The Final Exam will have 6-7 questions and the total marks will be 50. It is a restricted open book exam. The permitted materials for the exam are: Calculators (including programmable) Drawing instruments i.e. Rulers, Set Squares and Compasses Two (2) revision sheet (A4 double-sided). They must be handwritten original notes. Below are some sample questions. Question 1: Use Lagrange Multiplier method to solve the nonlinear programming problem: Minimize 2 x y Subject to x 2 y 2 1 Question 2: In the Simplex Method for solving linear programming problem, when a corner point is found, how to decide whether it is the optimal solution or not? How to improve the solution if the corner point is not the optimal solution? Please explain the process and give reasons why it works. Question 3: Formulate the problem as a linear programming problem. Solve this linear programming problem graphically. 1 Question 4: It is required to find the rectangular of the largest area (in the positive quadrant) that x2 can be transcribed within a given ellipse 1 x 22 1 and satisfy a prescribed linear 4 constraint 4 x1 3x2 6 Formulate the problem as a nonlinear programming problem. Solve this nonlinear programming problem graphically. Question 5: Explain the main idea of BFGS algorithm for unconstrained nonlinear programming problems. Use the following example to compute the search direction in the first step of BFGS algorithm assuming the starting point is x1 4, x2 3. x12 Minimise x 22 2 x 2 4 Question 6: Explain the main idea of SQP algorithm for constrained nonlinear programming problems. Question 7: Explain the principle, advantages and limitations of Genetic algorithm for nonlinear optimisation. Question 8: Explain the key idea of golden section method for solving one dimensional nonlinear programming problem. Question 9: Explain the key idea of branch and bound algorithm. Question 10: Use FOC and SOC to solve the following unconstrained optimization problem Minimize 2