Mathematics 530. Practice Problems. n + 1 }

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1 Department of Mathematical Sciences University of Delaware Prof. T. Angell October 19, 2015 Mathematics 530 Practice Problems 1. Recall that an indifference relation on a partially ordered set is defined by x y provided x y and y x. Show that is an equivalence relation. 2. With reference to the indifference relation in the preceeding problem we define an indifference set of x to be I x = {y y x}. Note that these are just the equivalence classes! Prove that (a) if y I x then I x = I y and y x. (b) If y I x then I x I y = and either x y or y x. 3. Identify the indifference sets for lexicographic order. 4. Let C R n and define the preference set of x to be P x = {y C x y} and the nonpreference set NP x = {y c y x}. Show that the assumption that these two sets are closed is equivalent to the assumption that it x 1 x 2 x 3, then any continuous curve with endpoints x 1 and x 2 passes through I x2. 5. Show, in detail, that the sequence x n = 3 n + ( 1) n 3 n does not converge. Find a convergent subsequence. 6. A sequence of real numbers {a n } n=1 is increasing provided a 1 a 2 a 3, decreasing if a 1 a 2 a 3, and monotonic if it is either of the two. Show that a bounded monotonic sequence of real numbers is convergent. HINT: Prove it for increasing using the notion of least upper bound. What changes are necessary to provd the result for decreasing sequences? 7. Find all accumulation points of the subset of R given by { 1 n + 1 } m m, n positive integers 8. Let A be a symmetric n n matrix. This matrix is said to be positive definite provided x, Ax 0 for all x R n x 0. Show that the mapping (x, y) y, Ax defines an inner product on R n provided that the matrix A is positive definite. 9. Let M mn (R) be the set of all m n matrices with real entries. Show that the pairing m n A, B = tr (A B) = a i,j b i,j, defines an inner product on the vector space M mn (R). What does the associated norm (called the Frobenius norm) look like? i=1

2 10. In R n, let {x 1, x 2,..., x k } be linearly independent vectors with k < n. For fixed y R n show that there are coefficients α 1, α 2, α k which minimize y k α i x i. 11. A norm is said to be strictly convex provided x = 1, y = 1, x+y = 2 implies x = y. Show that 2 is strictly convex. Is 1 also strictly convex? 12. If x R n and y R m then (x, y) R n+m. Therefore, if S and T are subsets of R n and R m respectively, we may identify S T with a subset of R n+m. Prove that if S and T are non-empty, then S T is bounded, or open, or closed, or compact, if and only if both S and T are bounded, or open, or closed, or compact, respectively. 13. Show that the closure of a bounded set in R n is bounded (and hence compact). 14. Which of the following sets are convex? (a) a slab : {x R n α a, x β}. (b) a rectangle : {x R n α i x i β i, i = 1,..., n}. (c) a wedge : {x R n a 1, x b 1, a 2, x b 2 }. (d) The set of points, U, closer to a given point x o than to a given set S, i.e. U = {x R n x x o x y, for all y S}. HINT: Look first at the set U y = {x R n x x o x y for fixed y S}. (e) For S 1, S 2 R n, S 1 convex, the set {x R n x + S 2 S 1 }. 15. If C is convex then α C is convex for all α R. 16. If C and D are convex, what can be said about the sets C + D and C D? 17. Identify specifically the convex hull of the set {x R n x 2 = x 2 1, 0 x 1 1}. 18. Let A be and m n matrix, b R m, and K R n a convex set. Prove or disprove that the set {x R n Ax b} K is a convex subset in R n. 19. If is the usual Euclidean norm in R n, and x o R n. If K is a convex subset of R n with x o K, define J : R n R by J(x) = x x o 2. (a) Find J(x). Hint: y 2 = y, y. (b) Explain why the minimizer of J over K is unique. (c) Explain the connection between this minimizing point and the projection of x o onto K? 20. Let C R n be the solution set of a quadratic inequality where A S n (R), b R n, and c R. C = {x R n x, Ax b, x + c 0} i=1

3 (a) Show that C is convex provided A 0. (b) Let H = { y, x + h = 0} be a hyperplane (where y 0). Show that C H is convex provided A + λ y, y 0 for some λ R. 21. If U R n is a subspace define the set U = {x R n x, u = 0 for all u U}. Show that U is a subspace of R n, that U U = {0}, and that each y R n can be written uniquely as y = u + v, u U, v U. Moreover show that y 2 = u 2 + v 2. Interpret in light of the Projection Theorem. 22. Let D R n and recall that the epigraph of a function f : D R is the subset of R n+1 given by epi (f) = {(x, z) R n R z f(x)}. Show that the epigraph of a continuous function on R n is a closed set. 23. Let f : R n R n R be continuous and for each x R n define the set F (x) := {y R n f(x, y) 0}. Then the map F is a correspondence. (a) Show that this correspondence has closed graph. (Recall, Gr(F ) = {(x, y) R 2n y F (x)}. (b) If h : R n R is continuous, is it true that the correspondence G(x) := {y R n x y h(x)} has closed graph? Choose a reasonable function h and draw an illustrative diagram. 24. Let A R n. Recall that a function f : A R is said to be lower semi-continuous at x o A provided f(x o ) lim inf k f(xk ), for all sequences x k x o. (see Defiinition B.4.11). Show that if f is lower semi-continuous at every point of A and that if A is compact, then there exists a point ˆx such that f(ˆx) = inf f(x). (Lower semi-continuous functions take on their minimimum value at a point of a compact set.) 25. Show that if f is as in the previous problem, and if there exists a γ R such that the level set for f, {x R n f(x) γ} is non-empty and bounded, then there exists a point ˆx A such that ˆx = inf f(x). HINT: Use B.4.13 and consider the cases γ = inf 26. Show that if X is a non-empty bounded subset of R n then cl(co (X)) = co (cl(x)). HINT: Use proposition and Corollary f(x) and γ > inf f(x) separately. 27. Let A be an m n matrix, b R m and let C R n and D R m be convex sets. Show: (a) A[C] + b = {Ax + b R m x C} is convex.

