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Herbert Warren
5 years ago
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338 Convex Analysis 2013 Let f : Q R be a strongly convex function with convexity parameter µ>0, where Q R n is a bounded, closed, convex set, which contains the origin. Let Q =conv(q, Q) andconsiderthefunction f given by f(x) =min{f(u)+f(v) :x = u v, u αq, v (1 α)q, 0 α 1} α,u,v for all x Q and f(x) =+ for all x Q. (a) [6 points] Show that f is strongly convex with parameter 1 2 µ on its domain, Q. (b) [2 points] Show that if a function g : C R on a convex set C is strongly convex with parameter µ>0 then the following holds: s y s x,y x µy x 2 for all x, y C, s x g(x) ands y g(y). (c) [2 points] Consider the conjugate function f ( ) of f( ) definedas f (s) =sup{s, x f(x) :x Q}. x Let x, y Q, s x f(x) ands y f(y) besuchthat f is differentiable at s x and s y. Show that f (s y ) f (s x ) 1 µ s y s x holds. Hint: You may use the previous parts even if you haven t proved them.

339 Discrete Mathematics Let H be a finite 3-uniform hypergraph on vertex set V and let C be a set of colors. For each v V aprescribedsetofcolorsc v C is given. A vertex coloring f : V C is proper with respect to (C v ) v V if every vertex v gets a color from its prescribed set and no edge of the hypergraph is monochromatic. (To be precise, a coloring is proper with respect to the prescribed sets if f(v) C v for all v V and for all e H there are u, v e such that f(u) = f(v).) Now, suppose the hypergraph H and the sets (C v ) v V have the property that for every vertex v there are at most d edges e that contain a vertex u such that C u C v =. Prove that there is a constant C (which does not depend on d or V ) suchthatif C v C d for all v V then there is a coloring that is proper with respect to (C v ) v V. 2. (a) State the definition of the 2-color van der Waerden number W (k). (b) Let N = W (t 2 ). Let f be a coloring of {1,...,N} with 2 colors. Prove that there exists a non-trivial t-term arithmetic progression {a + i d : i =0,...,t 1} which together with its difference d is monochromatic. I.e. f(d) =f(a + i d) for i =0,...,t 1, and d is non-zero.

340 Graph Theory 2013 Six professors, A, B, C, D, E, F, have been to the library on the day the rare tractate was stolen. Each entered once, stayed for some time, and then left. If two were in the library at the same time, then at least one of them saw the other. Detectives who questioned the professors gathered the following testimony: A said he saw B and C in the library B said he saw A and D C said he saw B and E D said he saw E and C E said he saw D and F F said he saw A and D One of the professors lied when he claimed to have seen one of his colleagues. Who was it?

341 Integer Programming 2013 Let n 2beanintegerandletS {0, 1} n have the property that, for every x {0, 1} n \ S, the set S contains the n vertices of the cube [0, 1] n that are adjacent to x. (i) Show that the following algorithm solves the problem min{cx : x S} : 1 if c j < 0, Let x be defined by x j = 0 otherwise. If x S, return x. Otherwise, let k be an index such that c k =min j c j. x j for j = k Let x be defined by x j = 1 x k for j = k. Return x. (ii) Prove that any formulation of the problem min{cx : x S} as an integer program min{cx : Ax b, x {0, 1} n } has a number of constraints at least as large as 2 n S. (iii) Let S be the set of 0, 1vectorsinR n that have an odd number of 1s. conv(s) hasatleast2 n 1 facets. Show that

342 Advanced Integer Programming 2013 Let A Q m n be a matrix, b Q m acolumnvector,c Q n arowvector,andd Q a scalar. Let P = {x R n : Ax = b, x 0}, Q = P {x R n : cx d}, R = P {x R n : x 1 x 2 =0} where x 1,x 2 are the first two components of vector x. (i) Let u R m and ν R n + be row vectors and u 0 R + ascalar. Showthatαx β is a valid inequality for Q for any α = ua ν + u 0 c β = ub + u 0 d. (ii) Conversely, is every valid inequality for Q of the form αx β where α and β are defined as in (i)? (iii) Let Q i = P {x R n : x i 0}. Applying (i) to Q 1 and Q 2, give sufficient conditions for an inequality αx β to be valid for R. (iv) Let ν R n + and u 0 R + such that there exists w R m, µ R n +, v 0 R + satisfying wa = ν µ u 0 e 1 + v 0 e 2 wb =0 where e 1 and e 2 are the first two unit vectors (ν,w,µ,e 1,e 2 are all row vectors). Show that the inequality n j=1 ν jx j u 0 x 1 0siavalidinequalityforR. (v) Assume that we have a tableau for problem Ax = b, x 0wherex 1 and x 2 are both basic and strictly positive: x 1 + j N a 1j x j = b 1 x 2 + j N a 2j x j = b 2, where N indexes the nonbasic variables and b 1 > 0, b 2 > 0. Prove that x 1 + a2j b 1 b j N 2 a + 1j x j 0 is a valid cut for R. b 1

343 Linear Programming 2013 (a) A unit simplex is {x R n : ex 1,x 0} where e =(1,...,1). Consider the problem of maximizing cx over a unit simplex, where c 1 > >c n > 0. State the optimal primal and dual solution. (b) Write the optimal tableau for the problem in (a). Add the constraint x 1 0and perform one iteration of the dual simplex method to restore feasibility. (c) Suppose that max{cx + dy : Ax + By b} is to be solved by Benders decomposition, with x in the master problem. Show that the projection of the feasible set onto x is described by the set of all possible Benders cuts arising from infeasible subproblems. (d) Use (c) to show that the projection of a unit simplex onto any subspace is a unit simplex. State explicitly the extreme ray solutions that yield the relevant Benders cuts. Be sure to treat nonnegativity constraints as part of the system Ax + By b and use all of them in the subproblem.

344 Networks and Matchings 2013 Let G be a connected bipartite graph with bipartition (A, B), each of whose edges is contained in a perfect matching. Show that A = B and for each nonempty proper subset X of A, N(X) > X. (Here N(X) isthesetofverticesadjacenttosomevertexinx).

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9.1 Linear Programs in canonical form

9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems