DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION Part I: Short Questions August 12, 2008 9:00 am - 12 pm General Instructions This examination is closed-book and consists of five equally-weighted questions. Do all five problems. Present your answers as clearly and concisely as you are able.
1. Consider the following LP: The points maximize 20x 1 + 32x 2 + 40x 3 subject to 3x 1 + 4x 2 + 6x 3 200 5x 1 + 6x 2 + 5x 3 260 8x 1 + 6x 2 + 5x 3 400 x 1, x 2, x 3 0 0 5 x = 35 y = 2 10 0 are an optimal primal-dual solution of the above LP. Without reconstructing the entire tableau: (a) Show that x and y are in fact optimal to their respective LPs. (b) Determine the range of values for the coefficient of x 1 in the objective (currently set at 20) such that x remains optimal. (c) Determine the range of values for the coefficient of x 1 in the second constraint (currently set at 5) such that x remains optimal. (d) Determine the range of values for the rhs in the third inequality (currently set at 400) such that y remains optimal. 2. A subvector of a vector a R n is a vector of the form (a k, a k+1,..., a l ), where k l. The subvector (a k, a k+1,..., a l ) is said to be increasing, if a i < a i+1 for all i with k i < l. You are given a R n with all components distinct. (a) Describe an O(n) dynamic programming algorithm to find the longest increasing subvector of a. Note that an O(n 2 ) algorithm is trivial to come up with, so such a solution will receive no credit. (b) Describe a directed graph, with O(n) nodes, in which finding longest paths will accomplish the same as running the algorithm in part (a).
3. Let n and k be positive integers with k n, and a R n with a 1 > a 2 > > a n. Denote by e the vector of all ones. Consider the LP min st. kz + e T u ze + u a u 0, z free. (P) (a) Write out the dual of (P). (b) Prove that every optimal solution of the dual has only integer components. (c) Determine explicitly the optimal value of (P). 4. Let S = {v 1,..., v m } be a set of vectors in R n. Show how one application of the Gauss-Jordan Reduction Method can be used to find a basis for both span(s) and null(s), and use this to show that dim(span(s))+dim(null(s)) = n. 5. Suppose that x 1,..., x 12 are 0 1 variables. (a) Using extra 0-1 variables, write constraints that will force the following: At least one out of and is true. x 1 + + x 5 2 (0.1) x 6 + + x 12 4 (0.2) (b) Using extra 0-1 variables, write constraints that will force the following: Exactly one out of (0.1) and (0.2) is true. For full credit, use the smallest possible big-m constants.
DEPARTMENT OF OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION Part II Long Questions August 12, 2008 1:00 pm - 4:30 pm General Instructions This examination is open-book and consists of two equally-weighted questions. You are to hand in the solutions to both problems. It is expected that your answers will be presented in a clear and concise form. Use of the internet is not permitted.
1. Let f be a concave differentiable function from R n to R, let g i, i = 1,..., m, be a set of convex differentiable functions from R n to R, and let b R m be an m-vector. Consider the nonlinear program (NLP b ) z b = max f(x) s.t. g i (x) b i, i = 1,..., m (a) Give the KKT conditions for a point x to be optimal to (NLP b ), using multipliers λ 1,..., λ m. (b) Now consider the nonlinear program (NLP c ) z c = max f(x) s.t. g i (x) c i, i = 1,..., m where c R m is unrelated to b. Give an upper bound for z c in terms of b, c, z b, and the associated multipliers λ i. Use the fact that a differentiable concave function f : R n R satisfies f(x) f( x) + (x x) f( x) and a differentiable convex function g : R n R satisfies g(x) g( x) + (x x) g( x) for each x R n, where denotes the gradient.
2. Consider a directed network G = (N, A) with m arcs, nonnegative capacities u e on each arc e A, and specified source and sink nodes s and t, respectively. We are interested in finding the maximum flow from s to t. To do this, let Γ be the set of p (s, t)-paths in G, and consider solving this problem by assigning flow f P to each of the p paths P Γ in such a way that capacities are respected and the maximum total flow is obtained. This results in LP max P Γ f P (F P ) P Γ, e P f P u e, e A f P 0, P Γ Since there may be prohibitively many (s, t)-paths in G for (F P ) to be solved explicitly, we would like to solve this problem using delayed column generation. (a) Write (F P ) in equality form by adding slack variables s e, e A. Give a feasible starting basis for this system, and explain why it is feasible. (b) After performing some Revised Simplex pivots, we get feasible basis ˆB with m elements made up of r path variables corresponding to paths P 1,..., P r and m r slack variables corresponding to arcs e 1,..., e m r. Write an explicit set of equations that will determine the shadow prices ŷ = (ŷ e : e A) corresponding to ˆB. (c) Next write the associated reduced costs associated with the shadow prices ŷ found in (b). Denote these reduced costs by γ P, P Γ and γ e, e A, respectively. Tell when these reduced costs indicate optimality of the tableau, in terms of the network G itself. (d) Given the reduced costs, describe an algorithm that in O( A 2 ) steps will either verify that the current solution is optimal; or, find an entering, and leaving variable of the next Revised Simplex pivot.