HW4

IE 312 Optimization Homework 4 Cover Sheet Instructor name: Danial Davarnia Team members: • Date assigned: Tuesday 10/24/2023 • Date due: Tuesday 11/8/2023, 5:00 PM • Homework submission must be made on Canvas. Email, hardcopy, or any other form of submission cannot be accepted. • This cover sheet must be filled out by all team members and submitted along with the homework answers on additional sheets. • Only one submission per team is required. • By submitting this homework with my name affixed above, – I understand that homework submitted at 5:01 PM or later will not be accepted, – I acknowledge that I am aware of the Iowa State University policy concerning academic misconduct (appended below), – I attest that the work I am submitting for this homework assignment is solely my own, and – I understand that suspiciously similar homework submitted by multiple individuals will be reported to the Dean of Students Office for investigation. • Academic Misconduct in any form is in violation of Iowa State University Student Disciplinary Regulations and will not be tolerated. This includes, but is not limited to: copying or sharing answers on tests or assignments, plagiarism, having someone else do your academic work or working with someone on homework when not permitted to do so by the instructor. Depending on the act, a student could receive an F grade on the test/assignment, F grade for the course, and could be suspended or expelled from the University. See the Conduct Code at www.dso.iastate.edu/ja for more details and a full explanation of the Academic Misconduct policies. 1

Some of the following questions are selected from the textbook “Introduction to Mathematical Programming.” For students that do not have access to the book, the pages that contain these questions are enclosed. 1. (10 points) Solve the following quadratic program using the Matlab quadprog function. Sub- mit the screenshots of your Matlab code and output. max 3 x 2 1 - 2 x 1 x 3 - 8 x 2 2 + 5 x 2 x 3 - 5 x 2 3 - 4 x 1 + 94 x 3 s . t . 3 x 1 - 6 x 2 + 4 x 3 ≤ 7 - x 1 - 3 x 2 + 4 x 3 = 10 5 x 1 - 4 x 2 + x 3 ≥ - 1 x 1 , x 2 , x 3 ≥ 0 2. (10 points) Page 502, Problem 2. Provide complete model formulation (including definition of variables), optimal solution, optimal objective value, and screenshots of Matlab intlinprog code and output. 3. (10 points) Page 503, Problem 6. Provide complete model formulation (including definition of variables), optimal solution, optimal objective value, and screenshots of Gusek code and output. 4. (10 points) Page 505, Problem 21. Provide complete model formulation (including definition of variables), optimal solution, optimal objective value, and screenshots of Gusek code and output. 5. Consider the following LP. max ζ = 3 x 1 + 2 x 2 (1) s . t . x 1 + 3 x 2 ≤ 6 (2) - x 1 + x 2 ≥ - 2 (3) x 1 , x 2 ≥ 0 . (4) a. (10 points) The optimal solution of (1)-(4) is x * 1 = 3 and x * 2 = 1. The objective function of the dual problem is { min 6 y 1 - 2 y 2 } . Use the weak and strong duality theorems to determine whether the following points are feasible, infeasible or optimal for the dual problem (do not use the full dual formulation). (a) y 1 = 0 and y 2 = 0. (b) y 1 = 3 and y 2 = 0. (c) y 1 = 1 . 25 and y 2 = - 1 . 75. (d) y 1 = 2 . 5 and y 2 = - 5 . 5. b. (5 points) What is the range of the parameter “2” in the objective function (1) that makes the optimal solution still optimal? c. (5 points) If the parameter “6” in Constraint (2) is decreased to 5.9, what would be the new optimal objective value? Derive your answer using the sensitivity analysis approach based on shadow price. 2