Optimization Methods. Final Examination. 1. There are 5 problems each w i t h 20 p o i n ts for a maximum of 100 points.

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1 5.93 Optimization Methods Final Examination Instructions:. There are 5 problems each w i t h 2 p o i n ts for a maximum of points. 2. You are allowed to use class notes, your homeworks, solutions to homework exercises, and the book by Bertsimas and Tsitsiklis. You are not allowed to use any other book. 3. You have 3 hours to work in the examination. 4. Please explain your work carefully. 5. G ood luck!

2 Problem (2 points) Please answer true or false. No explanation is needed. A correct answer is worth 2 points, no answer points, a wrong answer -.. A problem of maximizing a convex, piecewise-linear function over a polyhedron can be formulated as a linear programming problem. 2. The dual of the problem min x subject to x =, x x 2 has a nondegenerate optimal solution. 3. If there is a nondegenerate optimal basis, then there is a unique optimal basis. 4. An optimal basic feasible solution is strictly complementary. 5. The convergence of the primal-dual barrier interior point algorithm is aected by degeneracy. 6. Given a local optimum x for a nonlinear optimization problem it always satises the Kuhn-Tucker conditions when the gradients of the tight constraints and the gradients of the equality constraints at the point x are linearly independent. 7. In a linear optimization problem with multiple solutions, the primaldual barrier algorithm always nds an optimal basic feasible solution. 8. In the minimum cost ow problem with integral data all basic feasible solutions have i n tegral coordinates. 9. The convergence of the steepest descent method for quadratic problems min f (x) = x Qx highly depends on the condition number of the matrix Q. The larger the condition number the slower the convergence.. The convergence of the steepest descent method highly depends on the starting point. 2

3 Problem 2 (2 points) 3 Let f () be a concave function. Consider the problem n Let P = fx 2 R j Ax = b x g: X n minimize f (x j ) j= subject to Ax = b x : (a) Prove that if there exists an optimal solution, there exists an optimal solution which is an extrem e point of P. (b) Suppose we n o w add the constraints that x j 2 f g for all j, i.e., the problem becomes a - nonlinear integer optimization problem. Show that the problem can be reformulated as a linear integer optimization problem. Problem 3 (2 points) Let Q be nn matrices. The matrix is positive semidenite. minimize c subject to d x + (a) Write the Kuhn-Tucker conditions. x + x Qx 2 x 2 x a: (b) Propose an algorithm for the problem based on the Kuhn-Tucker conditions. (c) Suppose the matrix Q is also positive semidenite. Reformulate the problem as a semidenite optimization problem.

4 4 Problem 4 (2 points) (a) You are given points (x i a i), i = : : : m where x i are vectors in < n and a i are either or. The interpretation here is that point x i is of category or. We w ould like to decide whether it is possible to separate the points x i by a h yperplane f x = such that all points of category satisfy f x and all points of category satisfy f x >. Propose a linear optimization problem to nd the vector f. (b) You are given points (x i y i), i = : : : m, where x i are vectors in < n and y i 2 < are response variables. We w ould like to nd a hyperplane a x =, such that for all points a x i, y i x i, while if a x i >, yi 2 x i. More formally, for all those points x i with a x i, we will choose in order to minimize X jy i ; x i j i: a x i while for all those points x i with a x i >, we w ill choose 2 in order to minimize X i: a x i > jy i ; 2 x i j: Propose an integer programming problem to nd the vectors a 2.

5 Problem 5 (2 points) Consider the problem Z = min c x s:t: a x b a 2 x b 2 n x 2 f g with a a 2 c : We denote by Z LP the value of the LP relaxation. (a) Consider the relaxation: Z = m a x min c x ; (b 2 ; a 2 x) s:t: a x b x 2 f g n Indicate how y ou can compute the value of Z. What is the relation among Z Z Z LP. Please justify your answer. (b) Propose a dynamic programming algorithm for computing Z : 5

6 MIT OpenCourseWare J / 6.255J Optimization Methods Fall 29 For information about citing these materials or our Terms of Use, visit: