1. Given the following optimization problem - Minimize f(x) = (x 1) + (x 1) subject to g(x) = x + x 1 0 9(x) = x 0 (a) Plot the constrained optimization problem, including contours for F=1, F=4, F=9, F=16. Identify the constraints and the feasible region. Using the plot, identify the optimum. (b) Plot the pseudo-objective function contours F=1, F=4, F=9, F-16 for the exterior penalty function method when R=1. Identify the minimum of the pseudo- objective function on the plot. (c) Plot the pseudo-objective function contours F=1, F=4, F=9, F-16 for the exterior penalty function method when R=10. Identify the minimum of the pseudo- objective function on the plot. (d) Plot the pseudo-objective function contours F=1, F=4, F-9, F-16 for the exterior penalty function method when R=100. Identify the minimum of the pseudo- objective function on the plot. (e) Using the initial point (0, 0), perform two cycles of exterior penalty function. For n-D unconstrained optimization use DFP. (f) Using the initial point (1, -1), perform two cycles of interior penalty function (using the inverse penalty). For n-D unconstrained optimization use DFP. (g) Using the initial point (0, 0), perform two cycles of linear extended interior penalty function method. For n-D unconstrained optimization use DFP. (h) Using the initial point (0, 0), perform two cycles of the augmented lagrangian method presented in class. For n-D unconstrained optimization use DFP. 2. Solve the following optimization problem using fminbnd function of matlab Minimize f(x) = (x 1) 3. Solve the Arora 10.60 unconstrained optimization problem using fminunc function of matlab. 4. Solve the following optimization problem using fmincon function of matlab. Using the initial point (1, -1) and (0, 0) and compare the results. Minimize f(x) = (x - 1) + (x - 1) subject to g(x) = x + x 1 0 9(x) = x 0