Midterm Problems - Department of Mathematical Sciences...

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Department of Mathematical Sciences University of Delaware Prof. T. Angell November 10, 2010 Mathematics 530 Test Problems Exercise 1 : Determine the extreme points of the set S , where S is the set of all solutions to x 1 + x 2 4 2 x 1 + x 2 6 x 2 3 x 1 ,x 2 0 and write the point (9 / 7 , 1 / 2) as a convex combination of these extreme points. Exercise 2 : 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 | A x b } ∩ K is a convex subset in R n . Exercise 3 : Suppose that V is a real vector space and that f : V R satisfies the inequality f ((1 - λ ) x + λy ) (1 - λ ) f ( x ) + λf ( y ) , for all x,y V, 0 λ 1 . Show that the set of all minimizers for f is a convex set. (Be careful: what happens when f is not bounded below?) Exercise 4 : Let K R n be a convex set and f : K -→ R . Then f is called quasi- convex provided that, for all λ (0 , 1) , f ( (1 - λ ) x + λ y ) max { f ( x ) , f ( y ) } for all x , y K . (a) Show that every function satisfying the inequality in the previous problem is quasi- convex. (b) Show that every strictly increasing function of a single real variable is quasi-convex and find an example that shows that it need not be convex. (c) Show that the level sets of f , { x K : f ( x ) b,b R } , are convex if and only if the function f is quasi-convex.
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Exercise 5 : (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? Exercise 6 : Let A be a symmetric n × n matrix. The matrix A is said to be positive definite provided x > A x > 0 for all x R n , x 6 = 0. Show that the mapping { x , y } 7→ y > A x defines and inner product on R n provided that A is positive definite. HINT: the new inner product y * x := y > A x . Exercise 7
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Midterm Problems - Department of Mathematical Sciences...

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