1. Solve the following linear programming (LP) problem using MATLAB's LP solver or any other code available in the internet: Maximize f(x) = 2x1 +52 +4.5x3 + 3x4, subject to 1 + 1.2x2 + 5x48, 2x1 + 1.4x2 +0.923 3, x1 + 1.5x3 + 3x4 6, x2 + 5x3 2x4 4, -1 +8.2x2 +7.5x3 15x4 = 16, xi20, i= 1, ... ,4. (a) In the first attempt, the student forgets the final statement on variable bounds. Find the solution x* and respective f*. (b) The student realizes that some variable values in Item 1 has negative values. So, the student cor- rects the problem description with the above-mentioned variable bounds and re-solve the problem. Compare the new solution with that in Item (a). (c) Next, the student realizes that x3 and 24 must take integer values. First, the student approximates Item (b) solution to nearest integer value for 3 and 4. Finally, the student uses MATLAB's intlinprog () or another suitable code from the internet, which can handle integer restrictions on some variables within LP methodology. What is the new solution and how does it compare with the approximated solution and Item (b) solution? 2. Use Matlab's fmincon routine to solve the following nonlinear optimization problem: Minimize subject to f(x1, x2) = (x-4) + (x2 -3) 91(x1, x2) = 6x1 + x 25, 92(x1, x2) = -x1 + x2 3, -2 x 5, -2 x5. Use a starting solution (1,0)7. Which constraints are active at the optimal solutions? 3. [Optional for Section 730 Students:] Solve the above problem using the penalty function method with a fixed penalty parameter (R = 10) using MATLAB's fminsearch () or any other unconstrained optimization code from the internet: J Minimize f(x1, x2)+ R(9j(x1, x2)), j=1 where (a)= a if a > 0; zero, otherwise. Normalize 9 and 92 by dividing the left side by the constant on the right side. For example, 91(x1, x2) = (6x1 + x2-25) /25. Each variable bound can be suitably normalized. For example, -2 5 can be replaced by two normalized constraint functions: 93 (1, *2)= (-1-2)/2 and 94(x1, x2) = (x - 5)/5. Compare the solution with that found in Q2 above.