IE 310 Operations ResearchFinal ExamFall 2014Name:UIN:There are eleven problems.You have until 10:00 PM to finish and submit your solution.Please write neatly.There are empty pages on the back for your convenience. You can rip them off and use them as scratch papers.Problem 1a4ptb4ptc4ptProblem 2a1ptb1ptc6ptd1pte4ptProblem 310ptProblem 4a5ptb5ptProblem 5a8ptb3ptProblem 6a3ptb5ptc3ptProblem 7a5ptb4ptProblem 810ptProblem 9a2ptb2ptc2ptd2pte2ptProblem 105ptProblem 11a10ptb5ptTotal116pt1

•Problem 1.[Ch12. Convexity] There are three definitions of convexity of a functionf:R→R:(i) for allλ∈[0,1] and¯λ= 1-λ, and allx∈Randy∈R,λf(x) +¯λf(y)≥f(λx+¯λy)(ii)f(y)≥f(x) +f0(x)(y-x)(iii)f00(x)≥0Solve the following sub-problems.(a) [4pt] Show whetherf(x) =|x|is convex, concave or neither using definition (i).(b) [4pt] Show whetherf(x) = logxis convex, concave or neither using definition (ii) (consider only the regimewherex >0).(c) [4pt] Show whetherf(x) =|x|pfor somep≥1 is convex, concave or neither using definition (iii). Forp <0 isthe function convex, concave or neither?2

•Problem 2.[Ch12. Equality constrained optimization] We consider the following equality constrained NLP:minimizex41+x42subject tox21+x22= 1(a) [1pt] Write the Lagrangian functionh(x1, x2, λ1) of this NLP.(b) [1pt] Write the gradient of the Lagrangian∇h(x1, x2, λ1) in terms ofx1, x2, λ1.(c) [6pt] Find all the stationary points of this NLP by solving∇h(x1, x2, λ1) = 0.(d) [1pt] Find the optimal value of the above minimization.(e) [4pt] Find the optimal value of the following variation, where we now want to maximize the objective:maximizex41+x42subject tox21+x22= 13

•Problem 3.[Ch11.Branch and Bound, Ch8.Assignment Problem] [10pt] We need to carry out a series ofoperations by a team. There are four members of the team:A,B,C, andDand four operations (1,2,3, and 4) tobe carried out. Each team member can carry out exactly one operation. All four operations must be carried outsuccessfully for the overall project to succeed, however the cost of a particular team member carrying out a particularoperation varies, as shown in the table below. For example, if the team members were assigned to operations inthe orderABCD, then the overall cost of the project is 1 + 2 + 3 + 4 = 10.On the top of the constraints thateach member must carry out only one operation, we have the following extra constraints: (i)Bmust carry out anoperation afterAhas finished, and (ii)Cmust carry out an operation afterAhas finished.1234A1356B3215C2431D4764MemberOperationsUse “Branch and Bound” to find the arrangement of the team that gives the lowest total cost. Write the branchingtree for your method, and write the the cost of the candidate solution for each node, and whether the proposedsolution is feasible or infeasible. Write the final sequence of members corresponding to the minimum cost solutionand the value of the minimum cost. We are providing the table multiple times so that you can use it as you wish. If

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