Mathematica code to find the maxima of the following multidimensional function using optimal steepest ascent technique. Your function should take the initial starting point and number of iterations as input parameters. Plot the function and the search path on a same graph. [Hint use ContourPlot to plot the function ListLinePlot to draw the search path over the contour]?

f(x, y) = 2xy + 2x - x² - 2y²

2b. Randomized Steepest ascent

Now repeat the above exercise by moving in the steepest ascent direction with a step size h, which will be user defined. Again plot the function and the search path on a same graph. Which technique you think converges faster to the maxima for this given problem?.

Extra credit

Given a sphere of radius 10 units with 3 particles that interact with each other with in the sphere. The energy of the system for a given configuration of these particles is given by

where r_ij represents the distance between the particles i and j.

Use Mathematica in-built optimization library NMinimize to find out the minimum energy configuration state of this system. Note the particles cannot go outside the sphere neither two particles can occupy the same position with in the sphere. Can you think of a practical application of this simple demonstration.