4 (b) {x R n Ax + b D} is convex. 28. Let S R n be a convex set and y R n \ S. Show that co (S, x) = {(1 λ) x + λ y 0 λ 1, x S} is a convex set. Is it also a cone? Why? 29. Show that, for any X R n, X we have: (a) If A is an n n matrix then A[ co (X)] = co (A[X]). (b) con (X) = con ( co (X)). 30. In the plane, draw the cone C = {a j = (j, 1), j = 1, 2, 3,...} (a) Is C closed? (b) Let b = (0, 1). Is b C? If not, is there a point in C closest to b? 31. Prove: A set C R n is a convex cone if and only if C + C C and α C C for all real α > Show that the set R 2 {(x, y) R 2 x < 0, y x} is a cone, is not convex, and is not line-free. 33. Show that if C is a convex set and 0 A, then the cone K, generated by A is not pointed and that K {0} is line free. 34. Show that the set { ( x C = t ) R n R ( x t ) ( I ) ( x t ) 0, t 0 } is a convex cone. It is called the Lorentz cone. Show also that C = C. 35. Let S n (R) be the set of all real n n symmetric matrices. A matrix A S n is said to be positive semi-definite provided x, Ax 0 for all x R n. Show that the set of all positive semi-definite matrices forms a cone in the vector space of all n n real matrices, M nn (R). What it the partial order in M nn (R) induced by this cone? 36. Consider the matrix I = ( What is the vector space spanned by the columns of I and how does it differ from the cone generated by the columns of I? ). 37. Sketch the cone generated by the columns of the matrix ( ) A = and the cone generated by just the first and third columns of A. If b = (1, 0), does the system Ax = b have a solution with x 0?

5 38. Prove that a cone C containing the origin is convex if and only if the sum x + y of any two vectors x, y C is itself in the cone C. 39. (a) Sketch the convex polyhedron generated by the following set of points {(1, 0), (3, 2), (4, 3), ( 1, 2), ( 3, 2)}. (b) Find the equation of a supporting hyperplane to this polyhedron at the point (3, 2). Is this the only supporting hyperplane at that point? 40. Suppose that C and D are disjoint subsets of R n. Consider the set of all (a, b) R n+1 that satisfy both a, x b for all x C and a, x b for all x D. Show that this set of pairs (a, b) is a convex cone. (Note: it is the singleton {0} if there is no hyperplane that separates C and D.) 41. Suppose K R n is a non-empty, closed, convex cone. Show that if z R n, z K then there exists a K with a, z > Let A be an n n real matrix and C R m a non-empty, closed, convex cone. Define K = {x R n Ax C} and P := {A y, y C }. (a) Show that K is a closed, convex cone. (b) Show that K = cl(p ). HINT: To show that z cl(p ) use the preceeding problem. (c) Show that P = K. HINT: Use the Polar Cone Theorem. 43. Give an example of two closed convex sets that are disjoint but that cannot be strictly separated by a hyperplane. 44. (a) Express the closed convex set {x R 2 > x 1 x 2 1} as an intersection of halfspaces. (b) Let C = x R n x 1} (the unit ball of R n with respect to the l -norm), For each ˆx with ˆx = 1, identify the supporting hyperplane explicitly. Hint: Start by drawing a picture of the unit sphere. 45. In Economics, the Law of Constant Returns to Scale asserts that if the input levels required for the production of a produce P j are increased or decreased by a certain multiple, the output level of the product is increased or decreased by the same multiple. Let y i,j be the units of input I i that are required by the process P j to produce x j units of the output. For linear production models the relationship is described by coefficients a i,j, namely y i,j = a i,j x j, i = 1,..., m; j = 1,... n. The total output of the produced commodity resulting from the simultaneous operation of the n production processes is (O) x = n x j,

6 The amount y i of input I i required for this production program is or (I) y i = n a i,j x j, i = 1,..., m. To make sense, we also have (P 0 ) x j 0, j = 1,... n, (P I ) y j 0, i = 1,... m, y i = n y i,j as well as limitations on available inputs (B I ) y i b i, i = 1,..., m. The conditions (0), (I), (P 0 ), (P I ), and (B I ) constitute the mathematical formulation of the Linear Production Model. (a) The Technology Set T is the set of all input levels y = (y 1, y 2,... y m ) corresponding to possible output levels x = (x 1,..., x n ) satisfying conditions (0), (P 0 ), and (P I ) only. Show that T is a covex cone. (b) The set Q x = {y T y = Ax and x = n x j } is called the production isoquant. Show that the production isoquant is a convex set and that it is exactly the convex hull of the set { y (1),..., y (n)} where for given output level x, the vector y (k) is the vector of input levels required to produce the output level with exclusive use of the k th production process P k i.e., y (k) i = a i,j x, i = 1, 2,..., m. REMARK: The convexity of T has the economic interpretation of absence of external discontinuities which means that if a certain level of output is possible using two different sets of certain inputs, that same level of output can be maintained in infinitely many different ways by using different levels of the same inputs no additional inputs are necessary.